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A SHOET ACCOUNT
OF THE
HISTORY OF MATHEMATICS
j^&^m.
MACMILLAN AND CO., Limited
LONDON • BOMBAY • CALCUTTA
MELBOURNE
THE MACMILLAN COMPANY
NEW YORK • BOSTON • CHICAGO
DALLAS • SAN FRANCISCO
THE MACMILLAN CO. OF CANADA, Ltd.
TORONTO
A SHORT ACCOUNT
OF THE
HISTOEY OF MATHEMATICS
W. W. EOUSE BALL
FELLOW OF TRINITY COLLEGE, CAMBRIDGE
SIXTH EDITION
MACMILLAN AND CO., LIMITED
ST. MARTIN'S STREET, LONDON
1915
KJ
/• V ■- /
First Edition, i838.
Second Edition, 1893.
Third Edition, 1901.
Fourth Edition, 1908.
Fifth Edition, 1912.
Sixth Edition, 191 5.
PEEFACE TO THE FOUETH EDITION
The subject-matter of this book is a historical summary
of the development of mathematics, illustrated by the
lives and discoveries of those to whom the progress of the
science is mainly due. It may serve as an introduction
to more elaborate works on the subject, but primarily it
is intended to give a short and popular account of those
leading facts in the history of mathematics which many
who are unwilling, or have not the time, to study it
systematically may yet desire to know.
The first edition was substantially a transcript of
some lectures which I delivered in the year 1888 with
the object of giving a sketch of the history, previous to
the nineteenth century, that should be intelligible to any
one acquainted with the elements of mathematics. In
the second edition, issued in 1893, I rearranged parts
of it, and introduced a good deal of additional matter.
The third edition, issued in 1901, was revised, but not
materially altered ; and the present edition is practically
a reprint of this, save for a few small corrections and
additions.
346531
vi PREFACE
The scheme of arrangement will be gathered from the
table of contents at the end of this preface. Shortly it
is as follows. The first chapter contains a brief state-
ment of what is known concerning the mathematics of
the Egyptians and Phoenicians ; this is introductory to
the history of mathematics under Greek influence. The
subsequent history is divided into three periods : first,
that under Greek influence, chapters ii to vii ; second,
that of the middle ages and renaissance, chapters viii to
XIII ; and lastly that of modern times, chapters xiv to
XIX.
In discussing the mathematics of these periods I
have confined myself to giving the leading events in the
history, and frequently have passed in silence over men
or works whose influence was comparatively unimportant.
Doubtless an exaggerated view of the discoveries of those
mathematicians who are mentioned may be caused -by
the non -allusion to minor writers who preceded and
prepared the way for them, but in all historical sketches
this is to some extent inevitable, and I have done my
best to guard against it by interpolating remarks on the
progress of the science at different times. Perhaps also
I should here state that generally I have not referred
to the results obtained by practical astronomers and
physicists unless there was some mathematical interest
in them. In quoting results I have commonly made
use of modern notation ; the reader must therefore
recollect that, while the matter is the same as that
of any writer to whom allusion is made, his proof is
I
PREFACE vii
sometimes translated into a more convenient and familiar
language.
The greater part of my account is a compilation from
existing histories or memoirs, as indeed must be neces-
sarily the case where the works discussed are so numerous
and cover so much ground. When authorities disagree I
have generally stated only that view which seems to me
to be the most probable ; but if the question be one of
importance, I believe that I have always indicated that
there is a difference of opinion about it.
I think that it is undesirable to overload a popular
account with a mass of detailed references or the
authority for every particular fact mentioned. For the
history previous to 1758, I need only refer, once for all,
to the closely printed pages of M. Cantor's monumental
Vorlesungen ilber die Geschichte der Mathematik (here-
after alluded to as Cantor), which may be regarded
as the standard treatise on the subject, but usually
I have given references to the other leading
authorities on which I have relied or with which
I am acquainted. My account for the period sub-
sequent to 1758 is generally based on the memoirs or
monographs referred to in the footnotes, but the main
facts to 1799 have been also enumerated in a supple-
mentary volume issued by Prof. Cantor last year. I
hope that my footnotes will supply the means of studying
in detail the history of mathematics at any specified
period should the reader desire to do so.
My thanks are due to various friends and corre-
viii PREFACE
spondeiits who have called my attention to points in the
previous editions. I shall be grateful for notices of
additions or corrections which may occur to any of my
readers.
W. W. EOUSE BALL.
Trinity College, Cambridge,
May 1908.
NOTE TO THE SIXTH EDITION
The Fourth Edition was stereotyped in 1908, and since
then no material chancres have been made.
W. W. E. B.
Trinity College, Cambridge,
January 1915.
IX
TABLE OF CONTENTS.
PAGE
Preface v
Table of Contents ix
Chapter I. Egyptian and Phoenician Mathematics,
The history of mathematics begins with that of the Ionian Greeks 1
Greek indebtedness to Egyptians and Phoenicians .... 2
Knowledge of the science of numbers possessed by the Phoenicians 2
Knowledge of the science of numbers possessed by the Egyptians . 8
Knowledge of the science of geometry possessed by the Egyptians 5
Note on ignorance of mathematics shewn by the Chinese . . 9
jFtrst ^ertoK. Jlatfjematics untrer ffireefe Influence.
This period begins with the teaching of Thales, circ. 600 B.C., and ends
with the capture of Alexandria by the Mohammedans in or about 641 a.d.
The characteristic feature of this period is the development of geometry.
Chapter II. The Ionian and Pythagorean Schools.
Circ. 600 b.c.-400 b.c.
Authorities . . ' . . . . . . . .13
The Ionian School .......... 14
Thales, 640-550 b.c 14
His geometrical discoveries ...... 15
His astronomical teaching ...... 17
Anaximander. Anaximenes. Mamercus. Mandryatus . . 18
h
TABLE OF CONTENTS
The Pythagorean School
Pythagoras, 569-500 b.c
The Pythagorean teaching
The Pythagorean geometry
The Pythagorean theory of numbers
Epicharmus. Hippasus. Philolaus. Archippus.
Archytas, circ. 400 B.C.
His solution of the duplication of a cube
Theodorus. Timaeus. Bryso
Other Greek Mathematical Schools in the Fifth Cen
Oenopides of Chios
Zeno of Elea. Democritus of Abdera .
Lysis
ury B.C.
PAGE
19
19
20
22
24
28
28
29
30
30
30
31
Chaptek III. The Schools of Athens and Cyzicus.
CiEC. 420-300 B.C.
Authorities
Mathematical teachers at Athens prior to 420 b.c. .
Anaxagoras. The Sophists. Hippias (The quadratrix)
Antipho
Three problems in which these schools were specially interested
Hippocrates of Chios, circ. 420 B.c
Letters used to describe geometrical diagrams .
__^.:^^==^ Introduction in geometry of the method of reduction
The quadrature of certain lunes ....
The problem of the duplication of the cube
Plato, 429-348 b.c
Introduction in geometry of the method of analysis .
Theorem on the duplication of the cube .
EuDOXus, 408-355 b.c
Theorems on the golden section ....
Introduction of the method of exhaustions
Pupils of Plato and Eudoxus .
Menaechmus, circ. 340 b.c
Discussion of the conic sections ....
His two solutions of the duplication of the cube
Aristaeus. Theaetetus
Aristotle, 384-322 B.C. .
Questions on mechanics. Letters vised to indicate magnitudes
TABLE OF CONTENTS
Chapter IV. The First Alexandrian School.
CiRC. 300-30 B.C.
PAGE
Authorities . 50
Foundation of Alexandria .51
The Third Century before Christ 52
Euclid, circ. 330-275 B.c v . . .52'
Euclid's Elements ........ 53
The Elements as a text-book of geometry .... 54
The Elements as a text-book of the theory of numbers . 57
Euclid's other works 60"
Aristarchus, circ. 310-250 b.c 62
Alethod of determining the distance of the sun . . 62
Conon. Dositheus. Zeuxippus. Nicoteles ..... 64
Akchimedes, 287-212 b.c 64
His works on plane geometry ...... 67
His works on geometry of three dimensions ... 70
His two papers on arithmetic, and the "cattle problem " . 71
His works on the statics of solids and fluids . . .73
His astronomy ......... 76
The principles of geometry assumed by Archimedes . . 76
Apollonius, circ. 260-200 b.c 77
His conic sections ........ 77
His other works ........ 80
His solution of the duplication of the cube ... 81
Contrast between his geometry and that of Archimedes . 82
Eratosthenes, 275-194 b.c 83
The Sieve of Eratosthenes 83
The Second Century before Christ 84
Hypsicles (Euclid, book xiv). Nicomedes. Diodes ... 85
Perseus. Zenodorus 86
HiPPARCHUS, circ. 130 b.c 86
Foundation of scientific astronomy ..... 87
Foundation of trigonometry ...... 88
Hero of Alexandria, circ. 125 b.c 88
Foundation of scientific engineering and of land-surveying 88
Area of a triangle determined in terms of its sides . . 89
Features of Hero's works 91
xii TABLE OF CONTENTS
The First Century lefore Christ
Theodosius .....
Dionysodorus
End of the First Alexandrian School
Egypt constituted a Roman province
PAGE
91
91
92
92
92
Chapter V. The Second Alexandrian School.
30 B.C. -641 A.D.
Authorities . ... 93
The First Century after Christ ....... 94
Serenus. Menelaus 94
Mcomachus 94
Introduction of the arithmetic current in medieval Europe 95
The Second Century after Christ .95
Theon of Smyrna. Thymaridas ....... 95
Ptolemy, died in 168 96
The Almagest 96
Ptolemy's astronomy ....... 96
Ptolemy's geometry ........ 98
The Third Century after Christ 99
Pappus, circ. 280 99
The livva'ywyr], a synopsis of Greek mathematics . . 99
The Fourth Century after Christ 101
v,r3»»jyietrodorus. Elementary problems in arithmetic and algebra . 102
^^ Three stages in the development of algebra 103
DiOPHANTUP, circ. 320 (?) 103
Introduction of syncopated algebra in his Arithinetic . 104
The notation, methods, and subject-matter of the work . 104
His Porisms 110
Subsequent neglect of his discoveries .... 110
lamblichus 110
Theon of Alexandria. Hypatia Ill
Hostility of the Eastern Church to Greek science . . . .111
The Athenian School {in the Fifth Century) . . . . .111
Proclus, 412-485. Damascius. Eutocius 112
TABLE OF CONTENTS xiii
PAQE
Roman Mathematics . . . . . . . . .113
Nature and extent of the mathematics read at Rome . . , 113
Contrast between the conditions at Rome and at Alexandria . . 114
End of the Secoiid Alexandrian School . . . . . ,115
The capture of Alexandria, and end of the Alexandrian Schools . 115
Chaptee VL The Byzantine School. 641-1453.
Preservation of works of the great Greek Mathematicians . .116
Hero of Constantinople. Psellus. Planudes. Barlaam. Argyrus 117
Nicholas Rhabdas. Pachymeres. Moschopulus (Magic Squares) . 118
Capture of Constantinople, and dispersal of Greek Mathematicians 120
Chaptek VTI. Systems op Numeration and Primitive
Arithmetic.
Authorities 121
Methods of counting and indicating numbers among primitive races 121
Use of the abacus or swan- pan for practical calculation . . . 123
126
126
127
127
128
Methods of representing numbers in writing .
The Roman and Attic symbols for numbers .
The Alexandrian (or later Greek) symbols for numbers
Greek arithmetic .......
Adoption of the Arabic system of notation among civilized races
xiv TABLE OF CONTENTS
Seconli periotr. iWatljematics of tfje MVtlt aiges
antr of tfje iSitmimmct.
This period begins about the sixth century, and may be said to end
with the invention of analytical geometry and of the infinitesimal calculus.
The characteristic feature of this period is the creation or development of
modem arithmetic, algebra, and trigonometry.
Chapter VIII. The Rise of Learning in Western Europe.
CiRC. 600-1200.
PAOE
Authorities 131
JEdUfCation in the Sixth, Seventh, and Eighth Centuries . . .131
The Monastic Schools 131
Boethius, circ. 475-526 132
Medieval text-books in geometry and arithmetic . . 133
Cassiodorus, 490-566. Isidorus of Seville, 570-636 . . .133
The Cathedral and Conventual Schools 134
The Schools of Charles the Great . . . . . . .134
Alcuin, 735-804 . . .134
Education in the Ninth and Tenth Centuries .... 136
Gerbert (Sylvester IL), died in 1003 136
Bernelinus 139
The Early Medieval Universities 139
Rise during the twelfth century of the earliest universities . .139
Development of the medieval universities 140
Outline of the course of studies in a medieval university . . 141
Chapter IX. The Mathematics of the Arabs.
Authorities 144
Extent of Mathematics obtained from Greek Sources . . .144
The College of Scribes 145
TABLE OF CONTENTS xv
Extent of 3Iathematics obtained from the {Aryan) Hindoos
Akya-Bhata, circ. 530
His algebra and trigonometry (in his Aryahhathiya)
Brahmagupta, circ. 640
His algebra and geometry (in his Siddhanta) .
Bhaskara, circ. 1140
The Lilavati or arithmetic ; decimal numeration nsed
The Bija Ganita or algebra .....
Develojyment of Mathematics in Arabia .....
Alkarismi or Al-Khwarizmi, circ. 830 ....
His Al-gebr ice' I mukabala .....
His solution of a quadratic equation
Introduction of Arabic or Indian system of numeration
Tabit ibn Korra, 836-901 ; solution of a cubic equation
Alkayami. Alkarki. Development of algebra
Albategni. Albnzjani. Development of trigonometry .
Alhazen. Abd-al-gehl. Development of geometry .^-^y
Characteristics of the Arabian School .....
PAGE
146
147
147
148
148
150
150
153
155
155
156
157
158
158
159
161
161
162
Chapter X. Introduction of Arabian Works into Europe.
Circ. 1150-1450.
The Eleventh Century 165
Moorish Teachers. Geber ibn Aphla. Arzachel . . . .165
The Twelfth Century 165
Adelhard of Bath .165
Ben-Ezra. Gerard. John Hispalensis . . . .. • .166
The Thirteenth Century . . . . . . . . .167
Leonardo of Pisa, circ. 1175-1230 167
The Liber Abaci, 1202 167
The introduction of the Arabic numerals into commerce . 168
The introduction of the Arabic numerals into science . 168
The mathematical tournament 169
Frederick II., 1194-1250 170
xvi TABLE OF CONTENTS
PAGE
JoRDANUS, circ. 1220 171
His Be Numeris Datis ; syncopated algebra . . . 172
Holywood 174
Roger Bacon, 1214-1294 174
Campanus 177
The Fourteenth Century . . . . . . . .177
Bradwardine 177
Oresmus . . . . 178
The reform of the university curriculum ..... 179
The Fifteenth Century 180
Beldomandi 180
Chapter XI. The Development of Arithmetic.
Circ. 1300-1637.
Authorities . . . . . 182
The Boethian arithmetic . . 182
Algorism or modern arithmetic ....... 183
The Arabic (or Indian) symbols : history of 184
Introduction into Europe by science, commerce, and calendars . 186
Improvements introduced in algoristic arithmetic . . .188
(i) Simplification of the fundamental processes . . .188
(ii) Introduction of signs for addition and subtraction . .194
(iii) Invention of logarithms, 1614 195
(iv) Use of decimals, 1619 197
Chapter XII. The Mathematics of the Renaissance.
Circ. 1450-1637.
Authorities .
Effect of invention of printing. The renaissance .
Develofment of Syncopated Algebra and Trigonometry
Regiomontanus, 1436-1476
His De Triangulis (printed in 1496) .
Purbach, 1423-1461. Cusa, 1401-1464. Chuquet, circ.
Introduction and origin of symbols + and- .
TABLE OF CONTENTS xvii
Pacioli or Lucas di Burgo, circ. 1500 .....
His arithmetic and geometry, 1494 ....
Leonardo da Vinci, 1452-1519 . . . . . .
Diirer, 1471-1528. Copernicus, 1473-1543 ....
Record, 1510-1588 ; introduction of symbol for equality
Rudolff, circ. 1525. Riese, 1489-1559
Stifel, 1486-1567
His Arithmetica Integra, 1544 .....
Tartaglia, 1500-1559
His solution of a cubic equation, 1535
His arithmetic, 1556-1560
Cardan, 1501-1576
His Ars Magna, 1545 ; the third work printed on algebra
His solution of a cubic equation ....
Ferrari, 1522-1565 ; solution of a biquadratic equation .
Rheticus, 1514-1576. Maurolycus. Borrel. Xylander
Commandiho. Peletier. Romanus. Pitiscus. Ramus, 1515-157
Bombelli, circ. 1570
Development of SymloUc A Igebra
ViETA, 1540-1603
The In Artem ; introduction of symbolic algebra, 1591
Vieta's other works
Girard, 1590-1633 ; development of trigonometry and algebra
Napier, 1550-1617 ; introduction of logarithms, 1614 .
Briggs, 1556-1631 ; calculations of tables of logarithms .
Harriot, 1560-1621 ; development of analysis in algebra
Oughtred, 1574-1660
The Origin of the more Common Symbols in Algebra
PAGE
208
209
212
213
214
215
215
216
217
218
219
221
222
224
225
226
227
228
229
229
231
232
234
235
236
237
238
239
Chaptee XIII. The Close of the Rejjaissance.
Circ. 1586-1637.
Authorities . . 244
Development of Mechanics and Experimental Methods . . . 244
Stevinus, 1548-1620 244
Commencement of the modern treatment of statics, 1586 . 245
XVlll
TABLE OF CONTENTS
Galileo, 1564-1642
Commencement of the science of dynamics
Galileo's astronomy ....
Francis Bacon, 1561-1626. Guldinus, 1577-1643
Wright, 1560-1615 ; construction of maps
Snell, 1591-1626
Revival of Interest in Pure Geometry
Keplee, 1571-1630 . . . . .
His Paralipomena, 1604 ; principle of continuity
His Stereometria, 1615 ; use of infinitesimals .
Kepler's laws of planetary motion, 1609 and 1619
Desargues, 1593-1662 • .
His Brouillon project ; use of projective geometry
Mathematical Knowledge at the Close of the Renaissance .
PAGE
247
248
249
252
253
254
254
254
256
256
256
257
257
258
This period begins with the invention of analytical geometry and the
infinitesimal calculus. The mathematics is far more complex than that
produced in either of the preceding periods : but it may be generally de-
scribed as characterized by the development of analysis, and its application
to the phenomena of nature.
Chapter XIV. The History of Modern Mathematics.
Treatment of the subject 263
Invention of analytical geometry and the method of indivisibles . 264
Invention of the calculus . 265
Development of mechanics 265
Application of mathematics to physics 266
Recent development of pure mathematics 267
TABLE OF CONTENTS xix
Chapter XV. History of Mathematics from Descartes
TO HUYGENS. CiRC. 1635-1675.
PAGE
Authorities 268
Descartes, 1596-1650 268
His views on philosophy . . . . . . ,271
His invention of analytical geometry, 1637 . . . 272
His algebra, optics, and theory of vortices . . . 276
Cavalieri, 1598-1647 278
The method of indivisibles ...... 279
Pascal, 1623-1662 281
His geometrical conies 284
The arithmetical triangle 284
Foundation of the theory of probabilities, 1654 . . 285
His discussion of the cycloid 287
Wallis, 1616-1703 288
The Arithmetica Infinitorum, 1656 ..... 289
Law of indices in algebra . 289
Use of series in quadratures 290
Earliest rectification of curves, 1657 291
Wallis's algebra 292
Fermat, 1601-1665 293
His investigations on tlie theory of numbers . . . 294
His use in geometry of analysis and of infinitesimals . 298
Foundation of the theory of probabilities, 1654 . . 300
Huygens, 1629-1695 301
The Horoloyium Oscillatorium, 1673 .... 302
The undulatory theory of light ..... 303
Other Mathematicians of this Time 305
Bachet 305
Marsenne ; theorem on primes and perfect numbers . . . 306
Roberval. Van Schooten. Saint- Vincent 307
Torricelli. Hudde. Frenicle 308
De Laloubere. Mercator. Barrow ; the difi"erential triangle . 309
Brouncker ; continued fractions ....... 312
James Gregory ; distinction between convergent and divergent scries 313
Sir Christopher Wren . . .314
Hooke. Collins 315
Pell. Sluze. Viviani 316
Tschirnhausen. De la Hire. Roemer. Rolle .... 317
XX
TABLE OF CONTENTS
Chapter XVI. The Life and Woeks of Newton.
PAGE
Authorities ........... 319
Newton's school and undergraduate life ..... 320
Investigations in 1665-1666 on fluxions, optics, and gravitation . 321
His views on gravitation, 1666 ...... 321
Researches in 1667-1669 323
Elected Lucasian professor, 1669 ....... 324
Optical lectures and discoveries, 1669-1671 324
Emission theory of light, 1675 326
The Leibnitz Letters, 1676 327
Discoveries on gravitation, 1679 330
Discoveries and lectures on algebra, 1673-1683 . . . . 330
Discoveries and lectures on gravitation, 1684 ..... 333
The Principia, 1685-1686 .334
The subject-matter of the Principia ..... 335
Publication of the Principia 337
Investigations and work from 1686 to 1696 ..... 338
Appointment at the Mint, and removal to London, 1696 . . 339
Publication of the Optics, 1704 339
Appendix on classification of cubic curves . . . 340
Appendix on quadrature by means of infinite series . . 341
Appendix on method of fluxions 343
The invention of fluxions and the infinitesimal calculus . . . 347
Newton's death, 1727 348
List of his works . . . . , . . . . . 348
Newton's character ......... 349
Newton's discoveries 351
Chapter XVII. Leibnitz and the Mathematicians
OF the FrRST Half of the Eighteenth Century.
Authorities ..........
Leibnitz and the Bernoullis .......
Leibnitz, 1646-1716
His system of philosophy, and services to literature
The controversy as to the origin of the calculus
His memoirs on the infinitesimal calculus
His papers on various mechanical problems
Characteristics of liis work
353
353
353
355
356
362
363
365
TABLE OF CONTENTS
XXI
James Bernoulli, 1654-1705
John Bernoulli, 1667-1748
The younger Bernouillis
Development of Analysis on the Continent
L'Hospital, 1661-1704 •.
Varignon, 1654-1722. De Montmort. Nicole
Parent. Saurin. De Gua. Cramer, 1704-1752
Riccati, 1676-1754. Fagnauo, 1682-1766
Clairaut, 1713-1765 . .
D'Alembert, 1717-1783 ....
Solution of a partial differential equation of the second
Daniel Bernoulli, 1700-1782
English Mathematicians of the Eighteenth Century .
David Gregory, 1661-1708. Halley, 1656-1742 .
Ditton, 1675-1715
Brook Taylor, 1685-1731
Taylor's theorem
Taylor's physical researches
Cotes, 1682-1716
Demoivre, 1667-1754 ; development of trigonometry
Maclaurin, 1698-1746
His geometrical discoveries
The Treatise of Fluxions ....
His propositions on attractions .
Stewart, 1717-1785. Thomas Simpson, 1710-1761
order
367
368
369
369
370
371
372
373
374
376
377
378
379
380
380
381
382
382
383
384
385
386
387
388
Chapter XVIII. Lagrange, Laplace, and their
Contemporaries. Circ. 1740-1830.
Characteristics of the mathematics of the period
Development of Analysis and Mechanics . . . .
Euler, 1707-1783
The Introductio in Analysin Infinitorum, 1748
The Institutiones Calculi Differ entialis, 1755 .
The Institutiones Calculi Ditegralis, 1768-1770
The Anleitung zur Algebra, 1770
Euler's works on mechanics and astronomy
Lambert, 1728-1777
391
393
393
394
396
396
397
398
400
xxii TABLE OF CONTENTS
PAGE
Bezout, 1730-1783. Trembley, 1749-1811. Arbogast, 1759-1803 401
Lagrange, 1736-1813 401
Memoirs on various subjects ...... 403
The Mdcanique analytique, 1788 . . , . . 406
The Th6orie and Calcul des fondions, 1797, 1804 . . 410
The Risolution des Equations numeriques, 1798 . . . 410
Characteristics of Lagrange's work ..... 411
Laplace, 1749-1827 412
Memoirs on astronomy and attractions, 1773-1784 . . 413
Use of spherical harmonics and the potential . . . 413
Memoirs on problems in astronomy, 1784-1786 . . . 414
The M4canique celeste and Exposition du systeme du mmide 415
The Nebular Hypothesis 415
The Meteoric Hypothesis ....... 415
The TMorie analytique des iJrohahilites, 1812 . . , 418
The Method of Least Squares 418
Other researches in pure mathematics and in physics . 419
Characteristics of Laplace's work ..... 420
Character of Laplace . . . . . . .421
' Legendke, 1752-1833 421
His memoirs on attractions 422
The TMorie des nombres, 1798 423
Law of quadratic reciprocity 423
The Calcul integral and the Fonctions elliptiqucs . . 424
Pfaff, 1765-1825 425
Creation of Modern Geometry . . ... . . . . 425
Monge, 1748-1818 426
Lazare Carnot, 1753-1823. Poncelet, 1788-1867 . . . .428
DevelopTnent of Mathematical Physics . . . . . .429
Cavendish, 1731-1810 429
Rumford, 1753-1815. Young, 1773-1829 430
Dalton, 1766-1844 431
Fourier, 1768-1830 432
Sadi Carnot ; foundation of thermodynamics 433
PoissoN, 1781-1840 433
Ampere, 1775-1836. Fresnel, 1788-1827. Biot, 1774-1862 . . 436
Arago, 1786-1853 437
Introduction of Analysis into Erigland 438
Ivory, 1765-1842 439
The Cambridge Analytical School 439
"Woodhouse, 1773-1827 . .440
Peacock, 1791-1858. Babbage, 1792-1871. JohnHerschel, 1792-1871 441
TABLE OF CONTENTS xxiii
Chaptee XIX. Mathematics of the Nineteenth Century.
PAGE
Creation of new branches of mathematics ..... 444
Difficulty in discussing the mathematics of this century . . 445
Account of contemporary work not intended to be exhaustive . 445
Authorities 445
Gauss, 1777-1855 447
Investigations in astronomy -. 448
Investigations in electricity ...... 449
The Disquisitiones Arithmeticae, 1801 .... 452
His other discoveries ....... 453
. Comparison of Lagrange, Laplace, and Gauss . . . .454
Dirichlet, 1805-1859 454
Development of the Theory of Numbers 455
Eisenstein, 1823-1852 '. . 455
Henry Smith, 1826-1883 . . .456
Kuramer, 1810-1893 . .458
Notes on other writers on the Theory of Numbers . . . .459
Development of the Theory of Functions of Multiple Periodicity . 461
Abel, 1802-1829. Abel's Theorem 461
Jacobi, 1804-1851 462
RiEMANN, 1826-1866 464
Notes on other writers on Elliptic and Abelian Functions . . 465
Weierstrass, 1815-1897 466
Notes on recent writers on Elliptic and Abelian Functions . . 467
The Theory of Functions 467
Development of Higher Algebra ....... 468
Cauchy, 1759-1857 469
Argand, 1768-1822 ; geometrical interpretation of complex numbers 471
Sir William Hamilton, 1805-1865 ; introduction of quaternions . 472
Grassmann, 1809-1877 ; his non-commutative algebra, 1884 . 473
Boole, 1815-1864. De Morgan, 1806-1871 474
Galois, 1811-1832 ; theory of discontinuous substitution groups . 475
Cayley, 1821-1895 475
Sylvester, 1814-1897 476
Lie, 1842-1889 ; theory of continuous substitution groups . . 477
Hermite, 1822-1901 478
Notes on other writers on Higher Algebra 479
Development of Analytical Geometry ...... 480
Notes on some recent writers on Analytical Geometry . . . 481
Line Geometry 482
Analysis. Names of some recent writers on Analysis . . . 482
XXIV
TABLE OF CONTENTS
the subject
Development of Synthetic Geometry .
Steiner, 1796-1863
Von Staudt, 1798-1867 '
Other writers on modern Synthetic Geometry
Development of Non-Euclidean Geometry
Euclid's Postulate on Paraller Lines .
Hyperbolic Geometry. Elliptic Geometry
Congruent Figures ....
Foundations of Mathematics. Assumptions made
Kinematics .........
Development of the Theory of Jlechanics, treated Graphically
Development of Theoretical Mechanics, treated Analytically
Notes on recent writers on Mechanics ....
Development of Theoretical Astronomy . . . .
Bessel, 1784-1846
Leverrier, 1811-1877. Adams, 1819-1892 .
Notes on other writers on Theoretical Astronomy .
Recent Developments
Development of Mathematical Physics . .
PAGE
483
483
484
484
485
486
486
488
489
489
489
491
492
493
49»
494
495
497
Index .
Press Notices
499
523
CHAPTER I.
EGYPTIAN AND PHOENICIAN MATHEMATICS.
The history of mathematics cannot with certainty be traced
back to any school or period before that of the Ionian Greeks.
The subsequent history may be divided into three periods^, the
distinctions between which are tolerably well marked. The first
period is that of the history of mathematics under Greek influ-
ence, this is discussed in chapters ii to vii; the second is that
of the mathematics of the middle ages and the renaissance,
this is discussed in chapters viii to xiii ; the third is that of
modern mathematics, and this is discussed in chapters xiv to
XIX.
Although the history of mathematics commences with that
of the Ionian schools, there is no doubt that those Greeks who
first paid attention to the subject were largely indebted to
the previous investigations of the Egyptians and Phoenicians.
Our knowledge of the mathematical attainments of those races
is imperfect and partly conjectural, but, such as it is, it is here
briefly summarised. The definite history begins with the next
chapter.
On the subject of prehistoric mathematics, we may observe
in the first place that, though all early races which have left
records behind them knew something of numeration and
mechanics, and though the majority were also acquainted with
the elements of land-surveying, yet the rules which they
3E • . B
2 EGYPTIAN & PHOENICIAN MATHEMATICS [ch. i
possessed were in general founded only on the results of observa-
tion and experiment, and were neither deduced from nor did
they form part of any science. The fact then that various
nations in the vicinity of Greece had reached a high state of
civilisation does not justify us in assuming that they had studied
mathematics.
The only races with whom the Greeks of Asia Minor
(amongst whom our history begins) were likely to have come
into frequent contact were those inhabiting the eastern littoral
of the Mediterranean ; and Greek tradition uniformly assigned
the special development of geometry to the Egyptians, and
that of the science of numbers either to the Egyptians or to the
Phoenicians. I discuss these subjects separately.
First, as to the science of numbers. So far as the acquire-
ments of the Phoenicians on this subject are concerned it is
impossible to speak with certainty. The magnitude of the
commercial transactions of Tyre and Sidon necessitated a con-
siderable development of arithmetic, to which it is probable
the name of science might be properly applied. A Babylonian
table of the numerical value of the squares of a series of con-
secutive integers has been found, and this would seem to indicate
that properties of numbers were studied. According to Strabo
the Tyrians paid particular attention to the sciences of numbers,
navigation, and astronomy ; they had, we know, considerable
commerce with their neighbours and kinsmen the Chaldaeans ;
and Bockh says that they regularly supplied the weights and
measures used in Babylon. Now the Chaldaeans had certainly
paid some attention to arithmetic and geometry, as is shown
by their astronomical calculations ; and, whatever was the
extent of their attainments in arithmetic, it is almost certain
that the Phoenicians were equally proficient, while it is likely
that the knowledge of the latter, such as it was, was communi-
cated to the Greeks. On the whole it seems probable that the
( early Greeks were largely indebted to the Phoenicians for their
knowledge of practical arithmetic or the art of calculation, and
perliai)s also learnt from them a few properties of numbers. It
V.
CH. i] EARLY EGYPTIAN ARITHMETIC 3
may be worthy of note that Pythagoras was a Phoenician ; and
according to Herodotus, but this is more doubtful, Thales was
also of that race.
I may mention that the almost universal use of the abacus
or swan -pan rendered it easy for the ancients to add and
subtract without any knowledge of theoretical arithmetic.
These instruments will be described later in chapter vii ; it
will be sufficient here to say that they afford a concrete way
of representing a number in the decimal scale, and enable the
results of addition and subtraction to be obtained by a merely
mechanical process. This, coupled with a means of represent-
ing the result in writing, was all that was required for practical
purposes.
We are able to speak with more certainty on the arithmetic
of the Egyptians. About forty years ago a hieratic papyrus,^
forming part of the Rhind collection in the British Museum,
was deciphered, which has thrown considerable light on their
mathematical attainments. The manuscript was written by a
scribe named Ahmes at a date, according to Egyptologists,
considerably more than a thousand years before Christ, and it
is believed to be itself a copy, with emendations, of a treatise
more than a thousand years older. The work is called " direc-
tions for knowing all dark things," and consists of a collection of
problems in arithmetic and geometry ; the answers are given, but
in general not the processes by which they are obtained. It appears
to be a summary of rules and questions familiar to the priests.
The first part deals with the reduction of fractions of the
form 2/{2n+ 1) to a sum of fractions each of whose numerators
is unity : for example, Ahmes states that -^g is the sum of
2V5 js^ TTT' and 2 k i and -^\ is the sum of -^\, ^}^, and ^ ^^ .
In all the examples n is less than 50. Probably he had no
rule for forming the component fractions, and the answers
^ See Mn mathematisches Handhuch der alien Aegypter, by A. Eisenlohr,
second edition, Leipzig, 1891 ; see also Cantor, chap, i ; and A Short
History of Greek Mathematics, by J. Gow, Cambridge, 1884, arts. 12-14.
]3esides these authorities the papyrus has been discussed in memoirs by
L. Rodet, A. Favaro, V. Bobynin, and E. Weyr.
4 EGYPTIAN & PHOENICIAN MATHEMATICS [ch. i
given represent the accumulated experiences of previous writers :
in one solitary case, however, he has indicated his method, for,
after having asserted that | is the sum of J and J, he adds that
therefore two-thirds of one-fifth is equal to the sum of a half of
a fifth and a sixth of a fifth, that is, to —^ + -^q.
That so much attention was paid to fractions is explained by
the fact that in early times their treatment was found difficult.
The Egyptians and Greeks simplified the problem by reducing
a fraction to the sum of several fractions, in each of which the
numerator was unity, the sole exception to this rule being the
fraction f . This remained the Greek practice until the sixth
century of our era. The Romans, on the other hand, generally
kept the denominator constant and equal to twelve, expressing
the fraction (approximately) as so many twelfths. The Baby-
lonians did the same in astronomy, except that they used sixty
as the constant denominator ; and from them through the Greeks
the modern division of a degree into sixty equal parts is derived.
Thus in one way or the other the difficulty of having to consider
changes in both numerator and denominator was evaded. To-day
when using decimals we often keep a fixed denominator, thus
reverting to the Roman practice.
After considering fractions Ahmes proceeds to some examples
of the fundamental processes of arithmetic. In multiplication
he seems to have relied on repeated additions. Thus in one
numerical example, where he requires to multiply a certain
number, say a, by 13, he first multiplies by 2 and gets 2a, then
he doubles the results and gets 4a, then he again doubles the
result and gets 8a, and lastly he adds together a, 4a, and 8a.
Probably division was also performed by repeated subtractions, but,
as he rarely explains the process by which he arrived at a result,
this is not certain. After these examples Ahmes goes on to the
solution of some simple numerical equations. For example, he
says "heap, its seventh, its whole, it makes nineteen," by which
he means that the object is to find a number such that the sum
of it and one-seventh of it shall be together equal to 1 9 ; and he
gives as the answer 1 6 -f- J -f ^, which is correct.
CH. i] EARLY EGYPTIAN MATHEMATICS 5
The arithmetical part of the papyrus indicates that he had
some idea of algebraic symbols. The unknown quantity is
always represented by the symbol which means a heap ; addition
is sometimes represented by a pair of legs walking forwards,
subtraction by a pair of legs walking backwards or by a flight
of arrows ; and equality by the sign ^ .
The latter part of the book contains various geometrical
problems to which I allude later. He concludes the work with
some arithmetico-algebraical questions, two of which deal with
arithmetical progressions and seem to indicate that he knew
how to sum such series.
Second, as to the science of gecrnietry. Geometry is supposed
to have had its origin in land-surveying ; but while it is difficult
to say when the study of numbers and calculation — some know-
ledge of which is essential in any civilised state — became a
science, it is comparatively easy to distinguish between the
abstract reasonings of geometry and the practical rules of the
land-surveyor. Some methods of land-surveying must have
been practised from very early times, but the universal tradition
of antiquity asserted that the origin of geometry was to be
sought in Egypt. That it was not indigenous to Greece, and
that it arose from the necessity of surveying, is rendered the
more probable by the derivation of the word from yrj, the earth,
and fieTpGO), I measure. Now the Greek geometricians, as far as
we can judge by their extant works, always dealt with the
science as an abstract one : they sought for theorems which
should be absolutely true, and, at any rate in historical times,
would have argued that to measure quantities in terms of a
unit which might have been incommensurable with some of the
magnitudes considered would have made their results mere
approximations to the truth. The name does not therefore
refer to their practice. It is not, however, unlikely that it
indicates the use which was made of geometry among the
Egyptians from whom the Greeks learned it. This also agrees
with the Greek traditions, which in themselves appear probable^
for Herodotus states that the periodical inundations of the Nile
6 EGYPTIAN & PHOENICIAN MATHEMATICS [ch. i
(which swept away the landmarks in the valley of the river,
and by altering its course increased or decreased the taxable
value of the adjoining lands) rendered a tolerably accurate
system of surveying indispensable, and thus led to a systematic
study of the subject by the priests.
We have no reason to think that any special attention was
paid to geometry by the Phoenicians, or other neighbours of the
Egyptians. A small piece of evidence which tends to show that
the Jews had not paid much attention to it is to be found in
the mistake made in their sacred books, ^ where it is stated that
the circumference of a circle is three times its diameter : the
Babylonians ^ also reckoned that tt was equal to 3.
Assuming, then, that a knowledge of geometry was first
derived by the Greeks from Egypt, we must next discuss the
range and nature of Egyptian geometry.^ That some geo-
metrical results w^ere known at a date anterior to Ahmes's work
seems clear if we admit, as we have reason to do, that, centuries
before it was written, the following method of obtaining a right
angle was used in laying out the ground-plan of certain build-
ings. The Egyptians were very particular about the exact
orientation of their temples ; and they had therefore to obtain
with accuracy a north and south line, as also an east and west
line. By observing the points on the horizon where a star rose
and set, and taking a plane midway between them, they could
obtain a north and south line. To get an east and west line,
which had to be drawn at right angles to this, certain profes-
sional " rope-fasteners " were employed. These men used a
rope ABCD divided by knots or marks at B and (7, so that the
lengths AB, BC, CD were in the ratio 3:4:5. The length BG
was placed along the north and south line, and pegs P and Q
inserted at the knots B and C. The piece BA (keeping it
stretched all the time) was then rotated round the peg P, and
^ I. Kings, chap, vii, verse 23, and II. Chronicles, chap, iv, verse 2.
2 See J. Oppert, Journal Asiatique, August 1872, and October 1874.
^ See Eisenlohr ; Cantor, chap, ii ; Gow, arts. 75, 76 ; and Die
Geometrie der alien Aegypter, by E. Weyr, Vienna, 1884.
CH. i] EARLY EGYPTIAN GEOMETRY 7
similarly the piece CD was rotated round the peg Q, until the
ends A and D coincided ; the point thus indicated was marked
by a peg R. The result was to form a triangle PQR whose
sides RP^ FQ, QR were in the ratio 3:4:5. The angle of the
triangle at P would then be a right angle, and the line PR
would give an east and west line. A similar method is con-
stantly used at the present time by practical engineers for
measuring a right angle. The property employed can be
deduced as a particular case of Euc. i, 48 ; and there is reason
to think that the Egyptians were acquainted with the results of
this proposition and of Euc. i, 47, for triangles whose sides are
in the ratio mentioned above. They must also, there is little
doubt, have known that the latter proposition was true for an
isosceles right-angled triangle, as this is obvious if a floor be
paved with tiles of that shape. But though these are interest-
ing facts in the history of the Egyptian arts we must not press
them too far as showing that geometry was then studied as a
science. Our real knowledge of the nature of Egyptian geo-
metry depends mainly on the Rhind papyrus.
Ahmes commences that part of his papyrus which deals with
geometry by giving some numerical instances of the contents of
bams. Unluckily we do not know what was the usual shape
of an Egyptian barn, but where it is defined by three linear
measurements, say a, 6, and c, the answer is always given as
if he had formed the expression axbx{c + \c). He next
proceeds to find the areas of certain rectilineal figures ; if the
text be correctly interpreted, some of these results are wrong.
He then goes on to find the area of a circular field of diameter
12 — no unit of length being mentioned — and gives the result
as {d - \dY, where d is the diameter of the circle : this is
equivalent to taking 3 '1604 as the value of tt, the actual value
being very approximately 3'1416. Lastly, Ahmes gives some
problems on pyramids. These long proved incapable of inter-
pretation, but Cantor and Eisenlohr have shown that Ahmes
was attempting to find, by means of data obtained from the
measurement of the external dimensions of a building, the
8 EGYPTIAN & PHOENICIAN MATHEMATICS [ch. i
ratio of certain other dimensions which could not be directly
measured : his process is equivalent to determining the trigono-
metrical ratios of certain angles. The data and the results
given agree' closely with the dimensions of some of the existing
pyramids. Perhaps all Ahmes's geometrical results were intended
only as approximations correct enough for practical purposes.
It is noticeable that all the specimens of Egyptian geometry
which we possess deal only with particular numerical problems
and not with general theorems ; and even if a result be stated
as universally true, it was probably proved to be so only by a
wide induction. We shall see later that Greek geometry was
from its commencement deductive. There are reasons for think-
ing that Egyptian geometry and arithmetic made little or no
progress subsequent to the date of Ahmes's work ; and though
for nearly two hundred years after the time of Thales Egypt
was recognised by the Greeks as an important school of mathe-
matics, it would seem that, almost from the foundation of the
Ionian school, the Greeks outstripped their former teachers.
It may be added that Ahmes's book gives us much that idea
of Egyptian mathematics which we should ha^e gathered from
statements about it by various Greek and Latin authors, who
lived centuries later. Previous to its translation it was commonly
thought that these statements exaggerated the acquirements of
the Egyptians, and its discovery must increase the weight to be
attached to the testimony of these authorities.
We know nothing of the applied mathematics (if there were
any) of the Egyptians or Phoenicians. The astronomical attain-
ments of the Egyptians and Chaldaeans were no doubt consider-
able, though they were chiefly the results of observation : the
Phoenicians are said to have confined themselves to studying
what was required for navigation. Astronomy, however, lies
outside the range of this book.
I do not like to conclude the chapter without a brief mention
of the Chinese, since at one time it was asserted that they were
familiar with the sciences of arithmetic, geometry, mechanics,
optics, navigation, and astronomy nearly three thousand years
CH.i] EARLY CHINESE MATHEMATICS 9
ago, and a few writers were inclined to suspect (for no evidence
was forthcoming) that some knowledge of this learning had
filtered across Asia to the West. It is true that at a very early
period the Chinese were acquainted with several geometrical or
rather architectural implements, such as the rule, square, com-
passes, and level ; with a few mechanical machines, such as the
wheel and axle ; that they knew of the characteristic property
of the magnetic needle ; and were aware that astronomical events
occurred in cycles. But the careful investigations of L. A.
Sedillot ^ have shown that the Chinese made no serious attempt
to classify or extend the few rules of arithmetic or geometry
with which they were acquainted, or to explain the causes of
the phenomena which they observed.
The idea that the Chinese had made considerable progress
in theoretical mathematics seems to have been due to a mis-
apprehension of the Jesuit missionaries who went to China
in the sixteenth century. In the first place, they failed to
distinguish between the original science of the Chinese and
the views which they found prevalent on their arrival — the
latter being founded on the work and teaching of Arab or
Hindoo missionaries who had come to China in the course of
the thirteenth cgitury or later, and while there introduced a
knowledge of spherical trigonometry. In the second place,
finding that one of the most important government depart-
ments was known as the Board of Mathematics, they supposed
that its function was to promote and superintend mathematical
studies in the empire. Its duties were really confined to the
annual preparation of an almanack, the dates and predictions
in which regulated many affairs both in public and domestic
life. All extant specimens of these almanacks are defective
and, in many respects, inaccurate.
The only geometrical theorem with M^hich we can be certain
that the ancient Chinese were acquainted is that in certain cases
^ See .Boncompagni's Bulletino di hihliograjia e di storia delle scienze
matematiche e fisicJie for May, 1868, vol. i, pp. 161-166. On Chinese
mathematics, mostly of a later date, see Cantor, chap. xxxi.
10 EGYPTIAN & PHOENICIAN MATHEMATICS [ch. i
(namely, when the ratio of the sides is 3 : 4 : 5, or 1:1: ^^2)
the area of the square described on the hypotenuse of a right-
angled triangle is equal to the sum of the areas of the squares
described on the sides. It is barely possible that a few
geometrical theorems which can be demonstrated in the quasi-
experimental way of superposition were also known to them.
Their arithmetic was decimal in notation, but their knowledge
seems to have been confined to the art of calculation by means
of the swan -pan, and the power of expressing the results in
writing. Our acquaintance with the early attainments of the
Chinese, slight though it is, is more complete than in the case
of most of their contemporaries. It is thus specially instructive,
and serves to illustrate the fact that a nation may possess con-
siderable skill in the applied arts while they are ignorant of the
sciences on which those arts are founded.
From the foregoing summapy it will be seen that our know-
ledge of the mathematical attainments of those who preceded
the Greeks is very limited ; but we may reasonably infer that
from one source or another the early Greeks learned the use of
the abacus for practical calculations, symbols for recording the
results, and as much mathematics as is contained or implied in
the Khind papyrus. It is probable that thi^ sums up their
indebtedness to other races. In the next six chapters I shall
trace the development of mathematics under Greek influence.
11
FIRST PERIOD.
iiHatJematics untjcr ©reek Enfltunce.
This period begins with the teaching of Thales, circ. 600 B.C.,
and ends with the capture of Alexandria hy the Mohammedans
in or about 641 a.d. The characteristic feature of this period
is the development of Geometry.
It will be remembered that I commenced the last chapter by-
saying that the history of mathematics might be divided into
three periods, namely, that of mathematics under Greek influence,
that of the mathematics of the middle ages and of the renaissance,
and lastly that of modern mathematics. The next four chapters
(chapters ii, iii, iv and v) deal with the history of mathe-
matics under Greek influence : to these it will be convenient to
add one (chapter vi) on the Byzantine school, since through it
the results of Greek mathematics were transmitted to western
Europe ; and another (chapter vii) on the systems of numeration
which were ultimately displaced by the system introduced by the
Arabs. I should add that many of the dates mentioned in these
chapters are not known with certainty, and must be regarded as
only approximately correct.
13
CHAPTER 11.
THE IONIAN AND PYTHAGOREAN SCHOOLS. ^
CIRC. 600 B.C.-400 B.C.
With the foundation of the Ionian and Pythagorean schools we
emerge from the region of antiquarian research and conjecture
into the light of history. The materials at our disposal for
estimating the knowledge of the philosophers of these schools
previous to about the year 430 B.C. are, however, very scanty
Not only have all but fragments of the different mathematical
treatises then written been lost, but we possess no copy of the
history of mathematics written about 325 B.C. by Eudemus (who
was a pupil of Aristotle). Luckily Proclus, who about 450 a.d.
wrote a commentary on the earlier part of Euclid's Elements,
was familiar ^vith Eudemus's work, and freely utilised it in his
historical references. We have also a fragment of the General
View of Mathematics written by Geminus about 50 B.C., in which
the methods of proof used by the early Greek geometricians are
compared with those current at a later date. In addition to
these general statements we have biographies of a few of the
^ The history of these schools has been discussed by G. Loria in his Le Scienze
Esatte neir Antica Grecia, Modena, 1893-1900 ; by Cantor, chaps, v-viii ;
by G. J. Allman in his Greek Geometry from Tholes to Euclid, Dublin, 1889 ;
by J. Gow, in his Greek Mathematics, Cambridge, 1884 ; by C. A. Bret-
schneider in his Die Geometric unci die Geometer vor Eukleides, Leipzig, 1870 ;
and partially by H. Hankel in his posthumous Geschichte der Mathematik,
Leipzig, 1874.
14 IONIAN AND PYTHAGOREAN SCHOOLS [ch. ii
leading mathematicians, and some scattered notes in various
writers in which allusions are made to the lives and works of
others. The original authorities are criticised and discussed at
length in the works mentioned in the footnote to the heading of
the chapter.
The Ionian School.
Thales.^ The founder of the earliest Greek school of mathe-
matics and philosophy was Thales, one of the seven sages of
Greece, who was born about 640 B.C. at Miletus, and died in the
same town about 550 B.C. , The materials for an account of his
life consist of little more than a few anecdotes which have been
handed down by tradition.
During the early part of his life Thales was engaged partly
in commerce and partly in public affairs ; and to judge by two
stories that have been preserved, he was then as distinguished
for shrewdness in business and readiness in resource as he was
subsequently celebrated in science. It is said that once when
transporting some salt which was loaded on mules, one of the
animals slipping in a stream got its load wet and so caused
some of the salt to be dissolved, and finding its burden thus
lightened it rolled over at the next ford to which it came ; to
break it of this trick Thales loaded it with rags and sjjonges
which, by absorbing the water, made the load heavier and soon
effectually cured it of its troublesome habit. At another time,
according to Aristotle, when there was a prospect of an
unusually abundant crop of olives Thales got possession of all
the olive-presses of the district; and, having thus "cornered"
them, he was able to make his own terms for lending them out,
or buying the olives, and thus realized a large sum. These
tales may be apocryphal, but it is certain that he must have
had considerable reputation as a man of affairs and as a good
engineer, since he was employed to construct an embankment so
as to divert the river Halys in such a way as to permit of the
construction of a ford.
1 See Loria, book i, chap, ii ; Cantor, cliap. v ; Allmaii, cliap. i.
CH.ii] THALES 15
Probably it was as a merchant that Thales first went to
Egypt, but during his leisure there he studied astronomy and
geometry. He was middle-aged when he returned to Miletus ;
he seems then to have abandoned business and public life,
and to have devoted himself to the study of philosophy and
science — subjects which in the Ionian, Pythagorean, and
perhaps also the Athenian schools, were closely connected :
his views on philosophy do not here concern us. He continued
to live at Miletus till his death circ. 550 B.C.
We cannot form any exact idea as to how Thales presented
his geometrical teaching. We infer, however, from Proclus that
it consisted of a number of isolated propositions which were
not arranged in a logical sequence, but that the proofs were
deductive, so that the theorems were not a mere statement of
an induction from a large number of special instances, as
l^robably was the case with the Egyptian geometricians. The
deductive character which he thus gave to the science is his
chief claim to distinction.
The following comprise the chief propositions that can now
with reasonable probability be attributed to him ; they are
concerned with the geometry of angles and straight lines.
(i) The angles at the base of an isosceles triangle are equal
(Euc. I, 5). Proclus seems to imply that this was proved by
taking another exactly equal isosceles triangle, turning it over,
and then superposing it on the first — a sort of experimental
demonstration.
(ii) If two straight lines cut one another, the vertically
opposite angles are equal (Euc. i, 15). Thales may have
regarded this as obvious, for Proclus adds that Euclid was the
first to give a strict proof of it.
(iii) A triangle is determined if its base and base angles be
given (c/. Euc. i, 26). Apparently this was applied to find the
distance of a ship at sea — the base being a tower, and the base
angles being obtained by observation.
(iv) The sides of equiangular triangles are proportionals
(Euc. VI, 4, or perhaps rather Euc. vi, 2). This is said to
16 IONIAN AND PYTHAGOREAN SCHOOLS [ch. ii
have been used by Thales when in Egypt to find the height of
a pyramid. In a dialogue given by Plutarch, the speaker,
addressing Thales, says, "Placing your stick at the end of
the shadow of the pyramid, you made by the sun's rays two
triangles, and so proved that the [height of the] pyramid was
to the [length of the] stick as the shadow of the pyramid to
the shadow of the stick." It would seem that the theorem was
unknown to the Egyptians, and we are told that the king
Amasis, who was present, was astonished at this application of
abstract science.
(v) A circle is bisected by any diameter. This may have
been enunciated by Thales, but it must have been recognised as
an obvious fact from" the earliest times.
(vi) The angle subtended by a diameter of a circle at any
point in the circumference is a right angle (Euc. iii, 31).
This appears to have been regarded as the most remarkable
of the geometrical achievements of Thales, and it is stated that
on inscribing a right-angled triangle in a circle he sacrificed an
ox to the immortal gods. It has been conjectured that he may
have come to this conclusion by noting that the diagonals of a
rectangle are equal and bisect one another, and that therefore a
rectangle can be inscribed in a circle. If so, and if he went on
to apply proposition (i), he would have discovered that the
sum of the angles of a right-angled triangle is equal to two
right angles, a fact with which it is believed that he was
acquainted. It has been remarked that the shape of the tiles
used in paving floors may have suggested these results.
On the whole it seems unlikely that he knew how to draw a
perpendicular from a point to a line ; but if he possessed this
knowledge, it is possible he was also aware, as suggested by
some modern commentators, that the sum of the angles of any
triangle is equal to two right angles. As far as equilateral
and right-angled triangles are concerned, we know from
Eudemus that the first geometers proved the general property
separately for three species of triangles, and it is not unlikely
that they proved it thus. The area about a point can be filled
CH.li] THALES * 17
by the angles of six equilateral triangles or tiles, hence the
proposition is true for an equilateral triangle. Again, any two
equal right-angled triangles can be placed in juxtaposition so
as to form a rectangle, the sum of whose angles is four right
angles ; hence the proposition is true for a right-angled triangle.
Lastly, any triangle can be split into the sum of two right-
angled triangles by drawing a perpendicular from the biggest
angle on the opposite side, and therefore again the proposition
is true. The first of these proofs is evidently included in the
last, but there is nothing improbable in the suggestion that the
early Greek geometers continued to teach the first proposition
in the form above given.
Thales wrote on astronomy,- and among his contemporaries
was more famous as an astronomer than as a geometrician. A
story runs that one night, when walking out, he was looking so
intently at the stars that he tumbled into a ditch, on which an
old woman exclaimed, " How can you tell what is going on in
the sky when you can't see what is lying at your own feet ? "
— an anecdote which was often quoted to illustrate the un-
practical character of philosophers.
Without going into astronomical details, it may be mentioned
that he taught that a year contained about 365 days, and not
(as is said to have been previously reckoned) twelve months of
thirty days each. It is said that his predecessors occasionally
intercalated a month to keep the seasons in their customary
places, and if so they must have realized that the year contains,
on the average, more than 360 days. There is some reason to
think that he believed the earth to be a disc-like body floating
on water. He predicted a solar eclipse which took place at or
about ■ the time he foretold ; the actual date was either May 28,
585 B.C., or September 30, 609 B.C. But though this prophecy
and its fulfilment gave extraordinary prestige to his teaching,
and secured him the name of one of the seven sages of Greece,
it is most likely that he only made use of one of the EgyjDtian
or. Chaldaean registers which stated that solar eclipses recur
at intervals of about 18 years 11 days.
c
18 IONIAN AND PYTHAGOREAN SCHOOLS [ch. ii
Among the pupils of Thales were Anaximander, Anaximenes,
Mamercus, and Mandryatus. Of the three mentioned last we
know next to nothing. Anaximander was born in 611 B.C.,
and died in 545 B.C., and succeeded Thales as head of the
school at Miletus. According to Suidas he wrote a treatise on
geometry in which, tradition says, he paid particular attention
to the properties of spheres, and dwelt at length on the philo-
sophical ideas involved in the conception of infinity in space and
time. He constructed terrestrial and celestial globes.
Anaximander is alleged to have introduced the use of the
style or gnommi into Greece. This, in principle, consisted only
of a stick stuck upright in a horizontal piece of ground. It
was originally used as a sun-dial, in which case it was placed
at the centre of three concentric circles, so that every two
hours the end of its shadow passed from one circle to another.
Such sun-dials have been found at Pompeii and Tusculum. It
is said that he employed these styles to determine his meridian
(presumably by marking the lines of shadow cast by the style
at sunrise and sunset on the same day, and taking the plane
bisecting the angle so formed) ; and- thence, by observing the
time of year when the noon-altitude of the sun was greatest
and least, he got the solstices ; thence, by taking half the sum
of the noon-altitudes of the sun at the two solstices, he found
the inclination of the equator to the horizon (which determined
the altitude of the place), and, by taking half their difference,
he found the inclination of the ecliptic to the equator. There
seems good reason to think that he did actually determine the
latitude of Sparta, but it is more doubtful whether he really
made the rest of these astronomical deductions.
We need not here concern ourselves further with the
successors of Thales. The school he established continued to
flourish till about 400 b.c, but, as time went on, its members
occupied themselves more and more with philosophy and less
with mathematics. We know very little of the mathematicians
comprised in it, but they would seem to have devoted most of
their attention to astronomy. They exercised but slight in-
CH.ii] PYTHAGORAS 19
fluence on the further advance of Greek mathematics, which
was made almost entirely under the influence of the Pytha-
goreans, who not only immensely developed the science of
geometry, but created a science of numbers. If Thales was the
first to direct general attention to geometry, it was Pythagoras,
says Proclus, quoting from Eudemus, who "changed the study
of geometry into the form of a liberal education, for he ex-
amined its principles to the bottom and investigated its
theorems in an... intellectual manner"; and it is accordingly
to Pythagoras that we must now direct attention.
The Pythagorean School.
Pythagoras.^ Pythagoras was born at Samos about 569 B.C.,
perhaps of Tyrian parents, and died in 500 B.C. He was thus a
contemporary of Thales. The details of his life are somewhat
doubtful, but the following account is, I think, substantially
correct. He studied first under Pherecydes of Syros, and then
under Anaximander ; by the latter he was recommended to go
to Thebes, and there or at Memphis he spent some years.
After leaving Egypt he travelled in Asia Minor, and then
settled at Samos, where he gave lectures but without much
success. About 529 B.C. he migrated to Sicily with his mother,
and with a single disciple who seems to have been the sole fruit
of his labours at Samos. Thence he went to Tarentum, but
very shortly moved to Croton, a Dorian colony in the south of
Italy. Here the schools that he opened were crowded with
enthusiastic- audiences; citizens of all ranks, especially those
of the upper classes, attended, and even the women broke a law
which forbade their going to public meetings and flocked to hear
him. Amongst his most attentive auditors was Theano, the
^ See Loria, book i, chap, iii ; Cantor, chaps, vi, vii ; Allman, chap, ii ;
Hankel, pp. 92-111 ; Hoefer, Histoire des inatMmatiques, Paris, third edition,
1886, pp. 87-130 ; and various papers by S. P. Tannery. For an accoimt of
Pythagoras's life, embodying the Pythagorean traditions, see the biography
by lamblichns, of which there are two or three English translations. Those
who are interested in esoteric literature may like to see a modern attempt
to reproduce the Pythagorean teaching in Pythagoras, by E. Schure, Eug.
trans., London, 1906.
20 IONIAN AND PYTHAGOREAN SCHOOLS [ch. ii
young and beautiful daughter of his host Milo, whom, in spite
of the disparity of their ages, he married. She wrote a biography
of her husband, but unfortunately it is lost.
Pythagoras divided those who attended his lectures into
two classes, whom we may term probationers and Pythagoreans.
The majority were probationers, but it was only to the Pytha-
goreans that his chief discoveries were revealed. The latter
formed a brotherhood with all things in common, holding the
same philosophical and political beliefs, engaged in the same
pursuits, and bound by oath not to reveal the teaching or
secrets of the school ; their food was simple ; their discipline
severe; and their mode of life arranged to encourage self-
command, temperance, purity, and obedience. This strict
discipline and secret organisation gave the society a temporary
supremacy in the state which brought on it the hatred of various
classes; and, finally, instigated by his political opponents, the
mob murdered Pythagoras and many of his followers,
Though the political influence of the Pythagoreans was thus
destroyed, they seem to have re-established themselves at once
as a philosophical and mathematical society, with Tarentum as
their headquarters, and they continued to flourish for more than
a hundred years.
Pythagoras himself did not publish any books ; the assump-
tion of his school was that all their knowledge was held in
common and veiled from the outside world, and, further, that the
glory of any fresh discovery must be referred back to their
founder. Thus Hippasus (circ. 470 B.C.) is said to have been
drowned for violating his oath by publicly boasting that he had
added the dodecahedron to the number of regular solids enume-
rated by Pythagoras. Gradually, as the society became more
scattered, this custom was abandoned, and treatises containing
the substance of their teaching and doctrines were written.
The first book of the kind was composed, about 370 b.c, by
Philolaus, and we are told that Plato secured a copy of it. We
may say that during the early part of the fifth century before
Christ the Pythagoreans were considerably in advance of their
cH.ii] PYTHAGORAS 21
contemporaries, but by the end of that time their more
prominent discoveries and doctrines had become known to the
outside world, and the centre of intellectual activity was
transferred to Athens.
Though it is impossible to separate precisely the discoveries
of Pythagoras himself from those of his school of a later date,
we know from Proclus that it was Pythagoras who gave
geometry that rigorous character of deduction which it still
bears, and made it the foundation of a liberal education ; and
there is reason to believe that he was the first to arrange the
leading propositions of the subject in a logical order. It was
also, according . to Aristoxenus, the glory of his school that they
raised arithmetic above the needs of merchants. It was their
boast that they sought knowledge and not wealth, or in the
language of one of their maxims, " a figure and a step forwards,
not a figure to gain three oboli."
Pythagoras was primarily a moral reformer and philosopher,
but his system of morality and philosophy was built on a
mathematical foundation. His mathematical researches were,
however, designed to lead up to a system of philosophy whose
exposition was the main object of his teaching. The Pythago-
reans began by dividing the mathematical subjects with which
they dealt into four divisiqus : numbers absolute or arithmetic,
numbers applied or music, magnitudes at rest or geometry, and
magnitudes in motion or astronomy. This " quadrivium " was
long considered as constituting the necessary and sufficient
course of study for a liberal education. Even in the case of
geometry and arithmetic (which are founded on inferences
unconsciously made and common to all men) the Pythagorean
presentation was involved with philosophy; and there is no
doubt that their teaching of the sciences of astronomy,
mechanics, and music (which can rest safely only on the
results of conscious observation and experiment) was inter-
mingled with metaphysics even more closely. It will be con-
venient to begin by describing their treatment of geometry and
aritlimetic.
22 IONIAN AND PYTHAGOREAN SCHOOLS [ch. ii
First, as to their geometry. Pythagoras probably knew and
taught the substance of what is contained in the first two books
of Euclid about parallels, triangles, and parallelograms, and was
acquainted with a few other isolated theorems including some
elementary propositions on irrational magnitudes; but it is
suspected that many of his proofs were not rigorous, and in
particular that the converse of a theorem was sometimes assumed
without a proof. It is hardly necessary to say that we are un-
able to reproduce the whole body of Pythagorean teaching on
this subject, but we gather from the notes of Proclus on Euclid,
and from a few stray remarks in other writers, that it included
the following propositions, most of which are on the geometry
of areas.
(i) It commenced with a number of definitions, which prob-
ably were rather statements connecting mathematical ideas
with philosophy than explanations of the terms used. One
has been preserved in the definition of a point as unity having
position.
(ii) The sum of the angles of a triangle was shown to be
equal to two right angles (Euc. i, 32) ; and in the proof, which
has been preserved, the results of the propositions Euc. i, 13 and
the first part of Euc. i, 29 are quoted. The demonstration is
substantially the same as that in Euclid, and it is most likely
that the proofs there given of the two propositions last mentioned
are also due to Pythagoras himself.
(iii) Pythagoras certainly proved the properties of right-
angled triangles which are given in Euc. i, 47 and i, 48. We
know that the proofs of these propositions which are found in
Euclid were of Euclid's own invention ; and a good deal of
curiosity has been excited to discover what was the demon-
stration which was originally offered by Pythagoras of the first
of these theorems. It has been conjectured that not improbably
it may have been one of the two following.^
^ A collection of a hxmdred proofs of Euc. i, 47 was published in
the Anierican Mathematical Monthly Journal, vols, iii, iv. v. vi, 1896-
1899.
CH. ii] PYTHAGORAS 23
(a) Any square ABGD can be split up, as in Euc. ii, 4, into
two squares BK and DK and two equal rectangles AK and CK\
that is, it is equal to the square on FK^ the square on EK^ and
^^^
\
L^
\
four times the triangle AEF. But, if points be taken, G on
BC, H on CD, and E on DA, so that BG, CH, and Z)^ are
each equal to AF, it can be easily shown that EFGII is a
square, and that the triangles AEF, BFG, CGH, and DHE are
equal : thus the square ABCD is also equal to the square on
EF and four times the triangle AEF. Hence the square on EF
is equal to the sum of the squares on FK and EK.
{(3) Let ABC be a right-angled triangle, A being the right
angle. Draw AD perpendicular to BC, The triangles ABC
A
and DBA are similar,
.-. BC '.AB = AB :BD.
Similarly BC : AC = AC : DC.
Hence AB'- + AC^ = BC{BD + DC) = BC\
24 IONIAN AND PYTHAGOREAN SCHOOLS [ch. ii
This proof requires a knowledge of the results of Euc. ii, 2,
VI, 4, and vi, 17, with all of which Pythagoras was acquainted.
(iv) Pythagoras is credited by some writers with the discovery
of the theorems Euc. i, 44, and i, 45, and with giving a solution
of the problem Euc. ii, 14. It is said that on the discovery of
the necessary construction for the problem last mentioned he
sacrificed an ox, but as his school had all things in common the
liberality was less striking than it seems at first. The Pythagoreans
of a later date were aware of the extension given in Euc. vi, 25,
and AUman thinks that Pythagoras himself was acquainted with
it, but this must be regarded as doubtful. It will be noticed that
Euc. II, 14 provides a geometrical solution of the equation x^ = ah.
(v) Pythagoras showed that the plane about a point could be
completely filled by equilateral triangles, by squares, or by regular
hexagons — results that must have been familiar wherever tiles of
these shapes were in common use.
(vi) The Pythagoreans were said to have attempted the quad-
rature of the circle : they stated that the circle was the most
perfect of all plane figures.
(vii) They knew that there were five regular solids inscrib-
able in a sphere, which was itself, they said, the most perfect
of all solids.
(viii) From their phraseology in the science of numbers and
from other occasional remarks, it would seem that they were
acquainted with the methods used in the second and fifth books
of Euclid, and knew something of irrational magnitudes. In
particular, there is reason to believe that Pythagoras proved
that the side and the diagonal of a square were incommensur-
able, and that it was this discovery which led the early Greeks
to banish the conceptions of number and measurement from
their geometry. A proof of this proposition which may be that
due to Pythagoras is given below. ^
Next, as to their theory of numbers.'-^ In this Pythagoras
^ See below, page 60.
2 See tlie appendix ^r V arithmetique pythagorienne to S, P. Tannery's
La science hellene, Paris, 1887.
CH. Il]
PYTHAGOREAN GEOMETRY
25
was chiefly concerned with four different classes of problems
which dealt respectively with polygonal numbers, with ratio and
proportion, with the factors of numbers, and with numbers in
series ; but many of his arithmetical inquiries, and in particular
the questions on polygonal numbers and proportion, were treated
by geometrical methods.
Pythagoras commenced his theory of arithmetic by dividing
all numbers into even or odd : the odd numbers being termed
gnomons. An odd number, such as 2n + 1, was regarded as the
difference of two square numbers {n+iy and n'^ ; and the sum
of the gnomons from 1 to 2n+l was stated to be a square
number, viz. {n+Vf, its square root was termed a side. Pro-
ducts of two numbers were called plane, and if a product had
A ■ ' \
no exact square root it was termed an oblong. A product of
three numbers was called a solid number, and, if the three
numbers were equal, a cube. All this has obvious reference to
geometry, and the opinion is confirmed by Aristotle's remark
that when a gnomon is put round a square the figure remains
a square though it is increased in dimensions. Thus, in the
figure given above in which n is taken equal to 5, the
gnomon AKC (containing 11 small squares) when put round the
square AC (containing 5^ small squares) makes a square HL
(containing 6^ small squares). It is possible that several of
26 IONIAN AND PYTHAGOREAN SCHOOLS [ch. ii
the numerical theorems due to Greek writers were discovered
and proved by an analogous method : the abacus can be used
for many of these demonstrations.
The numbers {2n^ + 2n+l), (2n^ + 2n), and (2n+l) pos-
sessed special importance as representing the hypotenuse and
two sides of a right - angled triangle : Cantor thinks that
Pythagoras knew this fact before discovering the geometrical
proposition Euc. i, 47. A more general expression for such
numbers is (m^ + n^), 2mn, and (m^ -n^) \ it will be noticed
that the result obtained by Pythagoras can be deduced from
these expressions by assuming m = n + \ ; at a later time
Archytas and Plato gave rules which are equivalent to taking
7^ = 1 ; Diophantus knew the general expressions.
After this preliminary discussion the Pythagoreans pro-
ceeded to the four special problems already alluded to.
Pythagoras was himself acquainted with triangular numbers ;
polygonal numbers of a higher order were discussed by later
members of the school. A triangular number represents the
sum of a number of counters laid in rows on a plane ; the
bottom row containing n, and each succeeding row one less :
it is therefore equal to the sum of the series
that is, to \n{n+\). Thus the triangular number corre-
sponding to 4 is 10. This is the explanation of the language
of Pythagoras in the well-known passage in Lucian where the
merchant asks Pythagoras what he can teach him. Pythagoras
replies " I will teach you how to count." Merchant ^ " I know
that already." P^/^Aa^oras, " How do you count ? " Merchant,
" One, two, three, four—" Pythagoras, " Stop ! what you take
to be four is ten, a perfect triangle, and our symbol." As to
the work of the Pythagoreans on the factors of numbers we
know very little : they classified numbers by comparing them
with the sum of their integral subdivisors or factors, calling a
number excessive, perfect, or defective, according as it was
greater than, equal to, or less than the sum of these subdivisors.
CH.ii] PYTHAGOREAN GEOMETRY 27
These investigations led to no useful result. The third class of
problems which they considered dealt with numbers which
formed a proportion ; presumably these were discussed with the
aid of geometry as is done in the fifth book of Euclid. Lastly,
the Pythagoreans were concerned with series of numbers in
arithmetical, geometrical, harmonical, and musical progressions.
The three progressions first mentioned are well known; four
integers are said to be in musical progression when they are in
the ratio a : 2ab/(a + b) : ^ {a + b) : b, for example, 6, 8, 9, and
12 are in musical progression.
Of the Pythagorean treatment of the applied subjects of the
quadrivium, and the philosophical theories founded on them,
we know very little. It would seem that Pythagoras was much
impressed by certain numerical relations which occur in nature.
It has been suggested that he was acquainted with some of the
simpler facts of crystallography. It is thought that he was
aware that the notes sounded by a vibrating string depend on
the length of the string, and in particular that lengths which
gave a note, its fifth and its octave were in the ratio 2:3:4,
forming terms in a musical progression. It would seem, too,
that he believed that the distances of the astrological planets
from the earth were also in musical progression, and that the
heavenly bodies in their motion through space gave out
harmonious sounds : hence the phrase the harmony of the
spheres. These and similar conclusions seem to have suggested
to him that the explanation of the order and harmony of the
universe was to be found in the science of numbers, and that
numbers are to some extent the cause of form as well as
essential to its accurate measurement. He accordingly pro-
ceeded to attribute particular properties to particular numbers
and geometrical figures. For example, he taught that the cause
of colour was to be sought in properties of the number five,
that the explanation of fire was to be discovered in the nature
of the pyramid, and so on. I should not have alluded to this
were it not that the Pythagorean tradition strengthened, or
perhaps was chiefly responsible for the tendency of Greek
28 IONIAN AND PYTHAGOREAN SCHOOLS [ch. ii
writers to found the study of nature on philosophical con-
jectures and not on experimental observation — a tendency to
which the defects of Hellenic science must be largely attributed.
After the death of Pythagoras his teaching seems to have
been carried on by Epicharmus and Hippasus, and subse-
quently by Philolaus (specially distinguished as an astronomer),
ArcMppus, and Lysis. About a century after the murder of
Pythagoras we find Archytas recognised as the head of the
school.
Archytas.^ Archytas^ circ. 400 B.C., was one of the most
influential citizens of Tarentum, and was made governor of
the city no less than seven times. His influence among his
contemporaries was very great, and he used it with Dionysius
on one occasion to save the life of Plato. He was noted for the
attention he paid to the comfort and education of his slaves and
of children in the city. He was drowned in a shipwreck near
Tarentum, and his body washed on shore — a fit punishment, in
the eyes of the more rigid Pythagoreans, for his having departed
from the lines of study laid down by their founder. Several
of the leaders' of the Athenian school were among his pupils
and friends, and it is believed that much of their work was due
to his inspiration.
The Pythagoreans at first made no attempt to apply their
knowledge to mechanics, but Archytas is said to have treated it
with the aid of geometry. He is alleged to have invented and
worked out the theory of the pulley, and is credited with the
construction of a flying bird and some other ingenious mechanical
toys. He introduced various mechanical devices for construct-
ing curves and solving problems. These were objected to by
Plato, who thought that they destroyed the value of geometry
as an intellectual exercise, and later Greek geometricians con-
^ See Alltnan, chap. iv. A catalogue of the works of Archytas is given
by Fabricius iu his Bibliotlieca Graeca, vol. i, p. 833 : most of the fragments
on philosophy were published by Thomas Gale in his Opnscula Mytliologia,
Cambridge, 1670 ; and by Thomas Taylor as an Appendix to his translation
of lamblichus's Life of Pythagoras London, 1818. See also the references
given by Cantor, vol. i, p. 203.
CH. ii] ARCHYTAS 29
fined themselves to the use of two species of instruments,
namel}^, rulers and compasses. Archytas was also interested in
astronomy; he taught that the earth was a sphere rotating
round its axis in twenty -four hours, and round which the
heavenly bodies moved.
Archytas was one of the first to give a solution of the
problem to duplicate a cube, that is, to find the side of a cube
whose volume is double that of a given cube. This was one of
the most famous problems of antiquity. ^ The construction
given by Archytas is equivalent to the following. On the
diameter OA of the base of a right circular cylinder describe a
semicircle whose plane is perpendicular to the base of the
cylinder. Let the plane containing this semicircle rotate round
the generator through 0, then the surface traced out by the
semicircle will cut the cylinder in a tortuous curve. This curve
will be cut by a right cone whose axis is OA and semivertical
angle is (say) 60° in a point P, such that the projection of OP
on the base of the cylinder will be to the radius of the cylinder
in the ratio of the side of the required cube to that of the given
cube. The proof given by Archytas is of course geometrical ; ^
it will be enough here to remark that in the course of it he
shews himself acquainted with the results of the propositions
Euc. Ill, 18, Euc. Ill, 35, and Euc. xi, 19. To shew analytically
that the construction is correct, take OA as the axis of x, and
the generator through 0 as axis of z, then, with the usual
notation in polar co-ordinates, and if a be the radius of
the cylinder, we have for the equation of the surface described
by the semicircle, r = 2a sin 0 ; for that of the cylinder,
r sin ^ = 2a cos <^ ; and for that of the cone, sin ^ cos </> = J. These
three surfaces cut in a point such that sin^ ^ = J ? ^nd, therefore,
if p be the projection of OP on the base of the cylinder, then
p3 ^ (^ sin ^^3 ^ 2a^. Hence the volume of the cube whose side is
p is twice that of a cube whose side is a. I mention the problem
and give the construction used by Archytas to illustrate how
^ See below, pp. 37, 41, 42.
'^ It is printed by Allman, pp. 111-113.
30 IONIAN AND PYTHAGOREAN SCHOOLS [ch. ii
considerable was the knowledge of the Pythagorean school at
the time.
Theodoms. Another Pythagorean of about the same date as
Archytas was Theodorus of Cyrene, who is said to have proved
geometrically that the numbers represented by ^^3, ^5, ^6,
V7, V8, x/10, ^/ll, Jl% ^/13, V14, V15, and J\1 are in-
commensurable with unity. Theaetetus was one of his pupils.
Perhaps Timaeus of Lqcri and Bryso of Heraclea should be
mentioned as other distinguished Pythagoreans of this time. It
is believed that Bryso attempted_to find the area of a circle by
inscribing and circumscribing squares, and finally obtained
polygons between whose areas the area of the circle lay ; but it
is said that at some point he assumed that the area of the circle
was the arithmetic mean between an inscribed and a circum-
scribed polygon.
Other Greek Mathematical Schools in the Fifth Century B.C.
It would be a mistake to suppose that Miletus and Tarentum
were the only places where, in the fifth century, Greeks were
engaged in laying a scientific foundation for the study of mathe-
matics. These towns represented the centres of chief activity,
but there were few cities or colonies of any importance where
lectures on philosophy and geometry were not given. Among
these smaller schools I may mention those at Chios, Elea, and
Thrace.
The best known philosopher of the School of Chios was
Oenopides, who was born about 500 B.C., and died about 430
B.C. He devoted himself chiefly to astronomy, but he had
studied geometry in Egypt, and is credited with the solution of
two problems, namely, to draw a straight line from a given
external point perpendicular to a given straight line (Euc. i, 12),
and at a given point to construct an angle equal to a given angle
(Euc. I, 23).
Another important centre was at Elea in Italy. This was
founded in Sicily by Xenophanes. He was followed by
CH. ii] THE SCHOOLS OF CHIOS AND ELEA 31
Parmenides, Zeno, and Melissus. The members of the Eleatic
School were famous for the difficulties they raised in connection'
with questions that required the use of infinite series, such, for
example, as the well-known paradox of Achilles and the tortoise,
enunciated by Zeno, one of their most prominent members.
Zeno was born in 495 B.C., and was executed at Elea in 435 B.C.
in consequence of some conspiracy against the* state ; he was a
pupil of Parmenides, with whom he visited Athens, circ. 455-
450 B.C.
Zeno argued that if Achilles ran ten times as fast as a
tortoise, yet if the tortoise had (say) 1000 yards start it could
never be overtaken : for, when Achilles had gone the 1000
yards, the tortoise would still be 100 yards in front of him; by
the time he had covered these 100 yards, it would still be 10
yards in front of him ; and so on for ever : thus Achilles would
get nearer and nearer to the tortoise, but never overtake it. The
fallacy is usually explained by the argument that the time
required to overtake the tortoise, can be divided into an infinite
number of parts, as stated in the question, but these get smaller
and smaller in geometrical progression, and the sum of them all
is a finite time : after the lapse of that time Achilles would be
in front of the tortoise. Probably Zeno would have replied that
this argument rests on the assumption that space is infinitely
divisible, which is the question under discussion : he himself
asserted that magnitudes are not infinitely divisible.
These paradoxes made-the Greeks look with suspicion on the
use of infinitesimals, and ultimately led to the invention of the
method of exhaustions.
The Atomistic School, having its headquarters in Thrace, was
another important centre. This was founded by Leucippus,
>who was a pupil of Zeno. He was succeeded by Democritus
and Epicurus. Its most famous mathematician was Democritus,
born at Abdera in 460 B.C., and said to have died in 370 B.C.,
who, besides philosophical works, wrote on plane and solid
geometry, incommensurable lines, perspective, and numbers.
These works are all lost. From the Archimedean MS., discovered
32 THE ELEATIC AND ATOMISTIC SCHOOLS [ch. ii
by Heiberg in 1906, it would seem that Democritus enunciated,
but without a proof, the proposition that the volume of a
pyramid is equal to one-third that of a prism of an equal base
and of equal height.
But though several distinguished individual philosophers may
be mentioned who, during the fifth century, lectured at different
cities, they mostly seem to have drawn their inspiration from
Tarentum, and towards the end of the century to have looked to
Athens as the intellectual capital of the Greek world ; and it is
to the Athenian schools that we owe the next great advance in
mathematics.
33
CHAPTER III.
THE SCHOOLS OF ATHENS AND CYZICUS.^
cmc. 420 B.C.-300 b.c.
It was towards .the close of the fifth century before Christ that
Athens first became the chief centre of mathematical studies.
Several causes conspired to bring this about. During that
century she had become, partly by commerce, partly by appro-
priating for her own purposes the contributions of her allies, the
most wealthy city in Greece ; and the genius of her statesmen
had made her the centre on which the politics of the peninsula
turned. Moreover, whatever states disputed her claim to poli-
tical supremacy her intellectual pre-eminence was admitted by
all. There was no school of thought which had not at some
time in that century been represented at Athens by one or
more of its leading thinkers ; and the ideas of the new science,
which was being so eagerly studied in Asia Minor and Graecia
Magna, had been brought before the Athenians on various
occasions.
^ The history of these schools is discussed at length in G. Loria's Le
Scienze Esatte nelV Antica Grecia, Modena, 1893-1900 ; in G. J. Allman's
Greek Geometry from Tholes to Euclid, Dublin, 1889 ; and in J. Govv's Greek
Mathematics, Cambridge, 1884 ; it is also treated by Cantor, chaps, ix, x,
and xi ; by Hankel, pp. 111-156 ; and by C. A. Bretschneider in his Die
Geometric und die Geometer vor Eukleides, Leipzig, 1870 ; a critical account
of the original authorities is given by S. P. Tannery in his Geometric Grecque,
Paris, 1887, and other papers.
D
U THE SCHOOLS OF ATHENS AND CYZICUS [ch. hi
Anaxagoras. Amongst the most important of the philoso-
phers who resided at Athens and prepared the way for the
Athenian school I may mention Anaxagoras of Clazomenae,
who was almost the last philosopher of the Ionian school. He
was born in 500 B.C., and died in 428 B.C. He seems to have
settled at Athens about 440 B.C., and there taught the results of
the Ionian philosophy. Like all members of that school he was
much interested in astronomy. He asserted that the sun was
larger than the Peloponnesus : this opinion, together with some
attempts he had made to explain various physical phenomena
which had been previously supposed to be due to the direct
action of the gods, led to a prosecution for impiety, and he was
convicted, ^^^iilain prison he is said to have written_a^reatise
^,tha-quadrature .olihe circle.
The Sophists. The sophists can hardly be considered as
belonging to the Athenian school, any more than Anaxagoras
can ; but like him they immediately preceded and prepared the
way for it, so that it is desirable to devote a few words to them.
One condition for success in public life at Athens was the power
of speaking well, and as the wealth and power of the city in-
creased a considerable number of " sophists " settled there who
undertook amongst other things to teach the art of oratory.
Many of them also directed the general education of their pupils,
of which geometry usually formed a part. We are told that two
of those who are usually termed sophists made a special study
of geometry — these were Hippias of Elis and Antipho, and one
made a special study of astronomy — this was Meton, after whom
the metonic cycle is named.
Hippias. The first of these geometricians, Hippias of Elis
(circ. 420 B.C.), is described as an expert arithmetician, but he
is best known to us through his invention of a curve called the
quadratrix, by means of which an angle can be trisected, or
indeed divided in any given ratio. If the radius of a circle
rotate uniformly round the centre 0 from the position OA
through a right angle to OBj and in the same time a straigh't
line drawn perpendicular to OB move uniformly parallel to
CH. Ill]
THE QUADRATRIX
35
itself from the position OA to BC, the locus of their intersection
will be the quadratrix.
^
\
^\R
>
V ^^
A \
\R\
/
\
.
/ ^'-'
' \
\
Let OR and MQ be the position of these lines at any time \
and let them cut in P, a point on the curve. Then
angle A OF : angle AOB^ OM : OB.
Similarly, if OB! be another position of the radius,
angle AOP : angle AOB = OM' : OB.
.-. angle A OP : angle AOP' = 031 : OM ;
.-. angle AOF : angle rOP = OM' : MM.
Hence, if the angle A OP be given, and it be required to divide
it in any given ratio, it is siifficient to divide OM in that ratio
at J/', and draw the line JlP' ; then OP' will divide A OP in
the required ratio.
If OA be taken as the initial line, OP = r, the angle AOP=d,
and OA = a, we have ^ : ^tt = r sin ^ : a, and the equation of the
curve is 7rr = 2ad cosec 0.
Hippias_deyised__an instrument to construct the curve mechani-
cally ; but constructions which involved the use of any mathe-
matical instruments except a ruler and a pair of compasses were
objected to by Plato, and rejected by most geometricians of a
subsequent date.
36 THE SCHOOLS OF ATHENS AND CYZICUS [ch. hi
Antipho. The second sophist whom I mentioned was
Antipho (circ. 420 B.C.). He is one of the very few writers
among the ancients who attempted to find the area of a circle
by considering it as the limit of an inscribed regular polygon
with an infinite number of sides. He began by inscribing an
equilateral triangle (or, according to some accounts, a square) ;
on each side he inscribed in the smaller segment an isosceles
triangle, and so on ad infinitum. This method of attacking the
quadrature problem is similar to that described above as used by
Bryso of Heraclea.
No doubt there were other cities in Greece besides Athens
where similar and equally meritorious work was being done,
though the record of it has now been lost; I have mentioned
here the investigations of these three writers, chiefly because they
were the immediate predecessors of those who created the
Athenian school.
The Schools of Athens and Cyzicus. The history of the
Athenian school begins with the teaching of Hippocrates about
420 B.C. ; the school was established on a permanent basis by
the labours of Plato and Eudoxus ; and, together with the
neighbouring school of Cyzicus, continued to extend on the lines
laid down by these three geometricians until the foundation
(about 300 B.C.) of the university at Alexandria drew thither
most of the talent of Greece.
Eudoxus, who was amongst the most distinguished of the
Athenian mathematicians, is also reckoned as the founder of the
school at Cyzicus. The connection between this school and that
of Athens was very close, and it is now impossible to disentangle
their histories. It is said that Hippocrates, Plato, and
Theaetetus belonged to the Athenian school; while Eudoxus,
Menaechmus, and Aristaeus belonged to that of Cyzicus. There
was always a constant intercourse between the two schools, the
earliest members of both had been under the influence either of
Archytas or of his pupil Theodoras of Cyrene, and there was no
difference in their treatment of the subject, so that they may be
conveniently treated together.
CH. Ill] THE SCHOOLS OF ATHENS AND CYZICUS 37
Before discussing the work of the geometricians of these
schools in detail I may note that they were especially interested
in three problems : ^ namely (i), the duplication of a cube, that
is, the determination of the side of a cube whose volume is
double that of a given cube ; (ii) the trisection of an angle ; and
(iii) the squaring of a circle, that is, the determination of a
square whose area is equal to that of a given circle.
Now the first two of these problems (considered analytically)
require the solution of a cubic equation ; and, since a con-
struction by means of circles (whose equations are of the form'
x^ -\- y^ -\- ax ^-hy ^- c = 0) and straight lines (whose equations are
of the form a^ + /^^z + y = 0) cannot be equivalent to the solution
of a cubic equation, the problems are insoluble if in our con-
structions we restrict ourselves to the use of circles and straight
lines, that is, to Euclidean geometry. If the use of the conic
sections be permitted, both of these questions can be solved in
many ways. The third problem is equivalent to finding a
rectangle whose sides are equal respectively to the radius and to
the semiperimeter of the circle. These lines have been long
known to be incommensurable, but it is only recently that it has
been shewn by Lindemann that their ratio cannot be the root of
a rational algebraical equation. Hence this problem also is
insoluble by Euclidean geometry. The Athenians and Cyzicians
were thus destined to fail in all three problems, but the attempts
to solve them led to the discovery of many new theorems and
processes.
Besides attacking these problems the later Platonic school
collected all the geometrical theorems then known and arranged
them systematically. These collections comprised the bulk of
the propositions in Euclid's Elements^ books i-ix, xi, and xii,
together with some of the more elementary theorems in conic
sections.
Hippocrates. Hippocrates of Chios (who must be carefully
^ On these problems, solutions of them, and the authorities for their
history, see my Mathematical Recreations and Problems, London, ^ sixth
edition, 1914, chap. xii.
38 THE SCHOOLS OF ATHENS AND CYZICUS [ch. iii
distinguished from his contemporary, Hippocrates of Cos, the
celebrated physician) was one of the greatest of the Greek
geometricians. He was born about 470 B.C. at Chios, and began
life as a merchant. The accounts differ as to whether he was
swindled by the Athenian custom-house officials who were
stationed at the Chersonese, or whether one of his vessels was
captured by an Athenian pirate near Byzantium ; but at any
rate somewhere about 430 b.o. he came to Athens to try to
recover his property in the law courts. A foreigner was not
likely to succeed in such a case, and the Athenians seem only to
have laughed at him for his simplicity, first in allowing himself
to be cheated, and then in hoping to recover his money. While
prosecuting his cause he attended the lectures of various
philosophers, and finally (in all probability to earn a livelihood)
opened a school of geometry himself. He seems to have been
well acquainted with the Pythagorean philosophy, though there
is no sufficient authority for the statement that he was ever
initiated as a Pythagorean.
He wrote the first elementary text-book of geometry, a text-
book on which probably Euclid's Elements was founded ; and
therefore he may be said to have sketched out the lines on
which geometry is still taught in English schools. It is supposed
that the use of letters in diagrams to describe a figure was made
by him or introduced about this time, as he employs expressions
such as " the point on which the letter A stands " and " the line
on which AB \^ marked." Cantor, however, thinks that the
Pythagoreans had previously been accustomed to represent the
five vertices of the pentagram -star by the letters vy lO a;
and though this was a single instance, perhaps they may have
used the method generally. The Indian geometers never
employed letters to aid them in the description of their figures.
Hippocrates also denoted the square on a line by the word
SvvafXLS, and thus gave the technical meaning to the word
power which it still retains in algebra : there is reason to think
that this use of the word was derived from the Pythagoreans,
who are said to have enunciated the result of the pro-
CH. Ill] HIPPOCRATES 39
position Euc. i, 47, in the form that " the total power of the
sides of a right-angled triangle is the same as that of the
hypotenuse."
In this text -book Hippocrates introduced the method of
"reducing" one theorem to another, which being proved, the
thing proposed necessarily follows ; of this method the rediwtio
ad absurdiim is an illustration. No doubt the principle had
been used occasionally before, but he drew attention to it as
a legitimate mode of proof which was capable of numerous
applications. He elaborated the geometry of the circle : proving,
among other propositions, that similar segments of a circle
contain equal angles ; that the angle subtended by the chord of
a circle is greater than, equal to, or less than a right angle as
the segment of the circle containing it is less than, equal to, or
greater than a semicircle (Euc. iii, 31); and probably several
other of the propositions in the third book of Euclid. It is
most likely that he also established the propositions that [similar]
circles are to one another as the squares of their diameters
(Euc. XII, 2), and that similar segments are as the squares of
their chords. The proof given in Euclid of the first of these
theorems is believed to be due to Hippocrates.
The most celebrated discoveries of Hippocrates were, how-
ever, in connection with the quadrature of the circle and the
duplication of the cube, and owing to his influence these
problems played a prominent part in the history of the Athenian
school.
The following propositions will sufficiently illustrate the
method by w^hich he attacked the quadrature problem.
(a) He commenced by finding the area of a lune contained
between a semicircle and a quadrantal arc standing on the same
chord. This he did as follows. Let ABC be an isosceles right-
angled triangle inscribed in the semicircle ABOC, whose centre
is 0. On AB and ^C as diameters describe semicircles as in
the figure. Then, since by Euc. i, 47,
sq. on ^C = sq. on ^C-l-sq. on AB,
40 THE SCHOOLS OF ATHE]mA3ro^CJKP§^S^. Ill
therefore, by Euc. xii, 2, ^V^ -'Xx^ \
area J0 on ^C = area i-0 on ^C + area ^0 on ^^.
B O
Take away the common parts
,'. area A ABC = sum of areas of lunes AECD and AFBG.
Hence the area of the lune AECD is equal to half that of the
triangle ABC.
{P) He next inscribed half a regular hexagon ABCD in a
semicircle whose centre was 0, and on OA, AB, BC, and CD as
diameters described semicircles of which those on OA and AB
are drawn in the figure. Then ^i> is double any of the lines
OA, AB, BC, and CD,
CH. Ill] HIPPOCRATES 41
.-. sq. on AD = sum of sqs. on OA, AB, BC, and CD,
.'. area ^QABCD = sum of areas of |0s on OA, AB,BC, and CD.
Take away the common parts
.*. area trapezium ABCD = 3 lune AEBF + -J 0 on OA.
If therefore the area of this latter lune be known, so is that of
the semicircle described on OA as diameter. According to
Simplicius, Hippocrates assumed that the area of this lune was
the same as the area of the lune found in proposition (a) ; if
this be so, he was of course mistaken, as in this case he is deal-
ing with a lune contained between a semicircle and a sextantal
arc standing on the same chord ; but it seems more probable
that Simplicius misunderstood Hippocrates.
Hippocrates also enunciated various other theorems connected
with lunes (which have been collected by Bretschneider and by
Allman) of which the theorem last given is a typical example.
I believe that they are the earliest instances in which areas
bounded by curves were determined by geometry.
The other problem to which Hippocrates turned his attention
was the duplication of a cube, that is, the determination of
the side of a cube whose volume is double that of a given
cube.
This problem was known in ancient times as the Delian
problem, in consequence of a legend that the Delians had con-
sulted Plato on the subject. In one form of the story, which
is related by Philoponus, it is asserted that the Athenians in
430 B.C., when suffering from the plague of eruptive typhoid
fever, consulted the oracle at Delos as to how they could stop
it. Apollo replied that they must double the size of his altar
which was in the form of a cube. To the unlearned suppliants
nothing seemed more easy, and a new altar was constructed
either having each of its edges double that of the old one (from
which it followed that the volume was increased eightfold) or
by placing a similar cubic altar next to the old one. Where-
upon, according to the legend, the indignant god made the
pestilence worse than before, and informed a fresh deputation
42 THE SCHOOLS OF ATHENS AND CYZICUS [ch. hi
that it was useless to trifle with him, as his new altar must
be a cube and have a volume exactly double that of his old
one. Suspecting a mystery the Athenians applied to Plato,
who referred them to the geometricians, and especially to
Euclid, who had made a special study of the problem. The
introduction of the names of Plato and Euclid is an obvious
anachronism. Eratosthenes gives a somewhat similar account
of its origin, but with king Minos as the propounder of the
problem.
Hippocrates reduced the problem of duplicating the cube to
that of finding two means between one straight line (a), and
another twice as long (2a). If these means be x and y, we
have a'.x = x '.y = y :2a, from which it follows that x^ = 2a^.
It is in this form that the problem is usually presented now.
Hippocrates did not succeed in finding a construction for these
means.
Plato. The next philosopher of the Athenian school who
requires mention here was Plato. He was born at Athens in
429 B.C., and was, as is well known, a pupil for eight years of
Socrates ; much of the teaching of the latter is inferred from
Plato's dialogues. After the execution of his master in 399 B.C.
Plato left Athens, and being possessed of considerable wealth
he spent some years in travelling ; it was during this time that
he studied mathematics. He visited Egyj^t with Eudoxus, and
Strabo says that in his time the apartments they occupied at
Heliopolis were still shewn. Thence Plato went to Gyrene,
where he studied under Theodorus. Next he moved to Italy,
where he became intimate with Archytas the then head of the
Pythagorean school, Eurytas of Metapontum, and Timaeus of
Locri. He returned to Athens about the year 380 B.C., and
formed a school of students in a suburban gymnasium called
the " Academy." He died in 348 B.C.
Plato, like Pythagoras, was primarily a philosopher, and
perhaps his philosophy should be regarded as founded on the
Pythagorean rather than on the Socratic teaching. At any
rate it, like that of the Pythagoreans, was coloured with the
CH.iii] PLATO 43
idea that the secret of the universe is to be found in number
and in form ; hence, as Eudemus says, " he exhibited on every
occasion the remarkable connection between mathematics and
philosophy." All the authorities agree that, unlike many later
philosophers, he made a study of geometry or some exact
science an indispensable preliminary to that of philosophy.
The inscription over the entrance to his school ran " Let none
ignorant of geometry enter my door," and on one occasion an
applicant who knew no geometry is said to have been refused
admission as a student.
Plato's position as one of the masters of the Athenian
mathematical school rests not so much on his individual dis-
coveries and writings as on the extraordinary influence he
exerted on his contemporaries and successors. Thus the objec-
tion that he expressed to the use in the construction of curves
of any instruments other than rulers and compasses was at once
accepted as a canon which must be observed in such problems.
It is probably due to Plato that subsequent geometricians
began the subject with a carefuny_cQmpiIed__serie&-oi_ilfifijiitiQns,
jv^stiTlfltps, and ayioms. He also systematized the methods
which could be used in attacking mathematical questions, and
m particular directed attention to the jvalne-Of analysis. The
analytical method of proof begins by assuming that the theorem
or problem is solved, and thence deducing some result : if the
result be false, the theorem is not true or the problem is in-
capable of solution : if the result be true, and if the steps be
reversible, we get (by reversing them) a synthetic proof; but
if the steps be not reversible, no conclusion can be drawn.
Numerous illustrations of the method will be found in any
modern text-book on geometry. If the classification of the
methods of legitimate induction given by Mill in his work on
logic had been universally accepted and every new discovery in
science had been justified by a reference to the rules there laid
down, he would, I imagine, have occupied a position in refer-
ence to modern science somewhat analogous to that which Plato
occupied in regard to the mathematics of his time.
44 THE SCHOOLS OF ATHENS AND CYZICUS [ch.iii
The following is the only extant theorem traditionally attri-
buted to Plato. If CAB and DAB be two right-angled
triangles, having one side, AB, common, their other sides, AD
and BC, parallel, and their hypotenuses, AC and BD, at
right angles, then, if these hypotenuses cut in P, we have
PC:PB = PB:PA==PA: PD. This theorem was used in
duplicating the cube, for, if such triangles can be constructed
having PD = 2PC, the problem will be solved. It is easy
to make an instrument by which the triangles can be con-
structed.
Eudoxus.^ Of Eudoxus, the third great mathematician of
the Athenian school and the founder of that at Cyzicus, we
know very little. He was born in Cnidus in 408 B.C. Like
Plato, he went to Tarentum and studied under Archytas the
then head of the Pythagoreans. Subsequently he travelled
with Plato to Egyi^t, and then settled at Cyzicus, where he
founded the school of that name. Finally he and his pupils
moved to Athens. There he seems to have taken some part in
public affairs, and to have practised medicine ; but the hostility
of Plato and his own unpopularity as a foreigner made his
position uncomfortable, and he returned to Cyzicus or Cnidus
shortly before his death. He died while on a journey to Egypt
in 355 B.C.
His mathematical work seems to have been of a high order
of excellence. He discovered most of what we now know as
the fifth book of Euclid, and proved it in much the same form
as that in which it is there given. . .
He discovered some theorems on what was called "the
golden section." The problem to ._^'- C --^
cut a line ^^ in the golden section, A o^ H ^ B
that is, to divide it, say at H, in "
extreme and mean ratio (that is, so that AB : AH = AH : HB) is
solved in Euc. ii, 11, and probably was known to the Pytha-
^ The works of Eudoxus were discussed in considerable detail by
H. Kiinssberg of Dinkelsbiihl in 1888 and 1890 ; see also the authoi-ities
mentioned above in the footnote on p. 33.
CH.iii] EUDOXUS 45
goreans at an early date. If we denote A£ hy l, AH hj a, and
H£ by b, the theorems that Eudoxus proved are equivalent
to the following algebraical identities. (i) (a + ^l^ = 5(J^)^.
(ii) Conversely, if (i) be true, and AH be taken equal to a,
then AB will be divided at -^ in a golden section, (iii)
(b + la)2 = 5(1^2). (iv) l-^ + b^ = 3a^ (v) l + a:l = l:a, which
gives another golden section. These propositions were subse-
quently put by Euclid as the first five propositions of his
thirteenth book, but they might have been equally well placed
towards the end of the second book. All of them are obvious
algebraically, since l = a + h and a^ = bl.
Eudoxus further established the "method of exhaustions";
which depends on the proposition that "if from the greater
of two unequal magnitudes there be taken more than its half,
and from the remainder more than its half, and so on, there
will at length remain a magnitude less than the least of the
proposed magnitudes." This proposition was placed by Euclid
as the first proposition of the tenth book of his Elements, but
in most modern school editions it is printed at the beginning of
the twelfth book. By the aid of this theorem the ancient
geometers w^ere able to avoid the use of infinitesimals : the
method is rigorous, but awkward of application. A good illus-
tration of its use is to be found in the demonstration of Euc.
XII. .2, namely, that the square of the radius of one circle is to
the square of the radius of another circle as the area of the first
circle is to an area which is neither less nor greater than the
area of the second circle, and which therefore must be exactly
equal to it : the proof given by Euclid is (as was usual) com-
pleted by a reductio ad absiordum. Eudoxus applied the
principle to shew that the volume of a pyramid 'or a cone is
one-third that of the prism or the cylinder on the same base and
of the same altitude (Euc. xii, 7 and 10). It is believed that
he proved that the volumes of two spheres were to one another
as the cubes of their radii ; some writers attribute the proposi-
tion Euc. XII, 2 to him, and not to Hippocrates.
Eudoxus also considered certain curves other than the circle.
46 THE SCHOOLS OF ATHENS AND CYZICUS [ch. iii
There is no authority for the statement made in some old books
that these were conic sections, and recent investigations have
shewn that the assertion (which I repeated in the earlier editions
of this book) that they were plane sections of the anchor-ring is
also improbable. It seems most likely that they were tortuous
curves ; whatever they were, he applied them in explaining the
apparent motions of the planets as seen from the earth. '
Eudoxus constructed an orrery, and wrote a treatise on
practical astronomy, in which he supposed a number of moving
spheres to which the sun, moon, and stars were attached, and
which by their rotation produced the effects observed. In all
he required twenty-seven spheres. As observations became more
accurate, subsequent astronomers who accepted the theory had
continually to introduce fresh spheres to make the theory agree
with the facts. The work of Aratus on astronomy, which was
written about 300 B.C. and is still extant, is founded on that of
Eudoxus.
Plato and Eudoxus were contemporaries. Among Plato's
pupils were the mathematicians Leodamas, Neocleides,
Amyclas, and to their school also belonged Leon, Theudius
(both of whom wrote text-books on plane geometry), Cyzicenus,
Thasus, Hermotimus, Philippus, and Theaetetus. Among
the pupils of Eudoxus are reckoned Menaechmus, his brother
Dinostratus (who applied the quadratrix to the duplication and
trisection problems), and Aristaeus.
Menaechmus. Of the above-mentioned mathematicians
Menaechmus requires special mention. He was born about
375 B.C., and died about 325 B.C. Probably he succeeded
Eudoxus as head of the school at Cyzicus, where he acquired
great reputation as a teacher of geometry, and was for that
reason appointed one of the tutors of Alexander the Great.
In answer to his pupil's request to make his proofs shorter,
Menaechmus made the well-known reply that though in the
country there are private and even royal roads, yet in geometry
there is only one road for all.
Menaechmus was the first to discuss the conic sections, which
CH.iii] MENAECHMUS 47
were long called the Menaechmian triads. He divided them
into three classes, and investigated their properties, not by-
taking different plane sections of a fixed cone, but by keeping
his plane fixed and cutting it by different cones. He shewed
that the section of a right cone by a plane perpendicular to
a generator is an ellipse, if the cone be acute-angled ; a parabola,
if it be right-angled ; and a hyperbola, if it be obtuse-angled ;
and he gave a mechanical construction for curves of each class.
It seems almost certain that he was acquainted v^th the funda-
mental properties of these curves ; but some writers think that
he failed to connect them with the sections of the cone which he
had discovered, and there is no doubt that he regarded the
latter not as plane loci but as curves draw^n on the surface of a
cone.
He also shewed how these curves could be used in either of
the two following ways to give a solution of the problem
to duplicate a cube. In the first of these, he pointed out that
two parabolas having a common vertex, axes at right angles,
and such that the latus rectum of the one is double that of the
other will intersect in another point whose abscissa (or ordinate)
will give a solution ; for (using analysis) if the equations of the
parabolas be y^ = 2ax and x^=^ay^ they intersect in a point
whose abscissa is given by x^ = '2a^. It is probable that this
method was suggested by the form in which Hippocrates had cast
the problem ; namely, to find x and y so that a\x = x\y = y\2a^
whence we have a;^ = ay and y^ = 2ax.
The second solution given by Menaechmus was as follows.
Describe a parabola of latus rectum I. Next describe a rect-
angular hyperbola, the length of whose real axis is 4/, and
having for its asymptotes the tangent at the vertex of the para-
bola and the axis of the parabola. Then the ordinate and the
abscissa of the point of intersection of these curves are the
mean proportionals between I and 2Z. This is at once obvious
by analysis. The curves are x^ = ly and xy = 2l^. These
cut in a point determined by x^ = 2l^ and y^ = '\:l^. Hence
l'.x = x -.y^^y -.21.
48 THE SCHOOLS OF ATHENS AND CYZICUS [ch.iii
Aristaeus and Theaetetus. Of the other members of these
schools, Aristaeus and Theaetetus, whose works are entirely lost,
were mathematicians of repute. We know that Aristaeus wrote
on the five regular solids and on conic sections, and that Theae-
tetus developed the theory of incommensurable magnitudes.
The only theorem we can now definitely ascribe to the latter
is that given by Euclid in the ninth proposition of the tenth
book of the Elements, namely, that the squares on two commen-
surable right lines have one to the other a ratio which a square
number has to a square number (and conversely) ; but the
squares on two incommensurable right lines have one to the
other a ratio which cannot be expressed as that of a square
number to a square number (and conversely). This theorem
includes the results given by Theodorus.^
The contemporaries or successors of these mathematicians
wrote some fresh text-books on the elements of geometry and
the conic sections, introduced problems concerned with finding
loci, and systematized the knowledge already acquired, but they
originated no new methods of research.
Aristotle. An account of the Athenian school would be
incomplete if there were no mention of Aristotle, who was born
at Stagira in Macedonia in 384 B.C. and died at Chalcis in
Euboea in 322 B.C. Aristotle, however, deeply interested
though he was in natural philosophy, was chiefly concerned
with mathematics and mathematical physics as supplying illus-
trations of correct reasoning. A small book containing a few
questions on mechanics which is sometimes attributed to him
is of doubtful authority ; but, though in all probability it is
not his work, it is interesting, partly as shewing that the
principles of mechanics were beginning to excite attention, and
partly as containing the earliest known employment of letters
to indicate magnitudes.
The most instructive parts of the book are the dynamical
proof of the parallelogram of forces for the direction of the
resultant, and the statement, in effect, that if a be a force, P the
^ See above, p. 30.
CH. Ill] ARISTOTLE 49
mass to which it is applied, y the distance through which it is
moved, and 8 the time of the motion, then a wdll move ^yS
through 2y in the time S, or through y in the time JS : but the
author goes on to say that it does not follow that Ja will move
p through Jy in the time 8, because |a may not be able to move
/3 at all ; for 100 men may drag a ship 100 yards, but it does
not follow that one man can drag it one yard. The first part
of this statement is correct and is equivalent to the statement
that an impulse is proportional to the momentum produced, but
the second part is wrong.
The author also states the fact that what is gained in power
is lost in speed, and therefore that two weights which keep a
[weightless] lever in equilibrium are inversely proportional to
the arms of the lever ; this, he says, is the explanation why it
is easier to extract teeth with a pair of pincers than with the
fingers. Among other questions raised, but not answered, are
why a projectile should ever stop, and why carriages with large
wheels are easier to move than those with small.
I ought to add that the book contains some gross blunders,
and as a whole is not as able or suggestive as might be inferred
from the above extracts. In fact, here as elsewhere, the Greeks
did not sufficiently realise that the fundamental facts on which
the mathematical treatment of mechanics must be based can
be established only by carefully devised observations and
experiments. ^^^ '
50
CHAPTER IV.
THE FIRST ALEXANDRIAN SCHOOL. ^
CIRC. 300 B.C.-30 B.C.
The earliest attemi^t to found a university, as we understand
the word, was made at Alexandria. Richly endowed, supplied
with lecture rooms, libraries, museums, laboratories, and
gardens, it became at once the intellectual metropolis of the
Greek race, and remained so for a thousand years. It was
particularly fortunate in producing within the first century of its
existence three of the greatest mathematicians of antiquity —
Euclid, Ardlimedes^and ApoUonius. They laid down the lines
on which mathematics subsequently developed, and treated it as
a subject distinct from philosophy : hence the foundation of the
Alexandrian Schools is rightly taken as the commencement of a
new era. Thenceforward, until the destruction of the city by
the Arabs in 641 A.D., the history of mathematics centres
^ The history of tlie Alexandrian Schools is discussed by G. Loria in his
Le Scienze Esatte nclV Antica Grecia, Modena, 1893-1900 ; by Cautorj
chaps, xii-xxiii ; and by J. Gow in his History of Greek Mathematics,
Cambridge, 1884. The subject of Greek algebra is treated by E. H. F.
Nesselmann in his Die Algebra der Griechen, Berlin, 1842 ; see also L.
Matthiessen, Grundz'dge der antikcn und modernen Algebra der litter alen
Gleichungen, Leipzig, 1878. Tlie Greek treatment of the conic sections forms
the subject of Die Lehre von den Kegelschnitten in Altertum, by H. G.
Zeuthen, Copenhagen, 1886. The materials for the history of these schools
have been subjected to a searching criticism by S. P. Tannery, and most of
his papers are collected in his Geometrie Grecque, Paris, 1887.
CH. iv] THE FIRST ALEXANDRIAN SCHOOL 51
more or less round that of Alexandria; for this reason the
Alexandrian Schools are commonly taken to include all Greek
mathematicians of their time.
The city and university of Alexandria were created under the
following circumstances. Alexander the Great had ascended the
throne of Macedonia in 336 B.C. at the early age of twenty, and
by 332 B.C. he had conquered or subdued Greece, Asia Minor,
and Egypt. Following the plan he adopted whenever a com-
manding site had been left unoccupied, he founded a new city
on the Mediterranean near one mouth of the Nile ; and he him-
self sketched out the ground-plan, and arranged for drafts of
Greeks, Egyptians, and Jews to be sent to occupy it. The city
was intended to be the most magnificent in the world, and, the
better to secure this, its erection was left in the hands of
Dinocrates, the architect of the temple of Diana at Ephesus.
After Alexander's death in 323 b.c. his empire was divided,
and Egypt fell to the lot of Ptolemy, who chose Alexandria
as the capital of his kingdom. A short period of confusion
followed, but as soon as Ptolemy was settled on the throne, say
about 306 B.C., he determined to attract, so far as he was able,
learned men of all sorts to his new city ; and he at once began
the erection of the university buildings on a piece of ground
immediately adjoining his palace. The university was ready to
be opened somewhere about 300 B.C., and Ptolemy, who wished
to secure for its staff the most eminent philosophers of the time,
naturally turned to Athens to find them. The great library
which was the central feature of the scheme was placed under
Demetrius Phalereus, a' distinguished Athenian, and so rapidly
did it grow that within forty years it (together with the
Egyptian annexe) possessed about 600,000 rolls. The mathe-
matical department was placed under Euclid, who was thus the
first, as he was one of the most famous, of the mathematicians
of the Alexandrian school.
It happens that contemporaneously with the foundation
of this school the information on which our history is based
becomes more ample and certain. Many of the works of the
52 THE FIKST ALEXANDRIAN SCHOOL [ch. iv
Alexandrian mathematicians are still extant; and we have
besides an invaluable .treatise by Pappus, described below, in
which their best -known treatises are collated, discussed, and
criticized. It curiously turns out that just as we begin to be
able to speak with confidence on the subject-matter which was
taught, we find that our information as to the personality of
the teachers becomes vague ; and we know very little of the
lives of the mathematicians mentioned in this and the next
chapter, even the dates at which they lived being frequently
in doubt.
The third century before Christ.
Euclid.^ — This century produced three of the greatest
mathematicians of antiquity, namely Euclid, Archimedes, and
ApoUonius. The earliest of these was Euclid. Of his life we
know next to nothing, save that he was of Greek descent, and
was born about 330 B.C. ; he died about 275 B.C. It would
appear that he was well acquainted with the Platonic geometry,
but he does not seem to have read Aristotle's works ; and these
facts are supposed to strengthen the tradition that he was
educated at Athens. Whatever may have been his previous
training and career, he proved a most successful teacher when
settled at Alexandria. He impressed his own individuality on
the teaching of the new university to such an extent that to
his successors and almost to his contemporaries the name
Euclid meant (as it does to us) the book or books he wrote,
and not the man himself. Some of the mediaeval writers went
so far as to deny his existence, and -with the ingenuity of
^ Besides Loria, book ii, chap, i ; Cantor, chaps, xii, xiii ; and Gow, pp.
72-86, 195-221 ; see the articles Eudeides by A. De Morgan in Smith's
Dictionary of Greek and Roman Biography, London, 1849 ; the article on
Irrational Quantity by A. De Morgan in the Penny Cyclopaedia, London,
1839 ; Litter argeschichtliche Studien iiber Euklid, by J. L. Heiberg, Leipzig,
1882 ; and above all Euclid's Elements, translated with an introduction and
commentary by T. L. Heath, 3 volumes, Cambridge, 1908. The latest
complete edition of all Euclid's works is that by J. L. Heiberg and H. Menge,
Leipzig, 1883-96.
CH. iv] EUCLID 53
philologists they explained that the term was only a corruption
of vkXl a key, and St? geometry. The former word was presum-
ably derived from kXcls. I can only explain the meaning
assigned to St? by the conjecture that as the Pythagoreans
said that the number two symbolized a line, possibly a school-
man may have thought that it could be taken as indicative of
geometry.
From the meagre notices of Euclid which have come down
to us we find that the saying that there is no royal road in
geometry was attributed to Euclid as well as to Menaechmus ;
but it is an epigrammatic remark which has had many imitators.
According to tradition, Euclid was noticeable for his gentleness
and modesty. Of his teaching, an anecdote has been preserved.
Stobaeus, who is a somewhai doubtful authority, tells us that,
when a lad who had just begun geometry asked, " What do I
gain by learning all this stuff 1 " Euclid insisted that knowledge
was worth acquiring for its own sake, but made his slave give
the boy some coppers, "since," said he, "he must make a profit
out of what he learns."
Euclid was \h.e author of several works, but his reputation
rests mainly on his Elements. This treatise contains a systematic
exposition of the leading propositions of elementary metrical
geometry (exclusive of conic sections) and of the theory of
numbers. It was at once adopted by the Greeks as the standard
text-book on the elements of pure mathematics, and it is probable
that it was written for that purpose and not as a philosophical
attempt to shew that the results of geometry and arithmetic are
necessary truths.
The modern text^ is founded on an edition or commentary
prepared by Theon, the father of Hypatia (circ. 380 a.d.).
There is at the Vatican a copy (circ. 1000 a.d.) of an older text,
and we have besides quotations from the work and references to
it by numerous writers of various dates. From these sources we
^ Most of the modern text-books in English are founded on Simson's
edition, issued in 1758. Robert 'Simson, who was born in 1687 and died in
1768, was professor of mathematics at the University of Glasgow, and left
several valuable works on ancient geometry.
54 THE FIRST ALEXANDRIAN SCHOOL [ch. tv
gather that the definitions, axioms, and postulates were re-
arranged and slightly altered by subsequent editors, but that
the propositions themselves are substantially as Euclid wrote
them.
As to the matter of the work. The geometrical part is to a
large extent a compilation from the works of previous writers.
Thus the substance of books i and ii (except perhaps the treat-
ment of parallels) is probably due to Pythagoras ; that of book
III to Hippocrates ; that of book v to Eudoxus ; and the bulk of
books IV, VI, XI, and xii to the later Pythagorean or Athenian
schools. But this material was rearranged, obvious deductions
were omitted (for instance, the proposition that the perpendiculars
from the angular points of a triangle on the opposite sides meet
in a point was cut out), and in some cases new proofs sub-
stituted. Book X, which deals with irrational magnitudes, may
be founded on the lost book of Theaetetus ; but probably much
of it is original, for Proclus says that while Euclid arranged
the propositions of Eudoxus he completed many of those of
Theaetetus. The whole was presented as a complete and
consistent body of theorems.
The form in which the propositions are presented, consisting
of enunciation, statement, construction, proof, and conclusion,
is due to Euclid : so also is the synthetical character of the
work, each proof being written out as a logically correct train of
reasoning but without any clue to the method by which it was
obtained.
The defects of Euclid's Elements as a text-book of geometry
have been often stated ; the most prominent are these, (i) The
definitions and axioms contain many assumptions which are not
obvious, and in particular the postulate or axiom about parallel
lines is not self-evident. ^ (ii) No explanation is given as to
the reason why the proofs take the form in which they are
presented, that is, the synthetical proof is given but not the
^ We know, from the researches of Lobatschewsky and Riemaun, that it is
incapable of proof.
CH. iv] EUCLID 55
analysis by which it was obtained, (iii) There is no attempt
made to generalize the results arrived at ; for instance, the idea
of an angle is never extended so as to cover the case where it is
equal to or greater than two right angles : the second half of
the thirty-third proposition in the sixth book, as now printed,
appears to be an exception, but it is due to Theon and not to
Euclid, (iv) The principle of superposition as a method of
proof might be used more frequently with advantage, (v) The
classification is imperfect. And (vi) the work is unnecessarily
long and verbose. Some of those objections do not apply to
certain of the recent school editions of the ElemenU.
On the other hand, the propositions in Euclid are arranged
so as to form a chain of geometrical reasoning, proceeding from
certain almost obvious assumptions by easy steps to results of
considerable complexity. The demonstrations are rigorous, often
elegant, and not too difficult for a beginner. Lastly, nearly all
the elementary metrical (as opposed to the graphical) properties
of space are investigated, while the fact that for two thousand
years it was the usual text-book on the subject raises a strong
presumption that it is not unsuitable for the purpose.
On the Continent rather more than a century ago, Euclid
was generally superseded by other text -books. In England
determined efforts have lately been made with the same purpose,
and numerous other works on elementary geometry have been
produced in the last decade. The change is too recent to enable
us to say definitely what its effect may be. But as far as I can
judge, boys who have learnt their geometry on the new system
know more facts, but have missed the mental and logical training
which was inseparable from a judicious study of Euclid's
treatise.
I do not think that all the objections above stated can fairly
be urged against Euclid himself. He published a collection of
problems, generally known as the AeSo/xeva or Data. This
contains 95 illustrations of the kind of deductions which
frequently have to be made in analysis ; such as that, if one
of the data of the problem under consideration be that one
56 THE FIRST ALEXANDRIAN SCHOOL [ch. iv
angle of some triangle in the figure is constant, then it is
legitimate to conclude that the ratio of the area of the rectangle
under the sides containing the angle to the area of the triangle
is known [prop. 66]. Pappus says that the work was written
for those " who wish to acquire the power of solving problems."
It is in fact a gradual series of exercises in geometrical analysis.
In short the Elements gave the principal results, and were
intended to serve as a training in the science of reasoning, while
the Data were intended to develop originality.
Euclid also wrote a work called Hept Aiatpto-ewv or De
Divisionihus, known to us only through an Arabic translation
which may be itself imperfect. This is a collection of 36
problems on the division of areas into parts which bear to one
another a given ratio. It is not unlikely that this was only
one of several such collections of examples — jDossibly including
the Fallacies and the Pm^isms — but even by itself it shews that
the value of exercises and riders was fully recognized by Euclid.
I may here add a suggestion made by De Morgan, whose
comments on Euclid's writings were notably ingenious and
informing. From internal evidence he thought it likely that
the Elements were written towards the close of Euclid's life, and
that their present form represents the first draft of the proposed
work, which, with the exception of the tenth book, Euclid did
not live to revise. This opinion is generally discredited, and
there is no extrinsic evidence to support it.
The geometrical parts of the Elements are so well known
that I need do no more than allude to them. Euclid admitted
only those cofistructions which could be made by the use of a
ruler and compasses. ^ He also excluded practical work and
^ The ruler must be of imlimited length aud not graduated ; the compasses
also must be capable of being opened as wide as is desired. Lwenzo Mas-
cheroni (who was bora at Castagneta on May 14, 1750, and died at Paris
on July 30, 1800) set himself the task to obtain by means of constructions
made only with a pair of compasses the same results as Euclid had given.
Mascheroni's treatise on the geometry of the compass, which was published
at Pavia in 1795, is a curious to\ir de force : he was professor first at
Bergamo and afterwards at Pavia, and left numerous minor works. Similar
limitations have been proposed by other writers.
CH. iv] EUCLID 57
hypothetical constructions. The first four books and book
VI deal with plane geometry ; the theory of proportion (of
any magnitudes) is discussed in book v ; and books xi and
XII treat of solid geometry. On the hypothesis that the
Elements are the first draft of Euclid's proposed work, it is
possible that book xiii is a sort of appendix containing some
additional propositions which would have been put ultimately
in one or other of the earlier books. Thus, as mentioned
above, the first five propositions which deal with a line cut
in golden section might be added to the second book. The
next seven propositions are concerned wdth the relations be-
tween certain incommensurable lines in plane figures (such as
the radius of a circle and the sides of an inscribed regular
triangle, pentagon, hexagon, and decagon) which are treated by
the methods of the tenth book and as an illustration of them.
Constructions of the five regular solids are discussed in the last
six propositions, and it seems probable that Euclid and his
contemporaries attached great importance to this group of
problems. Bretschneider inclined to think that the thirteenth
book is a summary of part of the lost work of Aristaeus : but
the illustrations of the methods of the tenth book are due most
probably to Theaetetus.
Books VII, VIII, IX, and x of the Elements are given up
to the theory of numbers. The mere art of calculation or
XoyiGrTLKTi was taught to boys when quite young, it was stig-
matized by Plato as childish, and never received much atten-
tion from Greek mathematicians ; nor was it regarded as
forming part of a course of mathematics. We do not know
how it was taught, but the abacus certainly played a prominent
part in it. The scientific treatment of numbers was called
dpiOfxYjTiKyj, which I have here generally translated as the
science of numbers. It had special reference to ratio, pro-
portion, and the theory of numbers. It is with this alone that
most of the extant Greek works deal.
In discussing Euclid's arrangement of the subject, we must
therefore bear in mind that those who attended his lectures
58 THE FIRST ALEXANDRIAN SCHOOL [ch. iv
were already familiar with the art of calculation. The system
of numeration adopted by the Greeks is described later/ but
it was so clumsy that it rendered the scientific treatment of
numbers much more difficult than that of geometry; hence
Euclid commenced his mathematical course with plane geometry.
At the same time it must be observed that the results of the
second book, though geometrical in form, are capable of
expression in algebraical language, and the fact that numbers
could be represented by lines was probably insisted on at an
early stage, and illustrated by concrete examples. This
graphical method of using lines to represent numbers possesses
the obvious advantage of leading to proofs which are true for
all numbers, rational or irrational. It will be noticed that
among other propositions in the second book we get geometrical
proofs of the distributive and commutative laws, of rules for
multiplication, and finally geometrical solutions of the equations
a{a -x) = x^, that h x^-{-ax-a^ = 0 (Euc. ii, 11), and a;^ -ah = 0
(Euc. II, 14) : the solution of the first of these equations is
given in the form Ja^ + {\(if - \a. The solutions of the
equations ax^ -hx + c — O and ax^ -{-hx -c = Q are given later in
Euc. VI, 28 and vi, 29 ; the cases when a = 1 can be deduced
from the identities proved in Euc. ii, 5 and 6, but it is doubtful
if Euclid recognized this.
The results of the fifth book, in which the theory of propor-
tion is considered, apply to any magnitudes, and therefore are
true of numbers as well as of geometrical magnitudes. In the
opinion of many writers this is the most satisfactory way of
treating the theory of proportion on a scientific basis ; and it
was used by Euclid as the foundation on which he built the
theory of numbers. The theory of proportion given in this
book is believed to be due to Eudoxus. The treatment of the
same subject in the seventh book is less elegant, and is supposed
to be a reproduction of the Pythagorean teaching. This double
discussion of proportion is, as far as it goes, in favour of the
conjecture that Euclid did not live to revise the work.
^ See below, chapter vii.
CH.iv] EUCLID 59
In books VII, viii, and ix Euclid discusses the theory of
rational numbers. He commences the seventh book with some
definitions founded on the Pythagorean notation. In proposi-
tions 1 to 3 he shews that if, in the usual process for finding
the greatest common measure of two numbers, the last divisor
be unity, the numbers must be prime ; and he thence deduces
the rule for finding their g.c.m. Propositions 4 to 22 include
the theory of fractions, which he bases on the theory of pro-
portion; among other results he shews that ab = ba [prop. 16].
In propositions 23 to 34 he treats of prime numbers, giving
many of the theorems in modern text-books on algebra. In
propositions 35 to 41 he discusses the least common multiple of
numbers, and some miscellaneous problems.
The eighth book is chiefly devoted to numbers in continued
proportion, that is, in a geometrical progression ; and the cases
where one or more is a product, square, or cube are specially
considered.
In the ninth book Euclid continues the discussion of geo-
metrical progressions, and in proposition 35 he enunciates the
rule for the summation of a series of n terms, though the proof
is given only for the case where n is equal to 4. He also
develops the theory of primes, shews that the number of primes
is infinite [prop. 20], and discusses the properties of odd and
even numbers. He concludes by shewing that a number of the
form 2^-1(2^-1), where 2" - 1 is a prime, is a "perfect"
number [prop. 36]. t-. , :v ^ , y^
In the tenth book Euclid deals with certain irrational
magnitudes ; and, since the Greeks possessed no symbolism for
surds, he was forced to adopt a geometrical representation.
Propositions 1 to 21 deal generally with incommensurable
magnitudes. The rest of the book, namely, propositions 22 to
117, is devoted to the discussion of every possible variety of
lines which can be represented by J{ Ja± Jb), where a and b
denote commensurable lines. There are twenty-five species of
such lines, and that Euclid could detect and classify them all
is in the opinion of so competent an authority as Nesselmann
60 THE FIRST ALEXANDRIAN SCHOOL [ch. iv
the most striking illustration of his genius. No further advance
in the theory of incommensurable magnitudes was made until
the subject was taken up by Leonardo and Cardan after an
interval of more than a thousand years.
In the last proposition of the tenth book [prop. 117] the
side and diagonal of a square are proved to be incommensurable.
The proof is so short and easy that I may quote it. If possible
let the side be to the diagonal in a commensurable ratio,
namely, that of two integers, a and h. Suppose this ratio
reduced to its lowest terms so that a and h have no common
divisor other than unity, that is, they are prime to one another.
Then (by Euc. i, 47) h^ = ^a^ ; therefore IP- is an even number ;
therefore h is an even number ; hence, since a is prime to i, a
must be an odd number. Again, since it has been shewn that
h is an even number, h may be represented by In; therefore
i^nf = 2a2 ; therefore a?- = %i^ ; therefore a?- is an even number ;
therefore a is an even number. Thus the same number a must
be both odd and even, which is absurd ; therefore the side and
diagonal are incommensurable, Hankel believes that this proof
was due to Pythagoras, and this is not unlikely. This proposi-
tion is also proved in another way in Euc. x, 9, and for this
and other reasons it is now usually believed to be an interpola-
tion by some commentator on the Elements.
In addition to the Elements and the two collections of riders
above mentioned (which are extant) Euclid wrote the following
books on geometry : (i) an elementary treatise on conic sections
in four books ; (ii) a book on surface loci, probably confined
to curves on the cone and cylinder ; (iii) a collection of geo-
metrical fallacies, which were to be used as exercises in the
detection of errors ; and (iv) a treatise on porisms arranged in
three books. All of these are lost, but the work on porisms
was discussed at such length by Pappus, that some writers
have thought it possible to restore it. In particular, Chasles
in 1860 published what he considered to be substantially a re-
production of it. In this will be found the conceptions of cross
ratios and projection, and those ideas of modern geometry which
CH. IV] EUCLID 61
were used so extensively by Chasles and other writers of the
nineteenth century. It should be realized, however, that the
statements of the classical writers concerning this book are
either very brief or have come to us only in a mutilated
form, and De Morgan frankly says that he found them un-
intelligible, an opinion in which most of those who read them
will, I think, concur.
Euclid published a book on optics, treated geometrically,
which contains 61 propositions founded on 12 assumptions. It
commences with the assumption that objects are seen by rays
emitted from the eye in straight lines, " for if light proceeded
from the object we should not, as we often do, fail to perceive a
needle on the floor." A work called Catoptrica is also attributed
to him by some of the older writers ; the text is corrupt and the
authorship doubtful ; it consists of 31 propositions dealing with
reflexions in plane, convex, and concave mirrors. The geometry
of both books is Euclidean in form.
Euclid has been credited with an ingenious demonstration ^
of the principle of the lever, but its authenticity is doubtful.
He also wrote the Pfmenomena, a treatise on geometrical astro-
nomy. It contains references to the work of Autolycus ^ and to
some book on spherical geometry by an unknown writer. Pappus
asserts that Euclid also composed a book on the elements of
music : this may refer to the Sectio Canonis, which is by Euclid,
and deals with musical intervals.
To these works I may add the following little problem, which
occurs in the Palatine Anthology and is attributed by tradition
to Euclid. " A mule and a donkey were going to market laden
with wheat. The mule said, 'If you gave me one measure I
should carry twice as much as you, but if I gave you one we
^ It is given (from the Arabic) by F. Woepcke in the Journal Asiatique,
series 4, vol. xviii, October 1851, pp. 225-232.
2 Autolycus lived at Pitane in Aeolis and flonrished about 330 B.C. His
two works on astronomy, containing 43 propositions, are said to be the oldest
extant Greek mathematical treatises. They exist in manuscript at Oxford.
They were edited, with a Latin translation, by F. Hultsch, Leipzig, 1885.
62 THE FIRST ALEXANDRIAN SCHOOL [ch. iv
should bear equal burdens.' Tell me, learned geometrician,
what were their burdens." It is impossible to say whether the
question is due to Euclid, but there is nothing improbable in
the suggestion.
It will be noticed that Euclid dealt only with magnitudes,
and did not concern himself with their numerical measures, but
it would seem from the works of Aristarchus and Archimedes
that this was not the case with all the Greek mathematicians
of that time. As one of the works of the former is extant it
will serve as another illustration of Greek mathematics of this
period.
Aristarchus. Aristarchus of Samos, born in 310 B.C. and
died in 250 B.C., was an astronomer rather than a mathematician.
He asserted, at any rate as a working hypothesis, that the sun
was the centre of the universe, and that the earth revolved
round the sun. This view, in spite of the simple explanation
it ajfforded of various phenomena, was generally rejected by his
contemporaries. But his propositions ^ on the measurement of
the sizes and distances of the sun and moon were accurate in
principle, and his results were accepted by Archimedes in his
'^a/j.fjLLTrjs, mentioned below, as approximately correct. There
are 19 theorems, of which I select the seventh as a typical
illustration, because it shews the way in which the Greeks
evaded the difficulty of finding the numerical value of surds.
Aristarchus observed the angular distance between the moon
when dichotomized and the sun, and found it to be twenty-nine
thirtieths of a right angle. It is actually about 89° 21', but of
course his instruments were of the roughest description. He
then proceeded to shew that the distance of the sun is greater
than eighteen and less than twenty times the distance of the
moon in the following manner.
Let S be the sun, £J the earth, and 31 the moon. Then when
^ Ilepl fieyeduv Kal dwoaTrjfxdTwu 'HXiov Kai "ZeK-qvqs, edited by E. Nizze,
Stralsund, 1856. Latin translations were issued by F. Commandino in 1572
and by J. Wallis in 1688 ; and a French translation was published by
F. d'Urban in 1810 and 1823.
ARISTARCHUS
63
CH. iv]
the moon is dichotomized, that is, when the bright part which
we see is exactly a half-circle, the angle between 3IS and MB is
a right angle. With E as centre, and radii ES and B3f describe
circles, as in the figure above. Draw FA perpendicular to ES.
Draw EF bisecting the angle AES, and EG bisecting the angle
AFF, as in the figure. Let F3f (produced) cut ^i^ in //.
The angle A EM is by hypothesis ^\,-th of a right angle. Hence
we have
angle AEG : angle AEH= i rt. z. : J^ rt. z.= 15 : 2,
.'. AG : AH [ = tan AEG : tan AEH] > 15 : 2 (a).
Again FG'^ :AG^ = Er-. FA'- (Euc. vi, 3) = 2 : 1 (Euc. i, 47),
.-. FG-' :AG^ > 49 : 25,
.-. EG :AG^7:d,
.-. AF : AG :=- 12 : 5,
.-. AF .AG > 12 : 5 (/?).
Compounding the ratios (a) and (/?), we have
AF '.AH >\^ : 1.
But the triangles EMS and FAH are similar,
.-. ^^S': EM ^ 18 : 1.
64 THE FIKST ALEXANDRIAN SCHOOL [ch. iv
I will leave the second half of the proposition to amuse any
reader who may care to prove it : the analysis is straightforward.
In a somewhat similar way Aristarchus found the ratio of the
radii of the sun, earth, and moon.
We know very little of Conon and Dositheus, the immediate
successors of Euclid at Alexandria, or of their contemporaries
Zeuxippus and Nicoteles, who most likely also lectured there,
except that Archimedes, who was a student at Alexandria
probably shortly after Euclid's death, had a high opinion of
their ability and corresponded with the three first mentioned.
Their work and reputation has been completely overshadowed
by that of Archimedes.
Archimedes.^ Archimedes^ who probably was related to
the royal family at Syracuse, was born there in 287 B.C. and
died in 212 B.C. He went to the university of Alexandria
and attended the lectures of Conon, but, as soon as he had
finished his studies, returned to Sicily where he passed the
remainder of his life. He took no part in public affairs, but
his mechanical ingenuity was astonishing, and, on any difficulties
which could be overcome by material means arising^ his advice
was generally asked by the government.
Archimedes, like Plato, held that it was undesirable for a
philosopher to seek to apply the results of science to any prac-
tical use ; but in fact he did introduce a large number of new
inventions. The stories of the detection of the fraudulent
goldsmith and of the use of burning-glasses to destroy the
ships of the Roman blockading squadron will recur to most
readers. Perhaps it is not as well known that Hiero, who had
built a ship so large that he could not launch it off the sli^DS,
^ Besides Loria, book ii, chap, iii, Cantor, chaps, xiv, xv, and Gow,
pp. 221-244, see Quaestiones Archimedeae, by J. L. Heiberg, Copenhagen,
1879 ; and Marie, vol, i, pp. 81-134. The best editions of the extant works
of Archimedes are those by J. L. Heiberg, in 3 vols., Leipzig, 1880-81,
and by Sir Thomas L. Heath, Cambridge, 1897. In 1906 a manuscript,
previously unknown, was discovered at Constantinople, containing proposi-
tions on hydrostatics and on methods ; see Eine neue Schrift des Archimedes,
by J. L. Heiberg and H. G. Zeuthen, Leipzig, 1907, and the Method of
Archimedes, by Sir Thomas L. Heath, Cambridge, 1912.
">
CH.iv] ARCHIMEDES 65
applied to Archimedes. The difficulty was overcome by means
of an apparatus of cogwheels worked by an endless screw, but
we are not told exactly how the machine was used. It is said
that it was on this occasion, in acknowledging the compliments
of Hiero, that Archimedes made the well-known remark that had
he but a fixed fulcrum he could move the earth. ^^
Most mathematicians are aware that the Archhiiedean screw
was another of his inventions. It consists of a tube, open at
both ends, and bent into the form of a spiral like a corkscrew.
If one end be immersed in water, and the axis of the instrument
{i.e. the axis of the cylinder on the surface of which the tube
lies) be inclined to the vertical at a sufficiently big angle, and
the instrument turned round it, the water will flow along the
tube and out at the other end. In order that it may work, the
inclination of the axis of the instrument to the vertical must
be greater than the pitch of the screw. It was used in Egypt
to drain the fields after an inundation of the Nile, and was
also frequently applied to take water out of the hold of a
ship.
The story that Archimedes set fire to the Roman ships by
means of burning-glasses and concave mirrors is not mentioned
till some centuries after his death, and is generally rejected.
The mirror of Archimedes is said to have been made in the
form of a hexagon surrounded by rings of polygons ; and Buff'on ^
in 1747 contrived, by the use of a single composite mirror made
on this model, to set fire to wood at a distance of 150 feet,
and to melt lead at a distance of 140 feet. This was in April
and as far north as Paris, so in a Sicilian summer the use
of several such mirrors might be a serious annoyance to a
blockading fleet, if the ships were sufficiently near. It is
perhaps worth mentioning that a similar device is said to have
been used in the defence of Constantinople in 514 a.d., and is
alluded to by writers who either were present at the siege or
obtained their information from those who were engaged in it.
^ See Memoires de Vacademie royale des sciences for 1747, Paris, 1752,
pp. 82-101.
66 THE FIRST ALEXANDRIAN SCHOOL [ch. iv
But whatever be the truth as to this story, there is no doubt
that Archimedes devised the catapults which kept the Romans,
who were then besieging Syracuse, at bay for a considerable
time. These were constructed so that the range could be made
either short or long at pleasure, and so that they could be dis-
charged through a small loophole without exposing the artillery-
men to the fire of the enemy. So effective did they prove that
the siege was turned into a blockade, and three years elapsed
before the town was taken.
Archimedes was killed during the sack of the city which
followed its capture, in spite of the orders, given by the consul
Marcellus who was in command of the Romans, that his house
and life should be spared. It is said that a soldier entered his
study while he was regarding a geometrical diagram drawn in
sand on the floor, which was the usual way of drawing figures
in classical times. Archimedes told him to get off the diagram,
and not spoil it. The soldier, feeling insulted at having orders
given to him and ignorant of who the old man was, killed him.
According to another and more probable account, the cupidity
of the troops was excited by seeing his instruments, constructed
of polished brass which they supposed to be made of gold.
The Romans erected a splendid tomb to Archimedes, on which
was engraved (in accordance with a wish he had expressed) the
figure of a sphere inscribed in a cylinder, in commemoration of
the proof he had given that the volume of a sphere was equal
to two-thirds that of the circumscribing right cyjinder, and its
surface to four times the area of a great circle. Cicero ^ gives
a charming account of his efforts (which were successful) to
rediscover the tomb in 75 B.C.
It is difficult to explain in a concise form the works or dis-
coveries of Archimedes, partly because he wrote on nearly all
the mathematical subjects then known, "and partly because his
writings are contained in a series of disconnected monographs.
Thus, while Euclid aimed at producing systematic treatises
which could be understood by all students who had attained
^ See his Tusciilanarum Disjoutationum, v. 23.
CH. iv] ARCHIMEDES 67
a certain level of education, Archimedes ^3¥mt£_a number of
brilliant ^xssays addressed chiefly to the most educated mathe-
jmaticians of the day,-. The work for which he is perhaps now
best known is his treatment of the mechanics of solids and
fluids ; but he and his contemporaries esteemed his geometrical
discoveries of the quadrature of a parabolic area and of a
spherical surface, and his rule for finding the volume of a sphere
as more remarkable ; while at a somewhat later time his numerous
mechanical inventions excited most attention.
(i) On X)lane geometry the extant works of Archimedes are
three in number, namely, (a) the Measure of the Circle, {b) the
Quadrature of the Parabola, and (c) one on Spirals.
(a) The Measure of the Circle contains three propositions.
In the first proposition Archimedes proves that the area is the
same as that of a right-angled triangle whose sides are equal
respectively to the radius a and the circumference of the circle,
i.e. the area is equal to ^a (27ra). In the second proposition
he shows that 7ra^ : (2rt)2=ll : 14 very nearly; and next, in
the third proposition, that tt is less than 3y and greater than
3yf . These theorems are of course proved geometrically. To
demonstrate the two latter propositions, he inscribes in and
circumscribes about a circle regular polygons of ninety - six
sides, calculates their perimeters, and then assumes the circum-
ference of the circle to lie between them : this leads to the
result 6336 / 2017J < 7r< 14688 / 4673J, from which he deduces
the limits given above. It would seem from the proof that he
had some (at present unknown) method of extracting the square
roots of numbers approximately. The table which he formed
of the numerical values of the chords of a circle is essentially a
table of natural sines, and may have suggested the subsequent
work on these lines of Hipparchus and Ptolemy.
(b) The Quadrature of the Parabola contains twenty -four
propositions. Archimedes begins this work, which was sent to
Dositheus, by establishing some properties of conies [props. 1-5].
He then states correctly the area cut off from a parabola by any
chord, and gives a proof which rests on a preliminary mechanical
68 THE FIRST ALEXANDRIAN SCHOOL [ch. iv
experiment of the ratio of areas which balance when suspended
from the arms of a lever [props. 6-17]; and, lastly, he gives a
geometrical demonstration of this result [props. 18-24]. The
latter is, of course, based on the method of exhaustions, but for
brevity I will, in quoting it, use the method of limits.
Let the area of the parabola (see figure below) be bounded-
by the chord PQ. Draw VM the diameter to the chord
PQ, then (by a previous proposition), V is more remote from
P
PQ than any other point in the arc PVQ. Let the area of
the triangle P VQ be denoted by A. In the segments bounded
by VP and VQ inscribe triangles in the same way as the triangle
PVQ was inscribed in the given segment. Each of these tri-
angles is (by a previous proposition of his) equal to ^A, and
their sum is therefore J A. Similarly in the four segments left
inscribe triangles ; their sum will be y\A. Proceeding in this
way the area of the given segment is shown to be equal to the
limit of
A A A
^ + 4+i6+-+r«+-'
when n is indefinitely large.
CH. iv] ARCHIMEDES 69
The problem is therefore reduced to finding the sum of a
geometrical series. This he effects as follows. Let A, £, C,
..., J, Khe a, series of magnitudes such that each is one-fourth
of that which precedes it. Take magnitudes h, c, ..., k equal
resj^ectively to ^B, JC, ..., JA'. Then
Hence (^ + (7+... + /0 + (^ + c+...+^-) = l (A + B+... +J);
but, by hypothesis, {/) + c+ ...+j + k) = \{B + C +...+-'') + \K ;
„.. {B-\-C+...^K) + lK=\A.
.-. A + B+C-^ ...+K = ^.A-IK.
Hence the sum of these magnitudes exceeds four times the third
of the largest of them by one-third of the smallest of them.
Returning now to the problem of the quadrature of the
parabola A stands for A, and ultimately K is indefinitely
small ; therefore the area of the parabolic segment is four-thirds
that of the triangle PVQ, or two -thirds that of a rectangle
whose base is FQ and altitude the distance of V from PQ.
While discussing the question of quadratures it may be
added that in the fifth and sixth propositions of his work on
conoids and spheroids he determined the area of an ellipse.
(c) The work on Sjn^^als contains twenty-eight propositions
on the properties of the curve now known as the spiral of
Archimedes. It was sent to Dositheus at Alexandria accom-
panied by a letter, from which it appears that Archimedes had
previously sent a note of his results to Conon, who had died
before he had been able to prove them. The spiral is defined by
saying that the vectorial angle and radius vector both increase
uniformly, hence its equation is r = cd. Archimedes finds most
of its properties, and determines the area inclosed between the
curve and two radii vectores. This he does (in effect) by
saying, in the language of the infinitesimal calculus, that an
element of area is >^r^dd and <J(r + <ir)W: to effect the
sum of the elementary areas he gives two lemmas in which he
70 THE FIRST ALEXANDRIAN SCHOOL [ch. iv
sums (geometrically) the series a^ + {^af + (3a)2 + . • • + {naf
[prop. 10], and a + 2a + 3a + . . . + na [prop. 11].
{d) In addition to these he WTote a small treatise on
geometrical methods, and works on ^j)(X?'aZZe^ lines, triangles, the
iwojperties of right-angled triangles, data, the heptagon inscribed
in a circle, and systems of circles touching one another ; possibly
he wrote others too. These are all lost, but it is probable that
fragments of four of the propositions in the last-mentioned work
are preserved in a Latin translation from an Arabic manuscript
entitled Lemmas of Archimedes.
(ii) On geometry of three dimensions the extant works of
Archimedes are two in number, namely {a), the Sphere and
Cylinder, and (5) Conoids and Spheroids.
(a) The Sphere and Cylinder contains sixty propositions
arranged in two books. Archimedes sent this like so many
of his works to Dositheus at Alexandria; but he seems to
have played a practical joke on his friends there, for he pur-
posely misstated some of his results " to deceive those vain
geometricians who say they have found everything, but never
give their proofs, and sometimes claim that they have dis-
covered what is impossible." He regarded this work as his
masterpiece. It is too long for me to give an analysis of its
contents, but I remark in passing that in it he finds expressions
for the surface and volume of a pyramid, of a cone, and of
a sphere, as well as of the figures produced by the revolution
of polygons inscribed in a circle about a diameter of the circle.
There are several other propositions on areas and volumes of I
which perhaps the most striking is the tenth proposition of
the second book, namely, that " of all spherical segments whose
surfaces are equal the hemisphere has the greatest volume."
In the second proposition of the second book he enunciates the =
remarkable theorem that a line of length a can be divided ]
so that a-x : h — ia^ : 9^^ (where 6 is a given length), only
if b be less than Ja; that is to say, the cubic equation
oc^ - ax^ + ~a^ = 0 can have a real and positive root only if
a be greater than 3b. This proposition was required to com-
CH.iv] ARCHIMEDES 71
plete his solution of the problem to divide a given sphere by
a plane so that the volumes of the segments should be in a given
ratio. One very simple cubic equation occurs in the Arithmetic
of Diophantus, but with that exception no such equation appears
again in the history of European mathematics for more than a
thousand years.
(h) The Conoids and Spheroids contains forty propositions
on quadrics of revolution (sent to Dositheus in Alexandria)
mostly concerned with an investigation of their volumes.
(c) Archimedes also wrote a treatise on certain semi-regular
2MlyhedronSy that is, solids contained by regular but dissimilar
polygons. This is lost, but references to it are given by
Pappus.
(iii) On arithmetic Archimedes wrote two papers. One
(addressed to Zeuxippus) was on the principles of numeration ;
this is now lost. The other (addressed to Gelon) was called
■^a/x/xtTT^s {the sand-rechoner), and in this he meets an objection
which had been urged against his first paper.
The object of the first paper had been to suggest a con-
venient system by which numbers of any magnitude could
be represented ; and it would seem that some philosophers at
Syracuse had doubted whether the system was practicable.
Archimedes says people talk of the sand on the Sicilian shore
as something beyond the power of calculation, but he can
estimate it; and, further, he will illustrate the power of his
method by finding a superior limit to the number of grains of
sand which would fill the whole universe, i.e. a sphere whose
centre is the earth, and radius the distance of the sun. He
begins by saying that in ordinary Greek nomenclature it was
only possible to express numbers from 1 up to 10^ : these are
expressed in what he says he may call units of the first order.
If 10^ be termed a unit of the second order, any number from
10^ to 10^^ 'can be expressed as so many units of the second
order plus so many units of the first order. If 10^^ be a unit
of the third order any number up to 10^"* can be then expressed,
and so on. Assuming that 10,000 grains of sand occupy a
72 THE FIRST ALEXANDRIAN SCHOOL [ch. iv
sphere whose radius is not less than ^th of a finger-breadth,
and that the diameter of the universe is not greater than 10^*^
stadia, he finds that the number of grains of sand required to
fill the solar universe is less than 10^^.
Probably this system of numeration was suggested merely
as a scientific curiosity. The Greek system of numeration
with which we are acquainted had been only recently intro-
duced, most likely at Alexandria, and was sufficient for all the
purposes for which the Greeks then required numbers ; and
Archimedes used that system in all his papers. On the other
hand, it has been conjectured that Archimedes and Apollonius
had some symbolism based on the decimal system for their
own investigations, and it is possible that it was the one here
sketched out. The units suggested by Archimedes form a
geometrical progression, having 10^ for the radix. He inci-
dentally adds that it will be convenient to remember that the
product of the mth and nth. terms of a geometrical progression,
. whose first term is unity, is equal to the (m + ?i)th term of the
series, that is, that r'^ x r'^ = r'"+'\
To these two arithmetical papers I may add the following-
celebrated problem ^ which he sent to the Alexandrian mathe-
maticians. The sun had a herd of bulls and cows, all of
which were either white, grey, dun, or piebald : the number
of piebald bulls was less than the number of white bulls by
5/6ths of the number of grey bulls, it was less than the
number of grey bulls by 9/20ths of the number of dun bulls,
and it was less than the number of dun bulls by 13/42nds
of the number of white bulls ; the number of white cows was
7/12ths of the number of grey cattle (bulls and cows), the
number of grey cows was 9/20ths of the number of dun
cattle, the number of dun cows was ll/30ths of the number of
piebald cattle, and the number of piebald cows was 13/42nds
of the number of white cattle. The problem was to find the
^ See a memoir by B. Krumbiegel and A. Amthor, Zeitschrift fiir Mathe-
matik, Ahhandlungen zur Geschichte der Mathematik, Leipzig, vol. xxv, 1880,
pp. 121-136, 153-171.
CH. iv] ARCHIMEDES 73
composition of the herd. The problem is indeterminate, but
the solution in lowest integers is
white bulls, 10,366,482; white cows, 7,206,360;
grey bulls, 7,460,514; grey cows, 4,893,246;
dun bulls, 7,358,060; dun cows, 3,515,820;
piebald bulls, 4,149,387; piebald cows, 5,439,213.
In the classical solution, attributed to Archimedes, these num-
bers are multiplied by 80.
Nesselmann believes, from internal evidence, that the prob-
lem has been falsely attributed to Archimedes. It certainly
is unlike his extant work, but it was attributed to him among
the ancients, and is generally thought to be genuine, though
possibly it has come down to us in a modified form. It is
in verse, and a later copyist has added the additional con-
ditions that the sum of the white and grey bulls shall be a;^
square number, and the sum of the piebald and dun bulls a
triangular number.
It is perhaps worthy of note that in the enunciation the
fractions are represented as a sum of fractions whose numera-
tors are unity: thus Archimedes wrote 1/7 + 1/6 instead of
13/42, in the same way as Ahmes w^ould have done.
(iv) On mechanics the extant works of Archimedes are
two in number, namely, (a) his Mechanics, and (c) his Hydro-
statics.
{a) The Mechanics is a work on statics with special refer-
ence to the equilibrium of plane laminas and to properties of
their centres of gravity ; it consists of twenty-five propositions
in two books. In the first part of book i, most of the ele-
mentary properties of the centre of gravity are proved [props.
1-8]; and in the remainder of book i, [props. 9-15] and in
book II the centres of gravity of a variety of plane areas, such
as parallelograms, triangles, trapeziums, and parabolic areas
are determined.
As an illustration of the influence of Archimedes on the
history of mathematics, I may mention that the science of
L
74 THE FIRST ALEXANDRIAN SClTOOL [ch. iv
statics rested on his theory of the lever until 1586, when
Stevinus published his treatise on statics.
His reasoning is sufficiently illustrated by an outline of his
proof for the case of two weights, P and Q, placed at their centres
of gravity, A and B, on a weightless bar AB. He wants to shew
that the centre of gravity of P and Q is at a point 0 on the bar
such that P.O^ = (2.0J5.
On the line AB (produced if necessary) take points H and /i,
sothsitIIB = BK=AO; and a point L so that LA = OB. It
follows that LH will be bisected at A, UK at B, and LK at 0 ;
L IT K
A OB
also LH'.UK=^AH:HB=OB.AO = P:Q. Hence, by a
previous proposition, we may consider that the effect of P is the
same as that of a heavy uniform bar LH of weight P, and the
effect of Q is the same as that of a similar heavy uniform bar
HK of weight Q. Hence the effect of the weights is the same
as that of a heavy uniform bar LK. But the centre of gravity
of such a bar is at its middle point O.
{b) Archimedes also wrote a treatise on levers and perhaps,
on all the mechanical machines. The book is lost, but we
know from Pappus that it contained a discussion of how a
given weight could be moved with a given power. It was in
this work probably that Archimedes discussed the theory of
a certain compound pulley consisting of three or more simple
pulleys which he had invented, and which was used in some
public works in Syracuse. It is well known ^ that he boasted
that, if he had but a fixed fulcrum, he could move the whole
earth ; and a commentator of later date says that he added
he would do it by using a compound pulley.
(c) His work on floating bodies contains nineteen propositions
in two books, and was the first attempt to apply mathematical
reasoning to hydrostatics. The story of the manner in which
^ See above, p. 65.
CH. iv] ARCHIMEDES 75
his attention was directed to the subject is told by Vitruvius.
Hiero, the king of Syracuse, had given some gold to a goldsmith
to make into a crown. The crown was delivered, made up, and
of the proper weight, but it was suspected that the workman
had appropriated some of the gold, replacing it by an equal
weight of silver. Archimedes was thereupon consulted. Shortly
afterwards, when in the public baths, he noticed that his body
was pressed upwards by a force which increased the more com-
pletely he was immersed in the water. Recognising the value
of the observation, he rushed out, just as he was, and ran home
through the streets, shouting evpyjKa, evpi^Ka, " I have found it, I
have found it." There (to follow a later account) on making
accurate experiments he found that when equal weights of gold
and silver were weighed in water they no longer appeared equal :
each seemed lighter than before by the weight of the water it
displaced, and as the silver was more bulky than the gold its
weight was more diminished. Hence, if on a balance he weighed
the crown against an equal weight of gold and then immersed
the whole in water, the gold would outweigh the crown if any
silver had been used in its construction. Tradition says that
the goldsmith was found to be fraudulent.
Archimedes began the work by proving that the surface of
a fluid at rest is spherical, the centre of the sphere being at the
centre of the earth. He then proved that the pressure of the
fluid on a body, wholly or partially immersed, is equal to the
weight of the fluid displaced; and thence found the position
of equilibrium of a floating body, which he illustrated by
spherical segments and paraboloids of revolution floating on a
fluid. Some of the latter problems involve geometrical reason-
ing of considerable complexity.
The following is a fair specimen of the questions considered.
A solid in the shape of a paraboloid of revolution of height h
and latus rectum 4a floats in water, with its vertex immersed
and its base wholly above the surface. If equilibrium be
possible when the axis is not vertical, then the density of the
body must be less than {h - Sa^/h^ [book ii, prop. 4]. When
76 THE FIRST ALEXANDRIAN SCHOOL [ch. iv
it is recollected that Archimedes was unacquainted with trigono-
metry or analytical geometry, the fact that he could discover
and prove a proposition such as that just quoted will serve as an
illustration of his powers of analysis.
It will be noticed that the mechanical investigations of
Archimedes were concerned with statics. It may be added that
though the Greeks attacked a few problems in dynamics, they
did it with but indifferent success : some of their remarks were
acute, but they did not sufficiently realise that the fundamental
facts on which the theory must be based can be established only
by carefully devised observations and experiments. It was not
until the time of Galileo and Newton that this was done.
(v) We know, both from occasional references in his works
and from remarks by other writers, that Archimedes was largely
occupied in astronomical observations. He wrote a book, He/at
2^et/)o7rotia§, on the construction of a celestial sphere, which is
lost ; and he constructed a sphere of the stars, and an orrery.
These, after the capture of Syracuse, were taken by Marcellus to
Rome, and were preserved as curiosities for at least two or three
hundred years.
This mere catalogue of his works will show how wonderful
were his achievements ; but no one who has not actually read
some of his writings can form a just appreciation of his extra-
ordinary ability. This will be still further increased if we
recollect that the only principles used by Archimedes, in
addition to those contained in Euclid's Elements and Conic
sections, are that of all lines like
AGB, ADB, ... connecting two
points A and B, the straight line
is the shortest, and of the curved
^ ** lines, the inner one ADB is
shorter than the outer one ACB; together with two similar
statements for space of three dimensions.
In the old and medieval world Archimedes was reckoned
as the first of mathematicians, but possibly the best tribute to
his fame is the fact that those writers who have spoken most
CH.iv] ARCHIMEDES. APOLLONIUS 77
highly of his work and ability are those who have been them-
selves the most distinguished men of their own generation.
Apollonius.i ipj^g third great mathematician of this century
was Apollonius of Perga, who^ is chiefly p.p1phra.f.pd for having
produced a systematic jtreatise on the conic sections which not
only included all that was previously known about them, but
immensely extended the knowledge of these curves. This work
was accepted at once as the standard text-book on the subject,
and completely superseded the previous treatises of Menaech-
mus, Aristaeus, and Euclid which until that time had been in
general use.
We know very little of Apollonius himself. He was born
about 260 B.C., and died about 200 B.C. jle_studied^in Alex-
andria for many years, and probably lectured there ; he is
represented by Pappus as "vain, jealous of the reputation of
others, and ready to seize every opportunity to depreciate them."
It is curious that while we know next to nothing of his life, or
of that of his contemporary Eratosthenes, yet their nicknames,
which were respectively epsilon and heUx^ have come down to us.
Dr. Gow has ingeniously suggested that the lecture rooms at
Alexandria were numbered, and that they always used the rooms
numbered 5 and 2 respectively.
Apollonius spent some years at Pergamum in Pamphylia,
where a university had been recently established and endowed
in imitation of that at Alexandria. There he met Eudemus and
Attains, to whom he subsequently sent each book of his conies
as it came out with an explanatory note. He returned to
Alexandria, and lived there till his death, which was nearly
contemporaneous with that of Archimedes.
In his great work on conic sections Apollonius so thoroughly
investigated the properties of these curves that he left but little
^ In addition to Zeuthen's work and the other authorities mentioned in
the footnote on p. 51, see Litterargeschichtliclie Studien iiber Muklid, by
J. L. Heiberg, Leipzig, 1882. Editions of the extant works of Apollonius
were issued by .J. L. Heiberg in two volumes, Leipzig, 1890, 1893 ; and by
E. Halley, Oxford, 1706 and 1710: an edition of the conies was published by
T. L. Heath, Cambridge, 1896.
78 THE FIRST ALEXANDRIAN SCHOOL [ch. iv
for his successors to add. But hisjjXQols^aJifiUon^ and involved^
and I think most readers will be content to accept a short
analysis of his work, and the assurance that his demonstrations
are valid. Dr. Zeuthen believes that many of the properties
enunciated were obtained in the first instance by the use of
co-ordinate geometry, and that the demonstrations were trans-
lated subsequently into geometrical form. If this be so, we
must suppose that the classical writers were familiar with some
branches of analytical geometry — Dr. Zeuthen says the use of
orthogonal and oblique co-ordinates, and of transformations
depending on abridged notation — that this knowledge was
confined to a limited school, and was finally lost. This is a
mere conjecture and is unsupported by any direct evidence, but
it has been accepted by some writers as aff'ording an explanation
of the extent and arrangement of the work.
The treatise contained about four hundred propositions, and
was divided into eight books ; we have the Greek text of the
first four of these, and we also possess copies of the comment-
aries by Pappus and Eutocius on the whole work. In the ninth
century an Arabic translation was made of the first seven books,
which were the only ones then extant ; we have two manuscripts
of this version. The eighth book is lost.
In the letter to Eudemus which accompanied the first book
ApoUonius says that he undertook the work at the request of
Naucrates, a geometrician who had been staying with him at
Alexandria, and, though he had given some of his friends a
rough draft of it, he had preferred to revise it carefully before
sending it to Pergamum. In the note which accompanied the
next book, he asks Eudemus to read it and communicate it to
others who can understand it, and in j^articular to Philonides,
a certain geometrician whom the author had met at Ephesus.
The first four books deal with the elements of the subject,
and of these the first three are founded on Euclid's previous
work (which was itself based on the earlier treatises by
Menaechmus and Aristaeus). Heracleides asserts that much
of the matter in these books was stolen from an unpublished
CH. iv] APOLLONIUS 79
work of Archimedes, but a critical examination by Heiberg
has shown that this is improbable.
Apollonius begins by defining a cone on a circular base.
He then investigates the different plane sections of it, and
shows that they are divisible into three kinds of curves which
he calls ellipses, parabolas, and hyperbolas. He proves the
proposition that, if ^, A be the vertices of a conic, and if P be
any point on it, and FM the perpendicular drawn from P on
AA\ then (in the usual notation) the ratio MP^ \AM . MA' is
constant in an ellipse or hyperbola, and the ratio i/P- : AM
is constant in a parabola. These are the characteristic properties
on which almost all the rest of the work is based. He next
shows that, if A be the vertex, I the latus rectum, and ii AM
and MP be the abscissa and ordinate of any point on a conic
(see above figure), then 3IP^ is less than, equal to, or greater
than I. A3f according as the conic is an ellipse, parabola, or
hyperbola ; hence the names which he gave to the curves and
by which they are still known.
He had no idea of the directrix, and was not aware that
the parabola had a focus, but, with the exception of the pro-
positions which involve these, his first three books contain most
of the propositions which are found in modern text -books.
In the fourth book he develops the theory of lines cut
80 THE FIRST ALEXANDRIAN SCHOOL [ch. iv
harmonically, and treats of the points of intersection of
systems of conies. In the fifth book he commences with the
theory of maxima and minima ; applies it to find the centre of
curvature at any point of a conic, and the evolute of the curve ;
and discusses the number of normals which can be drawn from
a point to a conic. In the sixth book he treats of similar
conies. The seventh and eighth books were given up to a
discussion of conjugate diameters ; the latter of these was
conjecturally restored by E. Halley in his edition of 1710.
The verbose explanations make the book repulsive to most
modern readers ; but the arrangement and reasoning are
unexceptional, and it has been not unfitly described as the
crown of Greek geometry. It is the Avork on which the
reputation of Apollonius rests, and it earned for him the name
of "the great geometrician."
Besides this immense treatise he wrote numerous shorter
works; of course the books were written in Greek, but they
are usually referred to by their Latin titles : those about which
we now know anything are enumerated below. He was the
author of a work on the problem "given two co-planar straight
lines Aa and Bb, drawn through fixed points A and B ; to draw
a line Oab from a given point 0 outside them cutting them in
a and 6, so that Aa shall be to Bb in a given ratio." He reduced
the question to seventy - seven separate cases and gave an
appropriate solution, with the aid of conies, for each case ; this
was published by E, Halley (translated from an Arabic copy) in
1706. He also wrote a treatise De Sectione Spatii (restored by
E. Halley in 1706) on the same problem under the condition
that the rectangle Aa . Bb was given. He wrote another entitled
De Sectione Determinata (restored by R. Simson in 1749),
dealing with problems such as to find a point P in a given
straight line AB^ so that PA'^ shall be to PB in a given ratio.
He wrote another De Tactionibus (restored by Vieta in 1600)
on the construction of a circle which shall touch three given
circles. Another work was his De Inclinationibus (restored by
M. Ghetaldi in 1607) on the problem to draw a line so that the
CH. iv] APOLLONIUS 81
intercept between two given lines, or the circumferences of two
given circles, shall be of a given length. He was also the
author of a treatise in three books on plane loci, De Locis
Planis (restored by Fermat in 1637, and by R, Simson in
1746), and of another on the regular solids. And, lastly, he
wrote a treatise on unclassed incommensurables, being a com-
mentary on the tenth book of Euclid. It is believed that in
one or more of the lost books he used the method of conical
projections.
Besides these geometrical works he wrote on the methods of
arithmetical calcidation. All that we know of this is derived
from some remarks of Pappus. Friedlein thinks that it was
merely a sort of ready - reckoner. It seems, however, more
probable that Apollonius here suggested a system of numera-
tion similar to that proposed by Archimedes, but proceeding
by tetrads instead of octads, and described a notation for it.
It will be noticed that our modern notation goes by hexads,
a million = 10^ a billion = 10^2, a trillion = lO^^, etc. It is not
impossible that Apollonius also pointed out that a decimal
system of notation, involving only nine symbols, would facilitate
numerical multiplications.
•A-pollonijas. \ms^JJllierested . in astronomy, and wrote a book
on the stations and regy'essions of the planets of which Ptolemy
made some use in writing the Almagest. He also wrote a
treatise on the use and theory of the screw in statics.
This is a long list, but I should suppose that most of these
works were short tracts on special points.
Like so many of his predecessors, he too gave a construction
for finding two mean proportionals between two given lines, and
thereby duplicating the cube. It was as follows. Let OA and
OB be the given lines. Construct a rectangle OADB^ of
which they are adjacent sides. Bisect AB in C. Then, if
^\^ith C as centre we can describe a circle cutting OA produced
in a, and cutting OB produced in 6, so that aDh shall be a
straight line, the problem is effected. For it is easily shewn
that Oa.Aa^CA'^^Ca^.
G
82 THE FIRST ALEXANDRIAN SCHOOL
Similarly Ob.Bb + CB^ = Cb'^.
fcH. IV
Hence
That is.
Oa.Aa=Ob.Bb.
Oa: Ob = Bb : Aa.
But, by similar triangles,
BD:Bb = Oa:Ob = Aa:AD.
Therefore Oa :Bb = Bb :Aa = Aa: OB,
that is, Bb and Oa are the two mean proportionals between
OA and OB. It is impossible to construct the circle whose
centre is C by Euclidean geometry, but Apollonius gave a
mechanical way of describing it. This construction is quoted
by several Arabic writers.
In one of the most brilliant passages of his Apergu historique
Chasles remarks that, while Archimedes and Apollonius were
the most able geometricians of the old world, their works are
distinguished by a contrast which runs through the whole sub-
sequent history of geometry. 4^himede§, in attacking the
problem of the quadrature of curvilinear areas, established the
principles of the geometry which rests on measurements ; this
naturally gave rfse to the. infinitesimal calculus, and in fact the
method of exhaustions as used by Archimedes does not diiFer
in principle from the method of limits as used by Newton.
Apollonius, on the other hand, in investigating the properties of
conic sections by means of transversals involving the ratio of i
CH. iv] APOLLONIUS. ERATOSTHENES 83
rectilineal distances and of perspective, laid the foundations of
the geometry jjf, form and position.
Eratostlienes.^ Among the contemporaries of Archimedes
and ApoUonius I may mention Eratosthenes. Born at Cyrene
in 275 B.c, he was educated at Alexandria — perhaps at the
same time as Archimedes, of whom he was a personal friend —
and Athens, and was at an early age entrusted with the care of
the university library at Alexandria, a post which probably he
occupied till his death. He was the Admirable Crichton of his
age, and distinguished for his athletic, literary, and scientific
attainments : he was also something of a poet. He lost his
sight by ophthalmia, then as now a curse of the valley of the
Nile, and, refusing to live when he was no longer able to read,
he committed suicide in 194 B.C.
In science he was chiefly interested in astronomy and geodesy,
and he constructed various astronomical instruments which were
used for some centuries at the university. He suggested the
calendar (now known as Julian), in which every fourth year
contains 366 days ; and he determined the obliquity of the
ecliptic as 23° 51' 20". He measured the length of a degree on
the earth's surface, making it to be about 79 miles, which is too
long by nearly 10 miles, and thence calculated the circumference
of the earth to be 252,000 stadia. If we take the Olympic
stadium of 202 J yards, this is equivalent to saying that the
radius is about 4600 miles, but there was also an Egyptian
stadium, and if he used this he estimated the radius as 3925
miles, which is very near the truth. The principle used in the
determination is correct.
Of Eratosthenes's work in mathematics we have two extant
illustrations : one in a description of an instrument to duplicate
a cube, and the other in a rule he gave for constructing a table
of prime numbers. The former is given in many books. The
latter, called the " sieve of Eratosthenes," was as follows : write
^ The Avorks of Eratosthenes exist only in fragments. A collection of these
was published by G. Bernhardy at Berlin in 1822 : some additional fragments
were printed by E. Hillier, Leipzig, 1872.
84 THE FIRST ALEXANDRIAN SCHOOL [ch. iv
down all the numbers from 1 upwards; then every second
number from 2 is a multiple of 2 and may be cancelled; every
third number from 3 is a multiple of 3 and may be cancelled ;
every fifth number from 5 is a multiple of 5 and may be
cancelled; and so on. It has been estimated that it would
involve working for about 300 hours to thus find the primes in
the numbers from 1 to 1,000,000. The labour of determining
whether any particular number is a prime may be, however,
much shortened by observing that if a number can be ex-
pressed as the product of two factors, one must be less and the
other greater than the square root of the number, unless the
number is the square of a prime, in which case the two factors
are equal. Hence every composite number must be divisible by
a prime which is not greater than its square root.
The second century before Christ. '
The third century before Christ, which opens with the career
of Euclid and closes with the death of Apollonius, is the most
brilliant era in the history of Greek mathematics. But the
great mathematicians of that century were geometricians, and
under their influence attention was directed almost solely to that
branch of mathematics. With the methods they used, and to
which their successors were by tradition confined, it was hardly
possible to make any further great advance : to fill up a few
details in a work that was completed in its essential parts was
all that could be effected. It was not till after the lapse of
nearly 1800 years that the genius of Pescartes opened the way
to any further progress in geometry, and I therefore pass over
the numerous writers who followed Apollonius with but slight
mention. Indeed it may be said roughly that during the next
thousand years Pappus was the sole geometrician of great original
ability ; and during this long period almost the only other pure
mathematicians of exceptional genius were Hijpparchus_ and
Ptolemy^ who laid the foundations of trigonometry, and Dio-
phantus, who laid those of algebra.
CH.iv] HYPSICLES. NICOMEDES 85
Early in the second century, circ. 180 B.C., we find the names
of three mathematicians — Hypsicles, Nicomedes, and Diodes —
who in their own day were famous.
Hypsicles. The first of these was Hypsicles, who added a
fourteenth book to Euclid's Elements in which the regular solids
were discussed. In another small work, entitled Risings, we
find for the first time in Greek mathematics a right angle
divided in the Babylonian manner into ninety degrees ; possibly
Eratosthenes may have previously estimated angles by the
number of degrees they contain, but this is only a matter of
conjecture.
Nicomedes. The second was Nicomedes, who invented the
curve known as the conchoid or the shell-shai)ed curve. If from
a fixed point S a line be drawn cutting a given fixed straight
line in Q, and if P be taken on SQ so that the length QP is
constant (say d), then the locus of P is the conchoid. Its
equation may be put in the form r = asec^±^. It is easy
with its aid to trisect a given angle or to dui)licate a cube ; and
this no doubt was the cause of its invention.
Diodes. The third of these mathematicians was Diocles, the
inventor of the curve known as the cissoid or the ivy-shaped
curve, which, like the conchoid, was used to give a solution of
the duplication problem. He defined it thus: let AOA! and
BOB' be two fixed diameters of a circle at right angles to one
another. Draw two chords QQ' and RE parallel to BOB' and
equidistant from it. Then the locus of the intersection of ^^
and QQ' will be the cissoid. Its equation can be expressed in
the form ^2(2a -x) = a?^ The curve may be used to duplicate
the cube. For, if OA and OE be the two lines between which
it is required to insert two geometrical means, and if, in the
figure constructed as above, A!E cut the cissoid in P, and AP
cut OB in D, we have OD^=OA'^.OE. Thus OD is one
of the means required, and the other mean can be found at
once.
Diocles also solved (by the aid of conic sections) a problem
which had been proposed by Archimedes, namely, to draw a
86 THE FIRST ALEXANDRIAN SCHOOL [ch. iv
plane which will divide a sphere into two parts whose volumes
shall bear to one another a given ratio.
Perseus. Zenodorus. About a quarter of a century later,
say about 150 B.C., Perseus investigated the various plane sections
of the anchor-ring, and Zenodorus wrote a treatise on isoperi-
metrical figures. Part of the latter work has been preserved ;
one proposition which will serve to show the nature of the
problems discussed is that " of segments of circles, having equal
arcs, the semicircle is the greatest."
Towards the close of this century we find two mathematicians
who, by turning their attention to new subjects, gave a fresh
stimulus to the study of mathematics. These were Hipparchus
and Hero.
Hipparchus.^ Hipparchus was the most eminent of Greek
astronomers — his chief predecessors being Eudoxus, Aristarchus,
Archimedes, and Eratosthenes. Hipparchus is said to have been
born about 160 B.C. at Nicaea in Bithynia; it is probable that
he spent some years at Alexandria, but finally he took up his
abode at Rhodes where he made most of his observations.
Delambre has obtained an ingenious confirmation of the tradi-
tion which asserted that Hipparchus lived in the second century
before Christ. Hipparchus in one place says that the longitude
of a certain star t] Canis observed by him was exactly 90°, and
it should be noted that he was an extremely careful observer.
Now in 1750 it was 116° 4' 10", and, as the first point of Aries
regredes at the rate of 50*2" a year, the observation was made
about 120 B.C.
Except for a short commentary on a poem of Aratus dealing
with astronomy all his works are lost, but Ptolemy's great
treatise, the Almagest, described below, was founded on the
observations and writings of Hipparchus, and from the notes
^ See C. Manitius, Hipparchi in Arati et Eudoxi pTmenomena Commentarii,
Leipzig, 1894, and J. B. J. Delambre, Histoire de Vastronomie ancienne, Paris,
1817, vol. i, pp. 106-189. S. P. Tannery in his Recherches sur VMstoire de
Vastronomie ancienne, Paris, 1893, argues that the work of Hipparchus has
been overrated, but I have adopted the view of the majority of writers on the
subject.
CH. iv] HIPPARCHUS 87
there given we infer that the chief discoveries of Hipparchus
were as follows. He determined the duration of the year to
within six minutes of its true value. He calculated the inclina-
tion of the ecliptic and equator as 23° 51'; it was actually at
that time 23° 46'. He estimated the annual precession of the
equinoxes as 59" ; it is 5;0*2". He stated the lunar parallax as
57', which is nearly correct. He worked out the eccentricity of
the solar orbit as 1/24; it is very approximately 1/30. He
determined the perigee and mean motion of the sun and of the
moon, and he calculated the extent of the shifting of the plane
of the moon's motion. Finally he obtained the synodic periods
of the five planets then known. I leave the details of his
observations and calculations to writers who deal specially with
astronomy such as Delambre ; but it may be fairly said that
this work placed the subject for the first time on a scientific
To a;ccount for the lunar motion Hipparchus supposed the
moon to move with uniform velocity in a circle, the earth
occupying a position near (but not at) the centre of this circle.
This is equivalent to saying that the orbit is an epicycle of the
first order. The longitude of the moon obtained on this
hypothesis is correct to the first order of small quantities for a
few revolutions. To make it correct for any length of time
Hipparchus further supposed that the apse line moved forward
about 3° a month, thus giving a correction for eviction. He
explained the motion of the sun in a similar manner. This
theory accounted for all the facts which could be determined
with the instruments then in use, and in particular enabled him
to calculate the details of eclipses with considerable accuracy.
He commenced a series of planetary observations to enable
his successors to frame a theory to account for their motions ;
and with great perspicacity he predicted that to do this it
would be necessary to introduce epicycles of a higher order,
that is, to introduce three or more circles the centre of each
successive one moving uniformly along the circumference of the
preceding one.
88 THE FIRST ALEXANDRIAN SCHOOL [ch. iv
He also formed a list of 1080 of the fixed stars. It is said
that the sudden appearance in the heavens of a new and
brilliant star called his attention to the need of such a catalogue ;
and the appearance of such a star during his lifetime is confirmed
by Chinese records.
No further advance in the theory of astronomy was made
until the time of Copernicus, though the principles laid down
by Hipparchus were extended and worked out in detail by
Ptolemy.
Investigations such as these naturally led to trigonoDietry^
and Hipparchus must be credited with the invention of that
subject. It is known that in plane trigonometry he constructed
a table of chords of arcs, which is practically the same as one of
natural sines ; and that in spherical trigonometry he had some
method of solving triangles : but his works are lost, and we can
give no details. It is believed, however, that the elegant
theorem, printed as Euc. vi, d, and generally known as
Ptolemy's Theorem, is due to Hipparchus and was copied from
him by Ptolemy. It contains implicitly the addition formulae
for sin(^ ± B)- and cos(^ ± E) ; and Carnot showed how the
whole of elementary plane trigonometry could be deduced
from it.
I ought also to add that Hipparchus was the first to indicate
the position of a place on the earth by means of its latitude and
longitude.
Hero.^ The second of these mathematicians was Hero of
Alexandria, who placed engineering and land-surveying on a
scientific basis. He was a pupil of Ctesibus, who invented
^ See Recherches sur la vie et les ouvrages d^H^'on d' Alexandrie by T. H.
Martin in vol. iv of M4inoires pr4sent4s . . .d, Vacaddmie d' inscriptions, Paris,
1854 ; see also Loria, book iii, chap, v, pp. 107-128, and Cantor, chaps,
xviii, xix. On the work entitled Definitions, which is attributed to Hero,
see S. P. Tannery, chaps, xiii, xiv, and an article by G. Friedlein in
Boncompagni's Bulletino di hihliografixt, March 1871, vol. iv, pp. 93-126.
Editions of the extant works of Hero were published in Teubner's series,
Leipzig, 1899, 1900, 1903. An English translation of the livevixariKA. was
published by B. Woodcroft and J. G. Greenwood, London, 1851 : drawings
of the apparatus are inserted.
CH. iv] HERO 89
several ingenious machines, and is alluded to as if he were a
mathematician of note. It is not likely that Hero flourished
before 80 B.C., but the precise period at which he lived is
uncertain.
In pure mathematics Hero's principal and most characteristic
work consists of (i) some elementary geometry, with applications
to the determination of the areas of fields of given shapes ; (ii)
propositions on finding the volumes of certain solids, with
applications to theatres, baths, banquet-halls, and so on ; (iii) a
rule to find the height of an inaccessible object ; and (iv) tables
of weights and measures. He invented a solution of the
duplication problem which is practically the same as that which
Apollonius had already discovered. Some commentators think
that he knew how to solve a quadratic equation even when the
coefficients were not numerical ; but this is doubtful. He
proved the formula that the area of a triangle is equal to
{ s(s - a) {s- b) (s - c)y^\ where s is the semiperimeter, and a, b, c,
the lengths of the sides, and gave as an illustration a triangle
whose sides were in the ratio 13:14:15. He seems to have
been acquainted mth the trigonometry of Hipparchus, and the
values of cot27r/?i are computed for various values of n, but he
nowhere quotes a formula or expressly uses the value of the
sine; it is probable that like the later Greeks he regarded
trigonometry as forming an introduction to, and being an
integral part of, astronomy.
The following is the manner in which he solved ^ the problem
to find the area of a triangle ABC the length of whose sides are
a, b, r. Let s be the semiperimeter of the triangle. Let the
inscribed circle touch the sides in D, B, F^ and let 0 be its
centre. On BC produced take H so that CH= AF^ therefore
BH=s. Draw OK at right angles to OB^ and CK at right
angles to ^C ; let them meet in K. The area ABC or A is equal
to the sum of the areas OBC, OCA, OAB = \ar-{-\br-ithcr = sr,
^ In his Dioptra, Hultsch, part viii, pp. 235-237. It should be stated
that some critics thiuk that this is an interpolation, and is not due to
Hero.
90
THE FIRST ALEXANDRIAN SCHOOL [ch. iv
that is, is equal to BH . OD. He then shews that the angle
0^i^= angle CBK; hence the triangles OAF and CBK are
similar.
.-. BC'.CK=AF:OF=CH'.OD,
.'. BC'.CH=CK:OD = CL'.LD,
.'. BH'.CH^GD.LD,
.'. BH^ :CH.BH=CD.BD'.LD.BD--^ CD . BD : ODK
Hence
A = Bff. OB = {Off. BR. CD . BD}^= {{s - a)s{s - c)(s - h)}K
In applied mathematics Hero discussed the centre of gravity,
the five simple machines, and the problem of moving a given
weight with a given power ; and in one place he suggested a
way in which the power of a catapult could be tripled. He
also wrote on the theory of hydraulic machines. He described a
theodolite and cyclometer, and pointed out various problems in
surveying for which they would be useful. But the most
cH.iv] HERO 91
interesting of his smaller works are his IIi/ev/xaTtKa and
AvTo/xaTa, containing descriptions of about 100 small machines
and mechanical toys, many of which are ingenious. In the
former there is an account of a small stationary steam-engine
which is of the form now known as Avery's patent : it was in
common use in Scotland at the beginning of this century, but is
not so economical as the form introduced by Watt. There is
also an account of a double forcing pump to be used as a fire-
engine. It is probable that in the hands of Hero these instru-
ments never got beyond models. It is only recently that
general attention has been directed to his discoveries, though
Arago had alluded to them in his eloge on Watt.
All this is very different from the classical geometry and
arithmetic of Euclid, or the mechanics of Archimedes. Hero
did nothing to extend a knowledge of abstract mathematics ; he
learnt all that the text-books of the day could teach him, but he
was interested in science only on account of its practical appli-
cations, and so long as his results were true he cared nothing
for the logical accuracy of the process by which he arrived at
them. Thus, in finding the area of a triangle, he took the
square root of the product of four lines. The classical Greek
geometricians permitted the use of the square and the cube of
a line because these could be represented geometrically, but a
figure of four dimensions is inconceivable, and certainly they
would have rejected a proof which involved such a conception.
The first century before Christ.
The successors of Hipparchus and Hero did not avail them-
selves of the opportunity thus opened of investigating new
subjects, but fell back on the well-worn subject of geometry.
Amongst the more eminent of these later geometricians were
Theodosius and Dionysodorus, both of whom flourished about
50 B.C.
Theodosius. Theodosius was the author of a complete
92 CLOSE OF FIRST ALEXANDRIAN SCHOOL [ch. iv
treatise on the geometry of the sphere, and of two works on
astronomy.!
Dionysodoms. . Dionysodorus is known to us only by his
solution 2 of the problem to divide a hemisphere by a plane
parallel to its base into two parts, whose volumes shall be in a
given ratio. Like the solution by Diodes of the similar problem
for a sphere above alluded to, it was effected by the aid of conic
sections. Pliny says that Dionysodorus determined the length
of the radius of the earth approximately as 42,000 stadia,
which, if we take the Olympic stadium of 202|- yards, is a little
less than 5000 miles ; we do not know how it was obtained.
This may be compared with the result "given by Eratosthenes
and mentioned above.
End of the First Alexandrian ScJiool.
The administration of Egypt was definitely undertaken
by Rome in 30 b.c. The closing years of the dynasty of the
Ptolemies and the earlier years of the Roman occupation of
the country were marked by much disorder, civil and political.
The studies of the university were naturally interrupted, and
it is customary to take this time as the close of the first
Alexandrian school.
^ The work on the sphere was edited by I. Barrow, Cambridge, 1675,
and by E. Nizze, Berlin, 1852. The works on astronomy were published by
Dasypodiiis in 1572.
'^ It is reproduced in H. Suter's Oeschichte der mathematischen Wissen-
schaften, second edition, Zurich, 1873, p. 101.
93
CHAPTER V.
THE SECOND ALEXANDRIAN SCHOOL. ^
30 B.C.-641 A.D.
I CONCLUDED the last chapter by stating that the first school of
Alexandria may be said to have come to an end at about the
same time as the country lost its nominal independence. But,
although the schools at Alexandria suffered from the disturb-
ances which affected the whole Koman world in the transition,
in fact if not in name, from a republic to an empire, there was
no break of continuity; the teaching in the university was
never abandoned ; and as soon as order was again established,
students began once more to flock to Alexandria. This time of
confusion was, however, contemporaneous with a change in the
prevalent views of philosophy which thenceforward were mostly
neo-platonic or neo-pythagorean, and it therefore fitly marks the
commencement of a new j)eriod. These mystical opinions
reacted on the mathematical school, and this may partially
account for the paucity of good work.
Though Greek influence w^as still predominant and the
Greek language always used, Alexandria now became the in-
tellectual centre for most of the Mediterranean nations which
were subject to Rome. It should be added, however, that
the direct connection with it of many of the mathematicians
^ For aiitliorities, see footnote above on p. 50. All dates given hereafter
are to be taken as anno domini unless tbe contrary is expressly stated.
94 THE SECOND ALEXANDRIAN SCHOOL [ch. v
of this time is at least doubtful, but their knowledge was
ultimately obtained from the Alexandrian teachers, and they
are usually described as of the second Alexandrian school.
Such mathematics as were taught at Rome were derived from
Greek sources, and we may therefore conveniently consider
their extent in connection with this chapter.
The first century after Christ.
There is no doubt that throughout the first century after
Christ geometry continued to be that subject in science to
which most attention was devoted. But by this time it was
evident that the geometry of Archimedes and Apollonius was not
capable of much further extension ; and such geometrical treatises
as were produced consisted mostly of commentaries on the
writings of the great mathematicians of a preceding age. In
this century the only original works of any ability of which we
know anything were two by Serenus and one by Menelaus.
Serenus. Menelaus. Those by Serenus of Antissa or of
x^intinoe, circ. 70, are on the plane sections of the cone and
cylinder,'^ in the course of which he lays down the fundamental
proposition of transversals. That by Menelaus of Alexandria,
circ. 98, is on spherical trigonometry, investigated in the
Euclidean method. ^ The fundamental theorem on which the
subject is based is the relation between the six segments of the
sides of a spherical triangle, formed by the arc of a great circle
which cuts them [book iii, prop. 1]. Menelaus also wrote on
the calculation of chords, that is, on plane trigonometry ; this
is lost.
Nicomachus. Towards the close of this century, circ.
100, a Jew, Nicomachus, of Gerasa, published an Arithmetic,^
which (or rather the Latin translation of it) remained for a
^ These have been edited by J. L. Heiberg, Leipzig, 1896 ; and by
E. Halley, Oxford, 1710.
2 This was translated by E. Halley, Oxford, 1758.
2 The work has been edited by R, Hoche, Leipzig, 1866.
CH.v] THEON. THYMARIDAS 95
thousand years a standard authority on the subject. Geo-
metrical demonstrations are here abandoned, and the work is a
mere classification of the results then known, with numerical
illustrations : the evidence for the truth of the propositions
enunciated, for I cannot call them j)roofs, being in general an
induction from numerical instances. The object of the book
is the study of the i^roperties of numbers, and particularly of
tTieir ratios. Nicomachus commences with the usual distinc-
tions between even, odd, prime, and perfect numbers ; he next
discusses fractions in a somewhat clumsy manner; he then
turns to polygonal and to solid numbers ; and finally treats of
ratio, proportion, and the progressions. Arithmetic of this kind
is usually termed Boethian, and the work of Boethius on it was
a recognised text-book in the middle ages.
The second century after Christ.
Theon. Another text -book on arithmetic on much the
same lines as that of Nicomachus was produced by Theon of
Smyrna, circ. 130. It formed the first book of his work^ on
mathematics, written with the view of facilitating the study
of Plato's writings.
Thymaridas. Another mathematician, reckoned by some
writers as of about the same date as Theon, was Thymaridas^
who is worthy of notice from the fact that he is the earliest
known writer who explicitly enunciates an algebraical theorem.
He states that, if the sum of any number of quantities be
given, and also the sum of every pair which contains one of
them, then this quantity is equal to one {n - 2)th part of the
difi'erence between the sum of these pairs and the first given
sum. Thus, if
and if x^-\-x^ = S2^i x-^-\- x^ = s^, ..., and x-^ + Xn=-Sn,
then x^ = {s.y + s^+ ... -\-Sn- S)l(n - 2).
^ The Greek text of those parts which are now extant, with a French
translation, was issued by J, Dupuis, Paris, 1892.
96 THE SECOND ALEXANDRIAN SCHOOL [ch. v
He does not seem to have used a symbol to denote the unknown
quantity, but he always represents it by the same word, which
is an approximation to symbolism.
Ptolemy.^ About the same time as these writers Ptolemy
of Alexandria, who died in 168, produced his great work on
astronomy, which will preserve his name as long as the history
of science endures. This treatise is usually known as the
Almagest : the name is derived from the Arabic title al mid-
schisti, which is said to be a corruption of /JLeyia-rr) [/Aa^T^/xariKry]
(TvvTa^LS. The work is founded on the writings of Hipparchus,
and, though it did not sensibly advance the theory of the
subject, it presents the views of the older writer with a com-
pleteness and elegance which will always make it a standard
treatise. We gather from it that Ptolemy made observations
at Alexandria from the years 125 to 150; he, however, was
but an indifferent practical astronomer, and the observations
of Hipparchus are generally more accurate than those of his
expounder.
The work is divided into thirteen books. In the first book
Ptolemy discusses various preliminary matters ; treats of trigo-
nometry, plane or spherical ; gives a table of chords, that is,
of natural sines (which is substantially correct and is probably
taken from the lost work of Hipparchus) ; and explains the
obliquity of the ecliptic ; in this book he uses degrees, minutes,
and seconds as measures of angles. The second book is devoted
chiefly to phenomena depending on the spherical form of the
earth : he remarks that the explanations would be much
simplified if the earth were supposed to rotate on its axis
once a day, but states that this hypothesis is inconsistent with
known facts. In the third book he explains the motion of the
^ See the article Ptolemaeus Claudius, by A, De Morgan in Smith's
Dictionary of Greek and Roman Biography, London, 1849 ; S. P, Tannery,
Recherches sur I'histoire de V astronomie ancieyme, Paris, 1893 ; and
J. B. J. Delambre, Histoire de V astronomie ancientie, Paris, 1817, vol. ii.
An edition of all the works of Ptolemy which are now extant was
published at Bale in 1551. The Almagest with various minor works
was edited by M. Halma, 12 vols. Paris, 1813-28, and a new edition,
in two volumes, by J. L. Heiberg, Leipzig, 1898, 1903, 1907.
CH. v] PTOLEMY 97
sun round the earth by means of excentrics and epicycles : and
in the fourth and fifth books he treats the motion of the moon
in a similar way. The sixth book is devoted to the theory of
eclipses ; and in it he gives 3° 8' 30", that is 3 j^q, as the
approximate value of tt, which is equivalent to taking it equal
to 3'1416. The seventh and eighth books contain a catalogue
(probably copied from Hipparchus) of 1028 fixed stars deter-
mined by indicating those, three or more, that appear to be in
a plane passing through the observer's eye : and in another
work Ptolemy added a list of annual sidereal phenomena. The
remaining books are given up to the theory of the planets.
This work is a splendid testimony to the ability of its
author. It became at once the standard authority on astro-
nomy, and remained so till Copernicus and Kepler shewed
that the sun and not the earth must be regarded as the centre
of the solar system.
The idea of excentrics and epicycles on which the theories
of Hipparchus and Ptolemy are based has been often ridiculed
in modern times. No doubt at a later time, when more accu-
rate observations had been made, the necessity of introducing
epicycle on epicycle in order to bring the theory into accord-
ance with the facts made it very complicated. But De Morgan
has acutely observed that in so far as the ancient astronomers
supposed that it was necessary to resolve every celestial motion
into a series of uniform circular motions they erred greatly,
but that, if the hypothesis be regarded as a convenient way
of expressing known facts, it is not only legitimate but
convenient. The theory suffices to describe either the angular
motion of the heavenly bodies or their change in distance. The
ancient astronomers were concerned only with the former ques-
tion, and it fairly met their needs ; for the latter question it is
less convenient. In fact it was as good a theory as for their
purposes and with their instruments and knowledge it was
possible to frame, and corresponds to the expression of a given
function as a sum of sines or cosines, a method which is of,
frequent use in modern analysis.
H
98 THE SECOND ALEXANDRIAN SCHOOL [ch. v
In spite of the trouble taken by Delainbre it is almost
impossible to separate the results due to Hipparchus from
those due to Ptolemy. But Delambre and De Morgan agree
in thinking that the observations quoted, the fundamental
ideas, and the explanation of the apparent solar motion are due
to Hipparchus ; while all the detailed explanations and calcula-
tions of the lunar and planetary motions are due to Ptolemy.
The Almagest shews that Ptolemy w^as a geometrician of
the first rank, though it is with the application of geometry
to astronomy that he is chiefly concerned. He was also the
author of numerous other treatises. Amongst these is one on
pure geometry in which he proposed to cancel Euclid's postulate
on parallel lines, and to prove it in the following manner. Let
the straight line EFGII meet the two straight lines AB and
CD so as to make the sum of the angles BFG and FGD equal
to two right angles. It is required to prove that AB and CD
are parallel. If possible let them not be parallel, then they will
meet when produced say at M (or N). But the angle AFG is
the supplement of BFG, and is therefore equal to FGD :
similarly the angle FGC is equal to the angle BFG. Hence
the sum of the angles AFG and ,FGC is equal to two right
angles, and the lines BA and DC will therefore if produced
meet at JV (or M). But two straight lines cannot enclose a
space, therefore AB and CD cannot meet when produced, that
is, they are parallel. Conversely, ii AB and CD be parallel,
then AF and CG are not less parallel than FB and GD ; and
CH.v] PAPPUS , 99
therefore whatever be the sum of the angles AFG and FGG
such also must be the sum of the angles FGD and BFG. But
the sum of the four angles is equal to four right angles, and
therefore the sum of the angles BFG and FGD must be equal
to two right angles.
Ptolemy wrote another work to shew that there could not
be more than three dimensions in space : he' also discussed
orthographic and stereographic pi^ojections with special refer-
ence to the construction of sun-dials. He wrote on geography,
and stated that the length of one degree of latitude is 500
stadia. A book on sound is sometimes attributed to him, but
on doubtful authority.
Tlie third century after Christ.
Pappus. Ptolemy had shewn not only that geometry
could be applied to astronomy, but had indicated how new
methods of analysis like trigonometry might be thence de-
veloped. He found however no successors to take up the
work he had commenced so brilliantly, and we must look
forward 150 years before we find another geometrician of any
eminence. That geometrician was Papjms who lived and
taught at Alexandria about the end of the third century. We
know that he had numerous pupils, and it is probable that he
temporarily revived an interest in the study of geometry.
Pappus wrote several books, but the only one which has
come down to us is his Svi/aywy?},! a collection of mathe-
matical papers arranged in eight books of which the first and
part of the second have been lost. This collection was intended
to be a synopsis of Greek mathematics together with comments
and additional propositions by the editor. A careful com-
parison of various extant works with the account given of
them in this book shews that it is trustworthy, and we rely
largely on it for our knowledge of other works now lost. It
is not arranged chronologically, but all the treatises on the
1 It has been published by F. Hultsch, Berlin, 1876-8.
100 THE SECOND ALEXANDRIAN SCHOOL [ch. v
same subject are grouped together, and it is most likely that
it gives roughly the order in which the classical authors were
read at Alexandria. Probably the first book, which is now
lost, was on arithmetic. The next four books deal with
geometry exclusive of conic sections ; the sixth with astronomy
including, as subsidiary subjects, optics and trigonometry; the
seventh with analysis, conies, and porisms ; and the eighth with
mechanics.
The last two books contain a good deal of original work by
Pappus ; at the same time it should be remarked that in two or
three cases he has been detected in appropriating proofs from
earlier authors, and it is possible he may have done this in other
cases.
Subject to this suspicion we may say that Pappus's best
J work is in geometry. He discovered the directrix in the conic
, sections, but he investigated only a few isolated properties :
the earliest comprehensive account was given by Newton and
Boscovich. As an illustration of his power I may mention
that he solved [book vii, prop. 107] the problem to inscribe in
a given circle a triangle whose sides produced shall pass
through three collinear points. This question was in the
eighteenth century generalised by Cramer by supposing the
three given points to be anywhere; and was considered a
difficult problem.^ It w^as sent in 1742 as a challenge to
Castillon, and in 1776 he published a solution. Lagrange,
Euler, Lhulier, Fuss, and Lexell also gave solutions in 1780.
A few years later the problem was set to a Neapolitan lad
A. Giordano, who was only 16 but who had shewn marked
mathematical ability, and he extended it to the case of a
polygon of n sides which pass through n given points, and gave
a solution both simple and elegant. Poncelet extended it to
conies of any species and subject to other restrictions.
In mechanics Pappus shewed that the centre of mass of a
triangular lamina is the same as that of an inscribed triangular
^ For references to this problem see a note by H, Brocard in V Inter-
midiaire des mathfmaticiens, Paris, 1904, vol. xi, pp. 219-220.
cH.v] PAPPUS 101
lamina whose vertices divide each of the sides of the original
triangle in the same ratio. He also discovered the two
theorems on the surface and volume of a solid of revolution
which are still quoted in text-books under his name : these
are that the volume generated by the revolution of a curve
about an axis is equal to the product of the area of the curve
and the length of the path described by its centre of mass;
and the surface is equal to the product of the perimeter of
the curve and the length of the path described by its centre of
mass.
The problems above mentioned are but samples of many
brilliant but isolated theorems which were enunciated by
Pappus. His work as a whole and his comments shew that he
was a geometrician of power ; but it was his misfortune to
live at a time when but little interest was taken in geometry,
and w^hen th^- subject, as then treated, had been practically
exhausted. -^
Possibly a small tract ^ on multiplication and division of
sexagesimal fractions, which would seem to have been written
about this time, is due to Pappus.
The fourth century after Christ.
Throughout the second and third centuries, that is, from
the time of Nicomachus, interest in geometry had steadily
decreased, and more and more attention had been paid to the
theory of numbers, though the results were in no way com-
mensurate with the time devoted to the subject. It will
be remembered that Euclid used lines as symbols for any
magnitudes, and investigated a number of theorems about
numbers in a strictly scientific manner, but he confined him-
self to cases where a geometrical representation was possible.
There are indications in the works of Archimedes that he was
prepared to carry the subject much further : he introduced
^ It was edited by C. Henry, Halle, 1879, and is valuable as an illustration
of practical Greek arithmetic.
102 THE SECOND ALEXANDRIAN SCHOOL [ch.v
numbers into his geometrical discussions and divided lines by
lines, but he was fully occupied by other researches and had
no time to devote to arithmetic. Hero abandoned the geo-
metrical representation of numbers, but he, Nicomachus, and
other later writers on arithmetic did not succeed in creating
any other symbolism for numbers in general, and thus when
they enunciated a theorem they w^ere content to verify it by
a large number of numerical examples. They doubtless knew
how to solve a quadratic equation with numerical coefficients —
for, as pointed out above, geometrical solutions of the equa-
tions ax^ — hx-\-c = ^ and ax^ -f ^^ - c = 0 are given in Euc. vi,
28 and 29 — but probably this represented their highest attain-
ment.
It would seem then that, in spite of the time given to their
study, arithmetic and algebra had not made any sensible
advance since the time of Archimedes. The problems of this
kind which excited most interest in the third century may be
illustrated from a collection of questions, printed in the
Palatine Anthology, which was made by Metrodoms at the
beginning of the next century, about 310. ' Some of them are
due to the editor, but some are of an anterior date, and they
fairly illustrate the way in which arithmetic was leading up
to algebraical methods. The following are typical examples.
" Four pipes discharge into a cistern : one fills it in one day ;
another in two days; the third in three days; the fourth in
four days : if all run together how soon will they fill the
cistern?" "Demochares has lived a fourth of his life as a
boy ; a fifth as a youth ; a third as a man ; and has spent
thirteen years in his dotage : how old is he ? " " Make a crown
of gold,, copper, tin, and iron weighing 60 minae : gold and
copper shall be two- thirds of it ; gold and tin three-fourths of
it ; and gold and iron three-fifths of it : find the weights of
the gold, copper, tin, and iron which are required."' The
last is a numerical illustration of Thymaridas's theorem quoted
above.
It is believed that these problems were solved by rhetorical
CH. v] ARITHMETIC AND ALGEBRA 103
algebra, that is, by a process of algebraical reasoning expressed
in words and without the use of any symbols. This, according
to Nesselmann, is the first stage in the development of algebra,
and we find it used both by Ahmes and by the earliest Arabian,
Persian, and Italian algebraists : examples of its use in the
solution of a geometrical problem and in the rule for the solution
of a quadratic equation are given later. ^ On this view then a
rhetorical algebra had been gradually evolved by the Greeks,
or was then in process of evolution. Its development was
however very imperfect. Hankel, who is no unfriendly critic, ;
says that the results attained as the net outcome of the work
of six centuries on the theory of numbers are, whether we
look at the form or the substance, unimportant or even childish,
and are not in any way the commencement of a science.
In the midst of this decaying interest in geometry and these
feeble attempts at algebraic arithmetic, a single algebraist of
marked originality suddenly appeared who created what was
practically a new science. This was Diophantus who introduced
a system of abbreviations for those operations and quantities
which constantly recur, though in using them he observed all
the rules of grammatical syntax. The resulting science is called
by Nesselmann syncopated algebra : it is a sort of shorthand.
Broadly speaking, it may be said that European algebra did
not advance beyond this stage until the close of the sixteenth
century.
Modern algebra has progressed one stage further and is
entirely symbolic ; that is, it has a language of its own and a
system of notation which has no obvious connection with the
things represented, while the operations are performed according
to certain rules which are distinct from the laws of grammatical
construction.
Diophantus.-^ All that we know of Diojyhantus is that
1 See below, pp. 203, 210.
2 A critical edition of the collected works of Diophantus was edited by
S. P. Tannery, 2 vols., Leipzig, 1893 ; see also Diophantos of Alexandria,
by T. L. H©4|ii, Cambrid^, 1885 ; and Loria, book v, chap, v, pp. 95-158.
104 THE SECOND ALEXANDRIAN SCHOOL [ch. v
he lived at Alexandria, and that most likely he was not a
Greek. Even the date of his career is uncertain ; it cannot
reasonably be put before the middle of the third century, and
it seems probable that he was alive in the early years of the
fourth century, that is, shortly after the death of Pappus. He
was 84 when he died.
In the above sketch of [fhe lines on which algebra has de-
veloped I credited Diophantus with the invention of syncopated
algebra. This is a point on which opinions differ, and some
writers believe that he only systematized the knowledge which
was familiar to his contemporaries. In support of this latter
opinion it may be stated that Cantor thinks that there are traces
of the use of algebraic symbolism in Pappus, and Freidlein
mentions a Greek papyrus in which the signs / and 9 are used
for addition and subtraction respectively ; buT no other direct
evidence for the non-originality of Diophantus has been produced,
and no ancient author gives any sanction to this opinion.
Diophantus wrote a short essay on polygonal numbers ; a
treatise on algebra which has come down to us in a mutilated
condition ; and a work on porisms which is lost.
The Folygonal Numbers contains ten. propositions, and
was probably his earliest work. In this he reverts to the
classical system by which numbers are represented by lines, a
construction is (if necessary) made, and a strictly deductive
proof follows : it may be noticed that in it he quotes pro-
positions, such as Euc. ii, 3, and ii, 8, as referring to numbers
and not to magnitudes.
His chief work is his Arithmetic. This is really a treatise
on algebra; algebraic symbols are used, and the problems are
treated analytically. Diophantus tacitly assumes, as is done
in nearly all modern algebra, that the steps are reversible. He
applies this algebra to find solutions (though frequently only
particular ones) of several problems involving numbers. I
propose to consider successively the notation, the methods of
analysis employed, and the subject-matter of this work.
First, as to the notation. Diophantus always employed a
CH. v] DIOPHANTUS 105
symbol to represent the unknown quantity in his equations,
but as he had only one symbol he could not use more than
one unknown at a time.^ The unknown quantity is called
6 dpLdfji6<;, and is represented by g-' or g-^'. It is usually printed
as s. In the plural it is denoted by S9 or ss^^ This symbol
may be a corruption of aP, or perhaps it may be the final
sigma of this word, or possibly it may stand for the word a-iopos
a heap.^ The square of the unknown is called Svvafiis, and
denoted by 6" : the cube kvjSos, and denoted by k^ ; and so on
up to the sixth power.
The coefficients of the unknown quantity and its powers are
numbers, and a numerical coefficient is written immediately after
the quantity it multiplies : thus s'd = x, and ss"' ta = ss ta = 11a;:.
An absolute term is regarded as a certain number of units or
IxovdScs which are represented by /x^ : thus ix^d = 1, /x<'ta= 11.
There is no sign for addition beyond juxtaposition. Sub-
traction is represented by 7^, and this symbol affects all the
symbols that follow it^ Equality is represented by ^. Thus
K^d ssrj /p. 8°€ fx°a I sd
represents {x^ + 8^) - {bx^ + 1 ) = ^.
Diophantus also introduced a somewhat similar notation
for fractions involving the unknown quantity, but into the
details of this I need not here enter.
It will be noticed that all these symbols are mere abbre-
viations for words, and Diophantus reasons out his proofs,
writing these abbreviations in the middle of his text. In
most manuscripts there is a marginal summary in which the
symbols alone are used and which is really symbolic algebra ;
but probably this is the addition of some scribe of later times.
This introduction of a contraction or a symbol instead of a
word to represent an unknown quantity marks a greater advance
than anyone not acquainted with the subject would imagine,
and those who have never had the aid of some such abbreviated
^ See, however, below, page 108, example (iii), for an instance of how
he treated a problem involving two unknown quantities.
^ See above, page 5.
106 THE SECOND ALEXANDRIAN SCHOOL [ch. v
symbolism find it almost impossible to understand complicated
algebraical processes. It is likely enough that it might have
been introduced earlier, but for the unlucky system of numera-
tion adopted by the Greeks by which they used all the letters
of the alphabet to denote particular numbers and thus made it
impossible to employ them to represent any number.
Next, as to the knowledge of algebraic methods shewn in
the book. Diophantus commences with some definitions which
include an explanation of his notation, and in giving the symbol
for minus he states that a subtraction multiplied by a
subtraction gives an addition; by this he means that the
product of - 6 and -dm the expansion of {a -b) (c - d) is
+ hd, but in applying the rule he always takes care that the
numbers a, b, c, d are so chosen that a is greater than b and c
is greater than d.
The whole of the work itself, or at least as much as is now
extant, is devoted to solving problems which lead to equations.
It contains rules for solving a simple equation of the first
degree and a binomial quadratic. Probably the rule for solving
any quadratic equation was given in that part of the work which
is now lost, but where the equation is of the form ax^ + bx + c = 0
he seems to have multiplied by a and then "completed the
square " in much the same way as is now done : when the
roots are negative or irrational the equation is rejected as
"impossible," and even when both roots are positive he never
gives more than one, always taking the positive value of the
square root. Diophantus solves one cubic equation, namely,
x^ + x = 4^2 ^ 4 [book VI, prob. 19].
The greater part of the work is however given up to in-
determinate equations between two or three variables. When
the equation is between two variables, then, if it be of the first
degree, he assumes a suitable value for one variable and solves
the equation for the other. Most of his equations are of the
form ^^ = Ax'^-\-Bx + C. Whenever ^ or C is equal to zero,
he is able to solve the equation completely. When this is not
the case, then, if ^ = a^, he assumes ^ = ax + m; if (7 = 0^, he
cH.v] DIOPHANTUS . 107
assumes y = mx + c ; and lastly, if the equation can be put in the
form ip- = {ax =t h^ + c'^, he assumes y = mx : where in each case
m has some particular numerical value suitable to the problem
under consideration. A few particular equations of a higher
order occur, but in these he generally alters the problem so as
to enable him to reduce the equation to one of the above
forms.
The simultaneous indeterminate equations involving three
variables, or "double equations" as he calls them, which he
considers are of the forms y'^ = Ax^ 4- Bx + C and 2- = ax"^ + bx + c.
If A and a both vanish, he solves the equations in one of two
ways. It will be enough to give one of his methods which is
as follows : he subtracts and thus gets an equation of the form
y''" -z^ = mx + n ; hence, if y ± ^ = A, then y^z = {mx + ?i)/A. ; and
solving he finds y and z. His treatment of " double equations "
of a higher order lacks generality and depends on the particular
numerical conditions of the problem.
Lastly, as to the matter of the book. The problems he
attacks and the analysis he uses are so various that they cannot
be described concisely and I have therefore selected five typical
problems to illustrate his methods. What seems to strike his
critics most is the ingenuity with which he selects as his un-
known some quantity which leads to equations such as he can
solve, and the artifices by which he finds numerical solutions of
his equations.
I select the following as characteristic examples.
(i) Fimd four numhers^ the sum of every arrangement three
at a time being given; say 22, 24, 27, aTid 20 [book i,
prob. 17].
Let X be the sum of all four numbers ; hence the numbers
are a; - 22, x- 24, x - 27, and x - 20.
.-. x = {x-'21) + {x- 24) + {x- 27) + {x - 20).
.-. x = Z\.
.'. the numbers are 9, 7, 4, and 11.
108 THE SECOND ALEXANDRIAN SCHOOL [ch. v
(ii) Divide a nwniber^ such as 13 which is the sum of two
sqttares 4 and 9, into ttvo other squares [book ii, prob. 10].
He says that since the given squares are 2^ and 3^ he will
take (a? +2)- and {mx ~ 3)^ as the required squares, and will
assume m = 2.
.-. (^ + 2)2 + (2a; -3)2 = 13.
.-. ^ = 8/5.
.'. the required squares are 324/25 and 1/25.
(iii) Find two squares such that the sum of the product
and either is a square [book ii, prob. 29].
Let aj2 and y"^ be the numbers. Then x^y"^ + y'^ and x~y- + x^
are squares. The first will be a square ii x'^->r\ be a square,
which he assumes may be taken equal to (a?— 2),^ hence
a; = 3/4. He has now to make 9 (y^ + 1)/16 a square, to do this
he assumes that 9y2 + 9 = (3y - 4)2^ hence y = 7/24. Therefore
the squares required are 9/16 and 49/576.
It will be recollected that Diophantus had only one symbol
for an unknown quantity; and in this example he begins by
calling the unknowns x^ and 1, but as soon as he has found x
he then replaces the 1 by the symbol for the unknown quantity,
and finds it in its turn.
(iv) To find a [rational'\ right-angled triangle such that the
line bisecting an acute angle is rational [book vi, prob. 18].
His solution is as follows. Let ABC be the triangle of which
C is the right-angle. Let the bisector AD^bx^ and
let DC = Sx, hence AC=^4:X. Next let BC be a multiple of 3,
say 3, .-. BD = 3-3x, hence AB = i-4cx (by Euc. vi, 3).
CH. v] DIOPHANTUS 109
Hence (4 - 4^)2 = 32 + (ixY (Eiic. i, 47), .-. x = 7/32. Multiplying
by 32 we get for the sides of the triangle 28, 96, and 100 ; and
for the bisector 35.
(v) A man buys x measures of wine, some at 8 drachmae
a measure, the rest at 5. He pays for them a square number of
drachmae, sv^h that, if 60 be added to it, the resulting number
is x^-. Find the number he bought at each price [book v,
prob. 33].
The price paid was x^ - 60, hence Sx>x'^ - 60 and bx<x'^ - 60.
From this it follows that x must be greater than 11 and less
than 12.
Again x'^ - 60 is to be a square ; suppose it is equal to
(x - m)" then x = {m^ + 60)/2w, we have therefore
11<— 7^ <12
2m
.-. 19<7/i<21.
Diophantus therefore assumes that 7?t is equal to 20, which
gives him a:= llj ; and makes the total cost, i.e. x'^ - 60, equal
to 72 J drachmae.
He has next to divide this cost into two parts which shall
give the cost of the 8 drachmae measures and the 5 drachmae
measures respectively. Let these parts be y and z.
Then i, + J(72l-.) = i
^, „ 5x79 , 8x59
There! ore z = — r-^, and y = — y^ —
Therefore the number of 5 drachmae measures was 79/12, and
of 8 drachmae measures was 59/12.
From the enunciation of this problem it would seem that
the wine was of a poor quality, and Tannery ingeniously
suggested that the prices mentioned for such a wine are higher
than were usual until after the end of the second century. He
therefore rejected the view which was formerly held that
Diophantus lived in that century, but he did not seem to be
no THE SECOND ALEXANDRIAN SCHOOL [ch. v
aware that De Morgan had previously shewn that this opinion
was untenable. Tannery inclined to think that Diophantus
lived half a century earlier than I have supposed.
I mentioned that Diophantus wrote a third work entitled
Porisms. The book is lost, but we have the enunciations of
some of the propositions, and though we cannot tell whether
they were rigorously proved by Diophantus they confirm our
opinion of his ability and sagacity. It has been suggested that
some of the theorems which he assumes in his arithmetic were
proved in the porisms. Among the more striking of these
results are the statements that the difference of the cubes of two
numbers can be always expressed as the sum of the cubes of two
other numbers ; that no number of the form 4:7i - 1 can be
expressed as the sum of two squares ; and that no number of the
form 8n - 1 (or possibly '^in + 7) can be expressed as the sum
of three squares : to these we may perhaps add the proposition
that any number can be expressed as a square or as the sum of
two or three or four squares.
The writings of Diophantus exercised no perceptible influence
on Greek mathematics ; but his Arithmetic^ when translated into
Arabic in the tenth century, influenced the Arabian school, and
so indirectly affected the progress of European mathematics. An
imperfect copy of the original work was discovered in 1462; it
was translated into Latin and published by Xy lander in 1575;
the translation excited general interest, and by that time the
European algebraists had, on the whole, advanced beyond the
point at which Diophantus had left off.
lamblichus. lamblichus, circ. 350, to whom we owe a
valuable work on the Pythagorean discoveries and doctrines,
seems also to have studied the properties of numbers. He
enunciated the theorem that if a number which is equal to the
sum of three integers of the form '^n, 37i- 1, 37i-2 be taken,
and if the separate digits of. this number be added, and if the
separate digits of the result be again added, and so on, then the
final result will be 6 : for instance, the sum of 54, 53, and 52 is
159, the sum of the separate digits of 159 is 15, the sum of the
CH.v] HYPATIA. THE ATHENIAN SCHOOL 111
separate digits of 15 is 6. To any one confined to the usual
Greek numerical notation this must have been a difficult result
to prove : possibly it was reached empirically.
The names of two commentators will practically conclude the
long roll of Alexandrian mathematicians.
Theon. The first of these is Theon of Alexandria^ who
flourished about 370. He was not a mathematician of special
note, but we are indebted to him for an edition of Euclid's
Elements and a commentary on the Almagest ; the latter ^ gives
a great deal of miscellaneous information about the numerical
methods used by the Greeks.
Hsrpatia. The other was Hi/patia the daughter of Theon.
She was more distinguished than her father, and was the last
Alexandrian mathematician of any general reputation : she wrote
a commentary on the Comes of Apollonius and possibly some
other works, but none of her writings are now extant. She was
murdered at the instigation of the Christians in 415.
The fate of Hypatia may serve to remind us that the Eastern
Christians, as soon as they became the dominant party in the
state, showed themselves bitterly hostile to all forms of learning.
That very singleness of purpose which had at first so materially
aided their progress developed into a one-sidedness which refused
to see any good outside their own body ; and all who did not
actively assist them were persecuted. The final establishment of
Christianity in the East marks the end of the Greek scientific
schools, though nominally they continued to exist for two
hundred years more.
The Athenian School {in the fifth century).^
The hostility of the Eastern church to Greek science is further
illustrated by the fall of the later Athenian school. This school
1 It was translated with comments by M, Hahna and published at Paris
in 1821.
^ See Untersuchungen iiber die neu aufge/undene7i Scholien des Proklus,
by J. H. Knoche, Herford, 1865.
112 THE SECOND ALEXANDRIAN SCHOOL [ch.v
occupies but a small space in our history. Ever since Plato's
time a certain number of professional mathematicians had lived
at Athens ; and about the year 420 this school again acquired
considerable reputation, largely in consequence of the numerous
students who after the murder of Hypatia migrated there
from Alexandria. Its most celebrated members were Proclus,
Damascius, and Eutocius.
Proclus. Proclus was born at Constantinople in February
412 and died at Athens on April 17, 485. He wrote a com-
mentary 1 on the first book of Euclid's Elements, which contains
a great deal of valuable information on the history of Greek
mathematics : he is verbose and dull, but luckily he has pre-
served for us quotations from other and better authorities.
Proclus was succeeded as head of the school by Marinus, and
Marinus by Isidorus.
Damascius. Eutocius. Two pupils of Isidorus, who in
their turn subsequently lectured at Athens, may be mentioned
in passing. One of these, Damascius of Damascus, circ. 490,
is commonly said to have added to Euclid's Elements a fifteenth
book on the inscription of one regular solid in another, but his
authorship of this has been questioned by some writers. The other,
Eutocius, circ. 510, wrote commentaries on the first four books
of the Conies of ApoUonius and on various works of Archimedes.
This later Athenian school was carried on under great
difiiculties owing to the opposition of the Christians. Proclus,
for example, was repeatedly threatened with death because he
was "a philosopher." His remark, "after all my body does
not matter, it is the spirit that I shall take with me when
I die," which he made to some students who had offered to
defend him, has been often quoted. The Christians, after
several ineffectual attempts, at last got a decree from Justinian
in 529 that " heathen learning " should no longer be studied at
Athens. That date therefore marks the end of the Athenian school.
The church at Alexandria was less influential, and the city
was more remote from the centre of civil power. The schools
^ It has been edited by G. Friedlein, Leipzig, 1873,
CH.v] ROMAN MATHEMATICS 113
there were thus suffered to continue, though their existence was
of a precarious character. Under these conditions mathematics
continued to be read in Egypt for another hundred years, but
all interest in the study had gone.
Roman Mathematics}
I ought not to conclude this part of the history without any
mention of Roman mathematics, for it was through Rome that
mathematics first passed into the curriculum of medieval Europe,
and in Rome all modern history has its origin. There is, how-
ever, very little to say on the subject. The chief study of the
place was in fact the art of government, whether by law, by
persuasion, or by those material means on which all government
ultimately rests. There were, no doubt, professors who could
teach the results of Greek science, but there was no demand for
a school of mathematics. Italians who wished to learn more
than the elements of the science went to Alexandria or to places
which drew their inspiration from Alexandria.
The subject as taught in the mathematical schools at Rome
seems to have been confined in arithmetic to the art of calcula-
tion (no doubt by the aid of the abacus) and perhaps some of
the easier parts of the work of Nicomachus, and in geometry
to a few practical rules ; though some of the arts founded on
a knowledge of mathematics (especially that of surveying) were
carried to a high pitch of excellence. It would seem also that
special attention was paid to the representation of numbers by
signs. The manner of indicating numbers up to ten by the use
of fingers must have been in practice from quite early times, but
about the first century it had been developed by the Romans
into a finger- symbolism by which numbers up to 10,000 or
perhaps more could be represented : this would seem to have
been taught in the Roman schools. It is described by Bede,
and therefore would seem to have been known as far west as
^ The subject is discussed by Cantor, chaps, xxv, xxvi, and xxvii ; also
by Hankel, pp. 294-304.
I
114 THE SECOND ALEXANDRIAN SCHOOL [ce. v
Britain ; Jerome also alludes to it ; its use has still survived in
tlie Persian bazaars.
I am not acquainted with any Latin work on the principles
of mechanics, but there were numerous books on the practical
side of the subject which dealt elaborately with architectural
and engineering problems. We may judge what they were like
by the Mathematici Veteres, which is a collection of various
short treatises on catapults, engines of war, &c. : and by the
Keo-Tot, written by Sextus Julius Africarius about the end of
the second century, part of which is included in the Mathematici
Veteres, which contains, amongst other things, rules for finding
the breadth of a river when the opposite bank is occupied by an
enemy, how to signal with a semaphore, &c.
In the sixth century Boethius published a geometry containing
a few propositions from Euclid and an arithmetic founded on
that of Nicomachus ; and about the same time Cassiodorus
discussed the foundation of a liberal education which, after the
preliminary trivium of grammar, logic, and rhetoric, meant the
quadrivium of arithmetic, geometry, music, and astronomy.
These works were written at Rome in the closing years of
the Athenian and Alexandrian schools, and I therefore mention
them here, but as their only value lies in the fact that they
became recognized text-books in medieval education I postpone
their consideration to chapter viii.
Theoretical mathematics was in fact an exotic study at Rome ;
not only was the genius of the people essentially practical, but,
alike during the building of their empire, while it lasted, and under
the Goths, all the conditions were unfavourable to abstract science.
On the other hand, Alexandria was exceptionally well placed
to be a centre of science. From the foundation of the city to
its capture by the Mohammedans it was disturbed neither by
foreign nor by civil war, save only for a few years when the
rule of the Ptolemies gave way to that of Rome : it was wealthy,
and its rulers took a pride in endowing the university : and
lastly, just as in commerce it became the meeting-place of the
east and the west, so it had the good fortune to be the dwelling-
CH.v] EXD OF SECOND ALEXANDRIAN SCHOOL 115
place alike of Greeks and of various Semitic people ; the one
race shewed a peculiar aptitude for geometry, the other for
sciences which rest on measurement. Here too, however, as
time went on the conditions gradually became more unfavour-
able, the endless discussions on theological dogmas and the
increasing insecurity of the empire tending to divert men's
thoughts into other channels.
End of the Second Alexandrian School.
The precarious existence and unfruitful history of the last
two centuries of the second Alexandrian School need no record.
In 632 Mohammed died, and within ten years his successors
had subdued Syria, Palestine, Mesopotamia, Persia, and Egypt.
The precise date on which Alexandria fell is doubtful, but the
most reliable Arab historians give December 10, 641 — a date
which at any rate is correct within eighteen months.
With the fall of Alexandria the long history of Greek
mathematics came to a conclusion. It seems probable that
the greater part of the famous university library and museum
had been destroyed by the Christians a hundred or two
hundred years pre\iously, and what remained was unvalued
and neglected. Some two or three years after the first capture
of Alexandria a serious revolt occurred in Egypt, which was
ultimately put down with great severity. I see no reason to
doubt the truth of the account that after the capture of the
city the Mohammedans destroyed such university buildings and
collections as were still left. It is said that, when the Arab
commander ordered the library to be burnt, the Greeks made
such energetic protests that he consented to refer the matter to
the caliph Omar. The caliph returned the answer, " As to the
books you have mentioned, if they contain what is agreeable
with the book of God, the book of God is sufficient without
them ; and, if they contain what is contrary to the book of God,
there is no need for them ; so give orders for their destruction."
The account goes on to say that they were burnt in the public baths
of the city, and that it took six months to consume them all.
116
CHAPTER VI. • ^
THE BYZANTINE SCHOOL.
641-1453.
It will be convenient to consider the Byzantine school in
connection with the history of Greek mathematics. After the
capture of Alexandria by the Mohammedans the majority of the
philosophers, who previously had been teaching there, migrated
to Constantinople, which then became the centre of Greek learn-
ing in the East and remained so for 800 years. But though
the history of the Byzantine school stretches over so many
years — a period about as long as that from the Norman Con-
quest to the present day — it is utterly barren of any scientific
interest; and its chief merit is that it preserved for us the
works of the different Greek schools. The revelation of these
works to the West in the fifteenth century was one of the most
important sources of the stream of modern European thought,
and the history of the Byzantine school may be summed up by-
saying that it played the part of a conduit-jiipe in conyeyingto
us the results of an earlier and brighter age.
The time was one of constant war, and men's minds during
the short intervals of peace were mainly occupied with theo-
logical subtleties and pedantic scholarship. I should not have
mentioned any of the following writers had they lived in the
Alexandrian period, but in default of any others they may be
noticed as illustrating the character of the school. I ought also,
cH.vi] THE BYZANTINE SCHOOL 117
perhaps, to call the attention of the reader explicitly to the fact
that I am here departing from chronological order, and that the
mathematicians mentioned in this chapter were contemporaries
of those discussed in the chapters devoted to the mathematics
of the middle ages. The Byzantine school was so isolated that
I deem this the best arrangement of the subject.
Hero. One of the earliest members of the Byzantine school
was Hero of Constantinople, circ. 900, sometimes called the
younger to distinguish him from Hero of Alexandria. Hero
would seem to have written on geodesy and mechanics as applied
to engines of war.
During the t^nth century two emperors, Leo VI. and Con-
stantine VII., shewed considerable interest in astronomy and
mathematics, but the stimulus thus given to the study of these
subjects was only temporary.
Psellus. In the eleventh century Michael Psellns, born in
1020, wrote a pamphlet^ on the quadrivium : it is now in the
National Library at Paris.
In the fourteenth century we find the names of three monks
who paid attention to mathematics.
Planudes. Barlaam. Argynis. The first of the three
was Maximus Planudes? He wrote a commentary on the
first two books of the Arithmetic of Diophantus ; a work on
Hindoo arithmetic in which he used the Arabic numerals;
and another on proportions which is now in the National
Library at Paris. The next was a Calabrian monk named
Barlaam, who was born in 1290 and died in 1348. He
^vas the author of a work, Logistic, on the Greek methods
of calculation from which we derive a good deal of informa-
tion as to the way in which the Greeks treated numerical
fractions.^ Barlaam seems to have been a man of great
* It was printed at Bale in 1536. Psellus also wrote a Compendium
Maihe?naticum which was printed at Leyden in 1647.
^ His arithmetical commentary was published by Xylander, Bale, 1575 :
his work on Hindoo arithmetic, edited by C. J. Gerhardt, was published at
Halle, 1865.
^ Barlaam's Logistic, edited by Dasypodius, was published at Strassburg,
1572 ; another edition was issued at Paris in 1600,
118 THE BYZANTINE SCHOOL [ch. vi
intelligence. He was sent as an ambassador to the Pope at
Avignon, and acquitted himself creditably of a difficult mission ;
while there he taught Greek to Petrarch. He was famous at
Constantinople for the ridicule he threw on the preposterous
pretensions of the monks at Mount Athos who taught that those
who joined them could, by steadily regarding their bodies,
see a mystic light which was the essence of God. Barlaam
advised them to substitute the light of reason for that of their
bodies — a piece of advice which nearly cost him his life.
The last of these monks was Isaac Argyrus^ who died in 1372.
He wrote three astronomical tracts, the manuscripts of which
are in the libraries at the Vatican, Leyden, and Vienna : one
on geodesy, the manuscript of which is at the Escurial : one
on geometry, the manuscript of which is in the National Library
at Paris : one on the arithmetic of Nicomachus, the manuscript
of which is in the National Library at Paris : and one on
trigonometry, the manuscript of which is in the Bodleian at
Oxford.
Rhabdas. In the fourteenth or perhaps the fifteenth century
Nicholas Rhabdas of Smyrna wrote two papers ^ on arithmetic
which are now in the National Library at Paris. He gave an
account of the finger-symbolism ^ which the Romans had intro-
duced into the East and was then current there.
Pacliymeres. Early in the fifteenth century Pachynieres
wrote tracts on arithmetic, geometry, and four mechanical
machines.
Moschopulus. A few years later Emmanuel Moschojmlus^
who died in Italy circ. 1460, wrote a treatise on magic squares.
A magic square ^ consists of a number of integers arranged in
the form of a square so that the sum of the numbers in every
row, in every column, and in each diagonal is the same. If the
1 They have been edited by S. P. Tannery, Paris, 1886.
2 See above, page 113.
^ On the formation and history of magic squares, see my Mathematical
Recreations, London, sixth edition, 1914, chap. vii. On the work of
Moschopulus, see S. Giinther's Oeschichte der mathematischen Wisseu'
schaften, Leipzig, 1876, chap. iv.
CH. Vl]
MAGIC SQUARES
119
integers be the consecutive numbers from 1 to ?i-, the square is
said to be of the 7^th order, and in this case the sum of the
numbers in any row, column, or diagonal is equal to ^n{ji^ + 1).
Thus the first 16 integers, arranged in either of the forms given
below, form a magic square of the fourth order, the sum of
1
15
14
4
12
6
7
9
8
10
11
5
13
3
2
10
15
10
5
11
8
3
^
4
u\.\
14
1
2
13
7
12
the numbers in every row, every column, and each diagonal
being 34.
In the mystical philosophy then current certain metaphysical
ideas were often associated with particular numbers, and thus it
was natural that such arrangements of numbers should attract
attention and be deemed to possess magical properties. The
theory of the formation of magic squares is elegant, and several
distinguished mathematicians have written on it, but, though
interesting, I need hardly say it is not useful. Moschopulus
seems to have been the earliest European writer who attempted
to deal with the mathematical theory, but his rules apply only
to odd squares. The astrologers of the fifteenth and sixteenth
centuries w^ere much impressed by such arrangements. In
particular the famous Cornelius Agrippa (1486-1535) constructed
magic squares of the orders 3, 4, 5, 6, 7, 8, 9, which were asso-
ciated respectively with the seven astrological "planets," namely,
Saturn, Jupiter, Mars, the Sun, Venus, Mercury, and the Moon.
He taught that a square of one cell, in which unity was inserted,
represented the unity and eternity of God ; while the fact that
a square of the second order could not be constructed illustrated
the imperfection of the four elements, air, earth, fire, and water ;
and later writers added that it was symbolic of original sin. A
magic square engraved on a silver plate was often prescribed as
a charm against the plague, and one (namely, that in the first
120 THE BYZANTINE SCHOOL [ch.vi
diagram on the last page) is drawn in the picture of melancholy
painted about the year 1500 by Albrecht Diirer. Such charms
are still worn in the East.
Constantinople was captured by the Turks in 1453, and the
last semblance of a Greek school of mathematics then dis-
appeared. Numerous Greeks took refuge in Italy. In the
West the memory of Greek science had vanished, and even the
names of all but a few Greek writers were unknown ; thus the
books brought by these refugees came as a revelation to Europe,
and, as we shall see later, gave a considerable stimulus to the
study of science.
^
121
Ycv^^^
CHAPTER VIL
SYSTEMS OF NUMERATION AND PRIMITIVE ARITHMETIC.*
I HAVE in many places alluded to the Greek method of express-
ing numbers in writing, and I have thought it best to defer to
this chapter the whole of what I wanted to say on the various
systems of numerical notation which were displaced by the
system introduced by the Arabs.
First, as to symbolism and language. The plan of indicating
numbers by the digits of one or both hands is so natural that we
find it in universal use among early races, and the members of
all tribes now extant are able to indicate by signs numbers at
least as high as ten : it is stated that in some languages the
names for the first ten numbers are derived from the fingers used
to denote them. For larger numbers we soon, however, reach a
limit beyond which primitive man is unable to count, while as
far as language goes it is well known that many tribes have no
word for any nuniber higher than ten, and some have no word
for any number beyond four, all higher numbers being expressed
by the words plenty or heap : in connection with this it is worth
remarking that (as stated above) the Egyptians used the symbol
for the word heap to denote an unknown quantity in algebra.
The number five is generally represented by the open hand,
^ Tlie subject of this chapter has been discussed by Cantor and by Hankel.
See also thePhilosophy of Arithmetic by John Leslie, second edition, Edinburgh,
1820. Besides these authorities the article on Arithmetic by George Peacock
in the Encyclopaedia Metropolitana, Pure Sciences, London, 1845 ; E. B.
Tylor's Primitive Culture, London, 1873 ; Les signes mimSraux et I'arith-
nietique chez les peuples de V a^itiquite . . hy T. H. Martin, Rome, 1864 ; and
Die Zahlzeichen...\)y G. Friedlein, Erlangen, 1869, should be consulted.
122 SYSTEMS OF NUMERATION [ch. vii
I and it is said that in almost all languages the words five and
^-^and are derived from the same root. It is possible that in
early times men did not readily count beyond five, and things if
more numerous were counted by multiples of it. It may be that
the Roman symbol X for ten represents two "V's, placed apex
to apex, and, if so, this seems to point to a time when things were
counted by fives. ^ In connection with this it is worth noticing that
both in Java and among the Aztecs a week consisted of five days.
The members of nearly all races of which we have now any
knowledge seem, however, to have used the digits of both hands
to represent numbers. They could thus count up to and in-
cluding ten, and therefore were led to take ten as their radix of
notation. In the English language, for example, all the words
for numbers higher than ten are expressed on the decimal
system : those for 1 1 and 1 2, which at first sight seem to be
exceptions, being derived from Anglo-Saxon words for one and
ten and two and ten respectively.
Some tribes seem to have gone further, and by making use of
their toes were accustomed- to count by multiples of twenty.
The Aztecs, for example, are said to have done so. It may be
noticed that we still count some things (for instance, sheep) by
scores, the word score signifying a notch or scratch made on the
completion of the twenty ; while the French also talk of quatre-
vingts, as though at one time they counted things by multiples
of twenty. I am not, however, sure whether the latter argu-
ment is worth anything, for I have an impression that I have
seen the w^ord octante in old French books ; and there is no
question ^ that sep.tante and nonante were at one time common
words for seventy and ninety, and indeed they are still retained
in some dialects.
The only tribes of whom I have read who did not count in
terms either of five or of some multiple of five are the Bolans
of West Africa who are said to have counted by multiples of
^ See also the Odyssey, iv, 413-415, in which apparently reference is made
to a similar custom.
2 See, for example, V. M. de Kempten's Practique...d ciffrer, Antwerp,
1556.
cH.vii] SYSTEMS OF NUMERATION 123
seven, and the Maories who are said to have counted by-
multiples of eleven.
Up to ten it is comparatively easy to count, but primitive
people • find great difficulty in counting higher numbers ;
apparently at first this difficulty was only overcome by the
method (still in use in South Africa) of getting two men, one
to count the units up to ten on his fingers, and the other to
count the number of groups of ten so formed. To us it is
obvious that it is equally effectual to make a mark of some
kind on the completion of each group of ten, but it is alleged
that the members of many tribes never succeeded in counting
numbers higher than ten unless by the aid of two men.
Most races who shewed any aptitude for civilization pro-
ceeded fm'ther and invented a way of representing numbers by
means of pebbles or counters arranged in sets of ten ; and this
in its turn developed into the abacus or swan-pan. This instru-
ment was in use among nations so widely separated as the
Etruscans, Greeks, Egyptians, Hindoos, Chinese, and Mexicans ;
and was, it is believed, invented independently at several
different centres. It is still in common, use in Russia, China,
and Japan.
In its simplest form (see Figure 1, on the next page) the abacus
consists of a wooden board with a number of grooves cut in it,
or of a table covered with sand in which grooves are made with
the fingers. To represent a number, as many counters or pebbles
are put on the first groove as there are units, as many on the
second as there are tens, and so on. When by its aid a number
of objects are counted, for each object a pebble is put on the
first groove ; and, as soon as there are ten pebbles there, they
are taken off and one pebble put on the second groove ; and so
on. It was sometimes, as in the Aztec quipus, made with a
number of parallel wires or strings stuck in a piece of wood on
which beads could be threaded ; and in that form is called a
swan-pan. In the number represented in each of the instru-
ments drawn on the next page there are seven thousands, three
hundreds, no tens, and five units, that is, the number is 7305.
124
SYSTEMS OF NUMERATION
[CH. VII
Some races counted from left to right, others from right to left,
but this is a mere matter of convention.
The Roman abaci seem to have been rather more elaborate.
They contained two marginal grooves or • wires, one with four
beads to facilitate the addition of fractions whose denominators
1
u
(
J
1
3
or
Figure 1.
X
%^'- ^
nTTiTr
I n i i ? 1
I Hi! I
'' i i 4 :: i i
Figure 2.
■J^
Figure 3.
were four, and one with twelve beads for fractions whose
denominators w^ere twelve : but otherwise they do not differ in
principle from those described above. They were commonly
made to represent numbers up to 100,000,000. The Greek
abaci were similar to the Roman ones. The Greeks and Romans
Used their abaci as boards on which they played a game some-
thing like backgammon.
In the Russian tschotil (Figure 2) the instrument is intiproved
CH. vii] THE ABACUS 125
by having the wires set in a rectangular frame, and ten (or nine)
beads are permanently threaded on each of the wires, the wires
being considerably longer than is necessary to hold them. If
the frame be held horizontal, and all the beads be towards one
side, say the lower side of the frame, it is possible to represent
any number by pushing towards the other or upper side as
many beads on the first wire as there are units in the number,
as many beads on the second wire as there are tens in the
number, and so on. Calculations can be made somewhat more
rapidly if the five beads on each wire next to the upper side
be coloured diff'erently to those next to the lower side, and they
can be still further facilitated if the first, second, ..., ninth
counters in each column be respectively marked with symbols
for the numbers 1, 2, ..., 9. Gerbert^ is said to have intro-
duced the use of such marks, called apices, towards the close
of the tenth century.
Figure 3 represents the form of swan-pan or saroban in
common use in China and Japan. There the development is
carried one step further, and five beads on each wire are replaced
by a single bead of a difierent form or on a different division,
but apices are not used. I am told that an expert Japanese can,
•by the aid of a swan-pan, add numbers as rapidly as they can
be read out to him. It will be noticed that the instrument
represented in Figure 3 is made so that two numbers can be
expressed at the same time on it.
The use of the abacus in addition and subtraction is evident.
It can be used also in multiplication and division ; rules for these
processes, illustrated by examples, are given in various old works
I ^^on^arithmetic.2
P The abacus obviously presents a concrete way of representing
a number in the decimal system of notation, that is, by means
of the local value of the digits. Unfortunately the method of
writing numbers developed on different lines, and it was not
^ See below, ]>age 138.
- For example in K. Record's Grounde of Arfes, edition of 1610, London,
pp. 225-262.
126 SYSTEMS OF NUMERATION [ch.vii
until. about the thirteenth century of our era, when a symbol
zero used in conjunction with nine other symbols was introduced,
that a corresponding notation in writing was adopted in Europe.
Next, as to the means of representing numbers in writing.
In general we may say that in the earliest times a number
was (if represented by a sign and not a word) indicated by the
requisite number of strokes. Thus in an inscription from
Tralles in Caria of the date 398 B.C. the phrase seventh year is
represented by ereo? | | | | | | | . These strokes may have been
mere marks ; or perhaps they originally represented fingers,
since in the Egyptian hieroglyphics the symbols for the
numbers 1, 2, 3, are one, two, and three fingers respectively,
though in the later hieratic writing these symbols had become
reduced to straight lines. Additional symbols for 10 and 100
were soon introduced : and the oldest extant Egyptian and
Phoenician writings repeat the symbol for unity as many times
(up to 9) as was necessary, and then repeat the symbol for ten
as many times (up to 9) as was necessary, and so on. No
specimens of Greek numeration of a similar kind are in existence,
but there is every reason to believe the testimony of lamblichus
who asserts that this was the method by which the Greeks first
expressed numbers in writing.
This way of representing numbers remained in current use
throughout Roman history ; and for greater brevity they or
the Etruscans added separate signs for 5, 50, itc. The Roman
symbols are generally merely the initial letters of the names of
the numbers ; thus c stood for centum or 100, M for mille or
1000. The symbol v for 5 seems to have originally represented
an open palm with the thumb extended. The symbols l for 50
and D for 500 are said to represent the upper halves of the
symbols used in early times for c and m. The subtractive forms
like IV for iiii are probably of a later origin.
Similarly in Attica five was denoted by 11, the first letter of
TTcvre, or sometimes by T ; ten by A, the initial letter of ScKa ; a
hundred by H for cKarov ; a thousand by X for x^^'-^'- \ while
50 was represented by a A written inside a 11 ; and so on.
CH.vii] THE REPRESENTATION OF NUMBERS 127
These Attic symbols continued to be used for inscriptions and
formal documents until a late date.
This, if a clumsy, is a perfectly intelligible system ; but the
Greeks at some time in the third century before Christ abandoned
it for one which offers no special advantages in denoting a given
number, while it makes all the operations of arithmetic exceed-
ingly difficult. In this, which is* known from the place where it
was introduced as the Alexandrian system, the numbers from 1
to 9 are represented by the first nine letters of the alphabet ;
the tens from 10 to 90 by the next nine letters ; and the
hundreds from 100 to 900 by the next nine letters. To do this
the Greeks wanted 27 letters, and as their alphabet contained
only 24, they reinserted two letters (the digamma and koppa)
which had formerly been in it but had become obsolete, and
introduced at the end another symbol taken from the Phoenician
alphabet. Thus the ten letters a to t stood respectively for the
numbers from 1 to 10 ; the next eight letters for the multiples
of 10 from 20 to 90 ; and the last nine letters for 100, 200, etc.,
up to 900. Intermediate numbers like 1 1 were represented as
the sum of 10 and 1, that is, by the symbol la. This afforded
a notation for all numbers up to 999 ; and by a system of
suffixes and indices it was extended so as to represent numbers
up to 100,000,000.
There is no doubt that at first the results were obtained by
the use of the abacus or some similar mechanical method, and
that the signs were only employed to record the result ; the idea
of operating with the symbols themselves in order to obtain the
results is of a later growth, and is one with which the Greeks
never became familiar. The non-progressive character of Greek
arithmetic may be partly due to their unlucky adoption of the
Alexandrian system which caused them for most practical pur-
poses to rely on the abacus, and to supplement it by a table of
multiplications which was learnt by heart. The results of the
nuiltiplication or division of numbers other than those in the
multiplication table might have been obtained by the use of the
abacus, but in fact they were generally got by repeated additiolis
128 SYSTEMS OF NUMERATION [ch. vii
and subtractions. Thus, as late as 944, a certain mathema-
tician who in the course of his work wants to multiply 400 by
5 finds the result by addition. The same writer, when he wants
to divide 6152 by 15, tries all the multiples of 15 until he gets
to 6000, this gives him 400 and a remainder 152 ; he then
begins again with all the multiples of 15 until he gets to 150,
and this gives him 10 and a remainder 2. Hence the answer is
410 with a remainder 2.
A few mathematicians, however, such as Hero of Alexandria,
Theon, and Eutocius, multiplied and divided in what is essenti-
ally the same way as we do. Thus to multiply 18 by 13 they
proceeded as follows : —
ty + i7; = (i-|-y) (t + tj) 13 X 18 = (10 -f 3) (10-1-8)
= 6(1 + 7;) +7(1 + 7]) =10(10 + 8)-f3 (lO-i-8)
= p + Tr + X + K8 = 100 + 80 -f 30 -h 24
= (rA8 =234.
I suspect that the last step, in which they had to add four
numbers together, was obtained by the aid of the abacus.
These, however, were men of exceptional genius, and we must
recollect that for all ordinary purposes the art of calculation was
performed only by the use of the abacus and the multiplication
table, while the term arithmetic was confined to the theories of
ratio, proportion, and of numbers.
All the systems here described were more or less clumsy, and
they have been displaced among civilized races by the Arabic
system in which there are ten digits or symbols, namely, nine
for the first nine numbers and another for zero. In this system
an integral number is denoted by a succession of digits, each
digit representing the product of that digit and a power of ten,
and the number being equal to the sum of these products.
Thus, by means of the local value attached to nine symbols and
a symbol for zero, any number in the decimal scale of notation
can be expressed. The history of the development of the science
of arithmetic with this notation will be considered below in
chapter xi.
129
SECOND PERIOD.
^atljematica of tire iEitriik ^gea antr Utnaizzantt*
This period begins about the sixth century, and may he said
to end ivith the invention of analytical gemnetry and df the
infinitesimal calculus. The characteristic feature of this period
is the creation or development of modem arithmetic, algebra,
and trigonometry.
In this period I consider first, in chapter viii, the rise of
learning in Western Europe, and the mathematics of the middle
ages. Next, in chapter ix, I discuss the nature and history of
Hindoo and Arabian mathematics, and in chapter x their intro-
duction into Europe. Then, in chapter xi, I trace the subse-
quent progress of arithmetic to the year 1637. Next, in chapter
XII, I treat of the general history of mathematics during the
renaissance, from the invention of printing to the beginning of
the seventeenth century, say, from 1450 to 1637; this contains
an account of the commencement of the modern treatment of
arithmetic, algebra, and trigonometry. Lastly, in chapter
XIII, I consider the revival of interest in mechanics, exi)eri-
mental methods, and pure geometry which marks the last few
years of this period, and serves as a connecting link between the
mathematics of the renaissance and the mathematics of modern
times.
131
CHAPTER VIII.
THE RISE OF LEARNING IN WESTERN EUROPE.^
CIRC. 600-1200.
Education in the sixths seventh, and eighth centuries.
The first few centuries of this second period of our history are
singularly barren of interest ; and indeed it would be strange if
we found science or mathematics studied by those who lived in
a condition of perpetual war. Broadly speaking we may say
that from the sixth to the eighth centuries the only places of
study in western Europe were the Benedictine monasteries.
We may find there some slight attempts at a study of literature ;
but the science usually taught was confined to the use of the
abacus, the method of keeping accounts, and a knowledge of
the rule by which the date of Easter could be determined. Nor
was this unreasonable, for the monk had renounced the world,
and there was no reason why he should learn more science than
was required for the services of the Church and his monastery.
The traditions of Greek and Alexandrian learning gradually
died away. Possibly in Rome and a few favoured places copies
of the works of the great Greek mathematicians were obtain-
^ The mathematics of this period has been discussed by Cantor, by
S. Giinther, Geschichte des nuxthematischen Untemchtes im deutschen
Mittelalter, Berlin, 1887 ; and by H. Weissenborn, Gerhert, Beitrage zur
Kenntniss der Matliematik des Mittelalters, Berlin, 1888 ; and Zur Oeschickte
der Einfiihrung der jetzigen Ziffers, Berlin, 1892.
132 THE RISE OF LEARNING IN EUROPE [ch. viii
able, though with difficulty, but there were no students, the
books were unvalued, and in time became very scarce.
Three authors of the sixth century — Boethius, Cassiodorus,
and Isidorus — may be named whose writings serve as a con-
necting link between the mathematics of classical and of
medieval times. As their works remained standard text-books
for some six or seven centuries it is necessary to mention them,
but it should be understood that this is the only reason for
doing so ; they show no special mathematical ability. It will
be noticed that these authors were contemporaries of the later
Athenian and Alexandrian schools.
Boethius. Anicius Manlius Severinus Boethius^ or as the
name is sometimes written Boetms, born at Rome about 475
and died in 526, belonged to a family which for the two pre-
ceding centuries had been esteemed one of the most illustrious
in Rome. It was formerly believed that he was educated at
Athens : this is somewhat doubtful, but at any rate he was
exceptionally well read in Greek literature and science.
Boethius would seem to have wished to devote his life to
literary pursuits; but recognizing "that the world would be
happy only when kings became philosophers or philosophers
kings," he yielded to the pressure put on him and took an
active share in politics. He was celebrated for his extensive
charities, and, what in those days was very rare, the care that
he took to see that the recipients were worthy of them. He
was elected consul at an unusually early age, and took advantage
of his position to reform the coinage and to introduce the public
use of sun-dials, water-clocks, etc. He reached the height of
his prosperity in 522 when his two sons were inaugurated as
consuls. His integrity and attempts to protect the provincials
from the plunder of the public officials brought on him the
hatred of the Court. He was sentenced to death while absent
from Rome, seized at Ticinum, and in the baptistery of the
church there tortured by drawing a cord round his head till
the eyes were forced out of the sockets, and finally beaten
to death with clubs on October 23, 526. Such at least is the
CH.VIII] BOETHIUS. ISIDORUS 133
account that has come down to us. At a later time his merits
were recognized, and tombs and statvies erected in his honour by
the state.
Boethius was the last Roman of note who studied the
language and literature of Greece, and his works afforded to
medieval Europe some glimpse of the intellectual life of the
old world. His importance in the history of literature is thus
very great, but it arises merely from the accident of the time
at which he lived. After the introduction of Aristotle's works
in the thirteenth century his fame died away, and he has now
sunk into an obscurity which is as great as was once his
reputation. He is best known by his Consolatio, which was
translated by Alfred the Great into Anglo-Saxon. For our
purpose it is sufficient to note that the teaching of early
medieval mathematics was mainly founded on his geometry
and arithmetic.
(^His Geometry ^ consists of the enunciations (only) of the first
book of Euclid, and of a few selected propositions in the third
and fourth books, but with numerous practical applications to
finding areas, etc. He adds an appendix with proofs of the
first three propositions to shew that the enunciations may be
relied on. His Arithmetic is founded on that of Nicomachus.
Cassiodorus. A few years later another Roman, Magnus
Aurelius Cassiodorus, who was born about 490 and died in
566, published two works, De Institutione Divinarum Litte-
rarum and 'De Artibus ac Disciplinis, in which not only the
preliminary trivium of grammar, logic, and rhetoric were dis-
cussed, but also the scientific quadrivium of arithmetic, geometry,
music, and astronomy. These were considered standard works
during the middle ages; the former was printed at Venice
in 1598.
Isidoms. Isidorus, bishop of Seville, born in 570 and
died in 636, was the author of an encyclopaedic work in
twenty volumes called Origines, of which the third volume is
^ His works on geometry and arithmetic were edited by G. Friedlein,
Leipzig, 1867.
134 THE RISE OF LEARNING IN EUROPE [ch. viii
given up to the quadrivium. It was published at Leipzig in
1833.
The Cathedral and Conventiml Schools}
When, in the latter half of the eighth century, Charles the
Great had established his empire, he determined to promote
learning so far as he was able. He began by commanding
that schools should be opened in connection with every
cathedral and monastery in his kingdom ; an order which was
approved and materially assisted by the popes. It is interesting
to us to know that this was done at the instance and under the
direction of two Englishmen, Alcuin and Clement, who had
attached themselves to his court.
Alcuin. 2 Of these the more prominent was Alcuin, who
was born in Yorkshire in 735 and died at Tours in 804. He
was educated at York under archbishop Egbert, his " beloved
master," whom he succeeded as director of the school there.
Subsequently he became abbot of Canterbury, and was sent to
Rome by Offa to procure the pallium for archbishop Eanbald.
On his journey back he met Charles at Parma ; the emperor
took a great liking to him, and finally induced him to take up
his residence at the imperial court, and there teach rhetoric,
logic, mathematics, and divinity. Alcuin remained for many
years one of the most intimate and influential friends of Charles
and was constantly employed as a confidential ambassador ;
as such he spent the years 791 and 792 in England, and while
there reorganized the studies at his old school at York. In 801
he begged permission to retire from the court so as to be able
to spend the last years of his life in quiet : with difficulty he
obtained leave, and went to the abbey of St. Martin at Tours,
of which he had been made head in 796. He established a
^ See The Schools of Charles the Great and the Restoration of Education
in the Ninth Century by J. B. Mullinger, London, 1877.
2 See the life of Alcuin by F. Lorentz, Halle, 1829, translated by J. M.
Slee, London, 1837 ; Alcuin tmd sein Jahrhundert by K. Werner, Paderborn,
1876 ; and Cantor, vol. i, pp. 712-721.
cH.viii] • ALCUIN 135
school in connection with the abbey which became very
celebrated, and he remained and taught there till his death on
May 19, 804.
Most of the extant writings of Alcuin deal with theology
or history, but they include a collection of arithmetical pro-
positions suitable for the instruction of the young. The
majority of the propositions are easy problems, either determi-
nate or indeterminate, and are, I presume, founded on works
with which he had become acquainted when at Rome. The
following is one of the most difficult, and will give an idea of
the character of the work. ^ If one hundred bushels of corn be
distributed among one hundred people in such a manner that
each man receives three bushels, each woman two, and each
child half a bushel : how many men, women, and children
were there ? The general solution is (20 - 3n) men, bn women,
and (80 - 2n) children, where n may have any of the values
1, 2, 3, 4, 5, 6. Alcuin only states the solution for which
n = 3; that is, he gives as the answer 11 men, 15 women, and
74 children.
This collection however was the work of a man of excep-
tional genius, and probably we shall be correct in saying that
mathematics, if taught at all in a school, was generally con-
fined to the geometry of Boethius, the use of the abacus and
multiplication table, and possibly the arithmetic of Boethius;
while except in one of these schools or in a Benedictine cloister
it was hardly possible to get either instruction or opportunities
for study. It was of course natural that the works used should
come from Roman sources, for Britain and all the countries
included in the empire of Charles had at one time formed part
of the western half of the Roman empire, and their inhabitants
continued for a long time to regard Rome as the centre of
civilization, while the higher clergy kept up a tolerably constant
intercourse with Rome.
After the death of Charles many of his schools confined
themselves to teaching Latin, music, and theology, some
knowledge of which was essential to the worldly success of
136 THE RISE OF LEARNING IN EUROPE [ch.viii
tlie higher clergy. Hardly any science or mathematics was
taught, but the continued existence of the schools gave an
opportunity to any teacher whose learning or zeal exceeded
the narrow limits fixed by tradition ; and though there were
but few who availed themselves of the opportunity, yet the
number of those desiring instruction was so large that it
would seem as if any one who could teach was sure to attract
a considerable audience.
A few schools, where the teachers were of repute, became
large and acquired a certain degree of permanence, but even in
them the teaching was still usually confined to the trivium
and quadrivium. The former comprised the three arts of
grammar, logic, and rhetoric, but practically meant the art
of reading and writing Latin ; nominally the latter included
arithmetic and geometry with their applications, especially to
music and astronomy, but in fact it rarely meant more than
arithmetic sufficient to enable one to keep accounts, music for
the church services, geometry for the purpose of land-surveying,
and astronomy sufficient to enable one to calculate the feasts
and fasts of the church. The seven liberal arts are enumerated
in the line. Lingua^ tropus, ratio; numerus, tonus, angulus,
astra. Any student who got beyond the trivium was looked
on as a man of great erudition. Qui tria, qui septem, qui totum
scibile novit, as a verse of the eleventh century runs. The
special questions which then and long afterwards attracted
the best thinkers were logic and certain portions of transcen-
dental theology and philosophy.
We may sum the matter up by saying that during the
ninth and tenth centuries the mathematics taught was still
usually confined to that comprised in the two works of
Boethius together with the practical use of the abacus and the
multiplication table, tKough during the latter part of the time
a wider range of reading was undoubtedly accessible.
Gerbert.^ In the tenth century a man appeared who
^ Weissenborn, in the works already mentioned, treats Gerbert very fully ;
see also La Vie et les CEuvresde Qerhert, by A. Olleris, Clermont, 1867 ; Oer-
CH.viii] GERBERT 137
would in any age have been remarkable and who gave a great
stimulus to learning. This was Gerhert, an Aquitanian by
birth, who died in 1003 at about the age of fifty. His abilities
attracted attention to him even when a boy, and procured his
removal from the abbey school at Aurillac to the Spanish
march where he received a good education. He was in Rome
in 971, where his proficiency in music and astronomy excited
considerable interest^: but his interests were not confined to
these subjects, and he had already mastered all the branches of
the trivium and quadrivium, as then taught, except logic ; and
rt) learn this he moved to Rheims, which Archbishop Adalbero
had made the most famous school in Europe. Here he was at
once invited to teach, and so great was his fame that to him
Hugh Capet entrusted the education of his son Robert who
was afterwards king of France.
Gerbert was especially famous for his construction of abaci
and of terrestrial and celestial globes ; he was accustomed to
use the latter to illustrate his lectures. These globes excited
great admiration ; and he utilized this by offering to exchange
them for copies of classical Latin works, which seem already
to have become very scarce ; the better to effect this he ap-
pointed agents in the chief towns of Europe. To his efforts it
is believed we owe the preservation of several Latin works.
In 982 he received the abbey of Bobbie, and the rest of his life
was taken up with political affairs ; he became Archbishop of
Rheims in 991, and of Ravenna in 998 ; in 999 he was elected
Pope, when he took the title of Sylvester II. ; as head of the
Church, he at once commenced an appeal to Christendom to arm
and defend the Holy Land, thus forestalling Peter the Hermit by
a century, but he died on May 12, 1003, before he had time to
elaborate his plans. His library is, I believe, preserved in the
Vatican.
So remarkable a personality left a deep impress on his
hert von Aurillac, by K. Werner, second edition, Vienna, 1881 ; and Gerberti
...Opera mathematical edited by N. Bubnov, Berlin, 1899.
138 THE RISE OF LEARNING IN EUROPE [ch. viii
generation, and all sorts of fables soon began to collect around
his memory. It seems certain that he made a clock which
was long preserved at Magdeburg, and an organ worked by
steam which was still at Rheims two centuries after his death.
All this only tended to confirm the suspicions of his contem-
poraries that he had sold himself to the devil ; and the details
of his interviews with that gentleman, the powers he purchased,
and his effort to escape from his bargain when he was dying,
may be read in the pages of William of Malmesbury, Orderic
Vitalis, and Platina. To these anecdotes the first named
writer adds the story of the statue inscribed with the word's
" strike here," which having amused our ancestors in the Gesta
RoTiianorum has been recently told again in the Earthly
Paradise.
Extensive though his influence was, it must not be supposed
that Gerbert's writings shew any great originality. His mathe-
matical works comprise a treatise on arithmetic entitled De
Numerorum Divisione, and one on geometry. An improvement
in the abacus, attributed by some writers to Boethius, but which
is more likely due to Gerbert, is the introduction in every
column of beads marked by different characters, called apices,
for each of the numbers from 1 to 9, instead of nine exactly
similar counters or beads. These apices lead to a representation
of numbers essentially the same as the Arabic numerals. There
was however no symbol for zero ; the step from this concrete
system of denoting numbers by a decimal system on an abacus
to the system of denoting them by similar symbols in writing
seems to us to be a small one, but it would appear that Gerbert
did not make it. He found at Mantua a copy of the geometry
of Boethius, and introduced it into the medieval schools.
Gerbert's own work on geometry is of unequal ability; it includes
a few applications to land-surveying and the determination of
the heights of inaccessible objects, but much of it seems to be
copied from some Pythagorean text-book. In the course of it
he however solves one problem which was of remarkable
CH.VIII] ESTABLISHMENT OF UNIVERSITIES 139
difficulty for that time. The question is to find the sides of a
right-angled triangle whose hypotenuse and area are given.
He says, in effect, that if these latter be denoted respectively
by c and A^, then the lengths of the two sides will be
1{ v/c2 + U- + ^/c2 - 4A^} and i{Jc^ + 4/^2 - ^c' -^h^}.
Bemelinus. One of Gerbert's pupils, Bernelinus, published
a work on the abacus ^ which is, there is very little doubt, a
reproduction of the teaching of Gerbert. It is valuable as
indicating that the Arabic system of writing numbers was still
unknown in Europe.
The Early Medieval Universities.^
At the end of the eleventh century or the beginning of the
twelfth a revival of learning took place at several of these
cathedral or monastic schools ; and in some cases, at the same
time, teachers who were not members of the school settled in
its vicinity and, with the sanction of the authorities, gave
lectures which were in fact always on theology, logic, or civil
•law. As the students at these centres grew in numbers, it
became desirable to act together whenever any interest common
to all was concerned. The association thus formed was a sort
of guild or trades union, or in the language of the time a uni-
versitas magistrorum et scholarium. This was the first stage
in the development of the earliest medieval universities. In
some cases, as at Paris, the governing body of the university
was formed by the teachers alone, in others, as at Bologna, by
both teachers and students ; but in all cases precise rules for
the conduct of business and the regulation of the internal
economy of the guild were formulated at an early stage in its
history. The municipalities and numerous societies which
^ It is reprinted in Olleris's edition of Gerbert's works, pp. 311-326.
^ See the Universities of Europe in the Middle Ages by H. Rashdall,
Oxford, 1895 ; Die Universitdten des MittelaUers his 1400 by P. H. Denifle,
1885 ; and vol. i of the University of Cambridge by J. B. Mullinger,
Cambridge, 1873.
140 THE RISE OF LEARNING IN EUROPE [cii. viii
existed in Italy supplied plenty of models for the construction
of such rules, but it is possible that some of the regulations
were derived from those in force in the Mohammedan schools
at Cordova.
We are, almost inevitably, unable to fix the exact date of
the commencement of these voluntary associations, but they
existed at Paris, Bologna, Salerno, Oxford, and Cambridge
before the end of the twelfth century : these may be considered
the earliest universities in Europe. The instruction given at
Salerno and Bologna was mainly technical — at Salerno in medi-
cine, and at Bologna in law — and their claim to recognition as
universities, as long as they were merely technical schools, has
been disputed.
Although the organization of these early universities was
independent of the neighbouring church and monastic schools
they seem in general to have been, at any rate originally, asso-
ciated with such schools, and perhaps indebted to them for the
use of rooms, etc. The universities or guilds (self-governing
and formed by teachers and students), and the adjacent schools
(under the direct control of church or monastic authorities), con-
tinued to exist side by side, but in course of time the latter
diminished in importance, and often ended by becoming subject
to the rule of the university authorities. Nearly all the medieval
universities grew up under the protection of a bishop (or abbot),
and were in some matters subject to his authority or to that of
his chancellor, from the latter of whom the head of the univer-
sity subsequently took his title. The universities, however,
were not ecclesiastical organizations, and, though the bulk of
their members were ordained, their direct connection with the
Church arose chiefly from the fact that clerks were then the
only class of the community who were left free by the state to
pursue intellectual studies.
A universitas magistrorum et scholarium, if successful in
attracting students and acquiring permanency, always sought
special legal privileges, such as the right to fix the price of
provisions and the power to try legal actions in which its
CH.viii] EARLY EUROPEAN UNIVERSITIES 141
members were concerned. These privileges generally led to a
recognition of its power to grant degrees which conferred a right
of teaching anywhere within the kingdom. The university was
frequently incorporated at or about the same time. Paris
received its charter in 1200, and probably w^as the earliest
university in Europe thus officially recognized. Legal privileges
were conferred on Oxford in 1214, and on Cambridge in 1231 :
the development of Oxford and Cambridge follow^ed closely the
precedent of Paris on which their organization was modelled.
In the course of the thirteenth century universities were founded
at (among other places) Naples, Orleans, Padua, and Prague;
and in the course of the fourteenth century at Pa via and Vienna.
The title of university was generally accredited to any teaching
body as soon as it was recognized as a studium generate.
The most famous medieval universities aspired to a still
wider recognition, and the final step in their evolution was an
acknowledgment by the pope or emperor of their degrees as a
title to teach throughout Christendom — such universities were
closely related one with the other. Paris was thus recognized
in 1283, Oxford in 1296, and Cambridge in 1318.
The standard of education in mathematics has been largely
fixed by the universities, and most of the mathematicians of
subsequent times have been closely connected with one or more
of them ; and therefore I may be pardoned for adding a few
words on the general course of studies^ in a university in
medieval times.
The students entered when quite young, sometimes not being
more than eleven or twelve years old when first coming into
residence. It is misleading to describe them as undergraduates,
for their age, their studies, the discipline to which they were
subjected, and their position in the university shew that they
should be regarded as schoolboys. The first four years of their
residence were supposed to be spent in the study of the trivium,
^ For fuller details as to their organization of studies, their system of
instruction, and their constitution, see my History of the Study of Mathe-
matics at Cambridge, Cambridge, 1889.
142 THE RISE OF LEARNING IN EUROPE [ch. viii
that is, Latin grammar, logic, and rhetoric. In quite early
times, a considerable number of the students did not progress
beyond the study of Latin grammar — they formed an inferior
faculty and were eligible only for the degree of master of
grammar or master of rhetoric — but the more advanced students
(and in later times all students) spent these years in the study
of the trivium.
The title of bachelor of arts was conferred at the end of this
course, and signified that the student was no longer a schoolboy
and therefore in pupilage. The average age of a commencing
bachelor may be taken as having been about seventeen or
eighteen. Thus at Cambridge in the presentation for a degree
the technical term still used for an undergraduate is juvenis,
while that for a bachelor is vir. A bachelor could not take
pupils, could teach only under special restrictions, and probably
occupied a position closely analogous to that of an undergraduate
nowadays. Some few bachelors proceeded to the study of civil
or canon law, but it was assumed in theory that they next
studied the quadrivium, the course for which took three years,
and which included about as much science as was to be found
in the pages of Boethius and Isidorus.
The degree of master of arts was given at the end of this
course. In the twelfth and thirteenth centuries it was merely
a license to teach : no one sought it who did not intend to use
it for that purpose and to reside in the university, and only
those who had a natural aptitude for such work were likely to
enter a profession so ill-paid as that of a teacher. The degree
was obtainable by any student who had gone through the recog-
nized course of study, and shewn that he was of good moral
character. Outsiders were also admitted, but not as a matter
of course. I may here add that towards the end of the fourteenth
century students began to find that a degree had a pecuniary
value, and most universities subsequently conferred it only on
condition that the new master should reside and teach for at
least a year. Somewhat later the universities took a further
step and began to refuse degrees to those who were not Intel-
CH.viii] COURSE AT A MEDIEVAL UNIVERSITY 143
lectually qualified. This power was assumed on the precedent
of a case which arose in Paris in 1426, when the university
declined to confer a degree on a student — a Slavonian, one
Paul Nicholas — who had performed the necessary exercises in
a very indifferent manner : he took legal proceedings to compel
the university to grant the degree, but their right to withhold
it was established. Nicholas accordingly has the distinction
of being the first student who under modern conditions was
"plucked."
Athough science and mathematics were recognised as the
standard subjects of study for a bachelor, it is probable that,
until the renaissance, the majority of the students devoted most
of their time to logic, philosophy, and theology. The subtleties
of scholastic philosophy were dreary and barren, but it is only
just to say that they provided a severe intellectual training.
We have now arrived at a time when the results of Arab
and Greek science became known in Europe. The history of
Greek mathematics has been already discussed ; I must now
temporarily leave the subject of medieval mathematics, and
trace the development of the Arabian schools to the same date ;
and I must then explain how the schoolmen became acquainted
with the Arab and Greek text-books, and how their introduction
affected the progress of European mathematics.
lU
CHAPTER IX.
THE MATHEMATICS OF THE AEABs/
The story of Arab mathematics is known to us in its general
outlines, but we are as yet unable to speak with certainty on
many of its details. It is, however, quite clear that while part
of the early knowledge of the Arabs was derived from Greek
sources, part was obtained from Hindoo works ; and that it was
on those foundations that Arab science was built. I will begin
by considering in turn the extent of mathematical knowledge
derived from these sources.
Extent of Mathematics obtained from Greeh Sources.
According to their traditions, in themselves very probable,
the scientific knowledge of the Arabs was at first derived from
the Greek doctors who attended the caliphs at Bagdad. It is
^ The subject is discussed at leiigtli by Cantor, chaps, xxxii-xxxv ; by
Hankel, pp. 172-293 ; by A. von Krenier in Kulturgeschichte des Orientes
unter den Ghalifen, Vienna, 1877 ; and by H. Suter in his " Die Mathematiker
und Astronomen der Araber und ihre Werke," Zeitschrift fur Mathematik
tend Physik, Ahhandlungen zur Geschichte der Mathematik, Leipzig, vol. xlv,
1900. See also Matiriaux pour servir d Vhistoire comparee des sciences
mathematiques chez les Grecs et les Orientaiix, by L. A. Sedillot, Paris,
1845-9 ; and the following articles by Fr. Woepcke, Stir I' introduction
de V arithmetique Indienne en Occident, Rome, 1859 ; Sur Vhistoire des
sciences mathematiques chez les Orientaux, Paris, 1860 ; and Mhnoire sur la
propagation des chiffres Indiens, Paris, 1863.
CH. ix] THE MATHEMATICS OF THE ARABS 145
said that when the Arab conquerors settled in towns they
became subject to diseases which had been unknown to them
in their life in the desert. The study of medicine was then
confined mainly to Greeks and Jews, and many of these,
encouraged by the caliphs, settled at Bagdad, Damascus, and
other cities; their knowledge of all branches of learning was
far more extensive and accurate than that of the Arabs, and
the teaching of the young, as has often happened in similar
cases, fell into their hands. The introduction of European
science was rendered the more easy as various small Greek
schools existed in the countries subject to the Arabs : there
had for many years been one at Edessa among the Nestorian
Christians, and there were others at Antioch, Emesa, and
even at Damascus, which had preserved the traditions and some
of the results of Greek learning.
The Arabs soon remarked that the Greeks rested their
medical science on the works of Hippocrates, Aristotle, and
Galen ; and these books were translated into Arabic by order
of the caliph Haroun Al Raschid about the year 800. The
translation excited so much interest that his successor Al
Mamun (813-833) sent a commission to Constantinople to
obtain copies of as many scientific works as was possible, while
an embassy for a similar purpose was also sent to India. At
the same time a large staff of Syrian clerks was engaged, whose
duty it was to translate the works so obtained into Arabic and
Syriac. To disarm fanaticism these clerks were at first termed
the caliph's doctors, but in 851 they were formed into a college,
and their most celebrated member, Honein ibn Ishak, was made
its first president by the caliph Mutawakkil (847-861). Honein
and his son Ishak ibn Honein revised the translations before
they were finally issued. Neither of them knew much mathe-
matics, and several blunders were made in the works issued on
that subject, but another member of the college, Tabit ibn
Korra, shortly published fresh editions which thereafter became
the standard texts.
In this way before the end of the ninth century the Arabs
L
X
146 THE MATHEMATICS OF THE ARABS [ch. ix
obtained translations of the works of Euclid, Archimedes,
ApoUonius, Ptolemy, and others; and in some cases these
editions are the only copies of the books now extant. It is
curious, as indicating how completely Diophantus had dropped
out of notice, that as far as we know the Arabs got no manu-
script of his great work till 150 years later, by which time they
were already acquainted with the idea of algebraic notation and
processes.
Extent of Mathematics obtained from Hindoo Sources.
The Arabs had considerable commerce with India, and a
knowledge of one or both of the two great original Hindoo
works on algebra had been thus obtained in the caliphate of
Al Mansur (754-775), though it was not until fifty or sixty
years later that they attracted much attention. The algebra
and arithmetic of the Arabs were largely founded on these
treatises, and I therefore devote this section to the consideration
of Hindoo mathematics.
The Hindoos, like the Chinese, have pretended that they
are the most ancient people on the face of the earth, and
that to them all sciences owe their creation. But it is probable
that these pretensions have no foundation ; and in fact no
science or useful art (except a rather fantastic architecture and
sculpture) can be definitely traced back to the inhabitants of
the Indian peninsula prior to the Aryan invasion. This
invasion seems to have taken place at some time in the
latter half of the fifth century or in the sixth century,
when a tribe of Aryans entered India by the north-west
frontier, and established themselves as rulers over a large
part of the country. Their descendants, wherever they
have kept their blood pure, may still be recognised by
their superiority over the races they originally conquered;
but as is the case with the modern Europeans, they found
the climate trying and gradually degenerated. For the
first two or three centuries they, however, retained their
CH. ix] ARYA-BHATA 147
intellectual vigour, and produced one or two writers of great
ability.
Arya-Bhata. The earliest of these, of whom we have definite
information, is Arya-Bhata,'^ who was born at Patna in the year
476. He is frequently quoted by Brahmagupta, and in the
opinion of many commentators he created algebraic analysis,
though it has been suggested that he may have seen Diophantus's
Arithmetic. The chief work of Arya-Bhata with which we are
acquainted is his Aryahhathiya, which consists of mnemonic
verses embodying the enunciations of various rules and proposi-
tions. There are no proofs, and the language is so obscure and
concise that it long defied all efforts to translate it.
The book is divided into four parts : of these three are
devoted to astronomy and the elements of spherical trigono-
metry ; the remaining part contains the enunciations of thirty-
three rules in arithmetic, algebra, and plane trigonometry. It
is probable that Arya-Bhata regarded himself as an astronomer,
and studied mathematics only so far as it was useful to him in
his astronomy.
In algebra Arya-Bhata gives the sum of the first, second, and
third powers of the first n natural numbers ; the general solution
of a quadratic equation ; and the solution in integers of certain
indeterminate equations of the first degree. His solutions of
numerical equations have been supposed to imply that he was
acquainted with the decimal system of enumeration.
In trigonometry he gives a table of natural sines of the
angles in the first quadrant, proceeding by multiples of 3|°,
defining a sine as the semichord of double the angle. Assuming
that for the angle 3|° the sine is equal to the circular measure,
he takes for its value 225, i.e. the number of minutes in the
^ The subject of prehistoric Indian mathematics has been discussed by G.
Thibaut, Von Schroeder, and H. Vogt. A Sanskrit text of the Aryabhathiya,
edited by H. Kern, was published at Leyden in 1874 ; there is also an
article on it by the same editor in the Jottrnal of the Asiatic Society, London,
1863, vol. XX, pp. 371-387 ; a French translation by L. Rodet of that part
which deals with algebra and trigonometry is given in the Journal Asiatique,
1879, Paris, series 7, vol. xiii, pp. 393-434.
148 THE MATHEMATICS OF THE ARABS [ch. ix
angle. He then enunciates a rule which is nearly unintelligible,
but probably is the equivalent of the statement
sin (n-{-l)a- sin na = sin na-sin(n-l)a- sin na cosec a,
where a stands for 3|° ; and working with this formula he
constructs a table of sines, and finally finds the value of sin 90°
to be 3438. This result is correct if we take 3*1416 as the
value of TT, and it is interesting to note that this is the number
which in another place he gives for tt. The correct trigono-
metrical formula is
sin (n+l)a- sin na = sin na — sin{n-l)a-4: sin na sin^ |a.
Arya-Bhata, therefore, took 4 sin^ Ja as equal to cosec a, that is,
he supposed that 2 sin a = 1 + sin 2a : using the approximate
values of sin a and sin 2 a given in his table, this reduces to
2(225) = 1 + 449, and hence to that degree of approximation his
formula is correct. A considerable proportion of the geometrical
propositions which he gives is wrong.
Brahmagupta. The next Hindoo writer of note is Brakrn'a-
gupta, who is said to have been born in 598, and probably was
alive about 660. He wrote a work in verse entitled Brahmcu-
Sphuta-Siddhanta, that is, the Siddhanta, or system of Brahma
in astronomy. In this, two chapters are devoted to arithmetic,
algebra, and geometry.^
The arithmetic is entirely rhetorical. Most of the problems
are worked out by the rule of three, and a large proportion of
them are on the subject of interest.
In his algebra, which is also rhetorical, he works out the
fundamental propositions connected with an arithmetical pro-
gression, and solves a quadratic equation (but gives only the
positive value to the radical). As an illustration of the prob-
lems given I may quote the following, which was reproduced in
slightly different forms by various subsequent writers, but I
replace the numbers by letters. " Two apes lived at the top of
^ These two chapters (chaps, xii and xviii) were translated by H. T. Cole-
brooke, and published at London in 1817.
CH.ix] BEAHMAGUPTA 149
a cliff of height A, whose base was distant mh from a neighbour-
ing village. One descended the cliff and walked to the village,
the other flew up a height x and then flew in a straight line to
the village. The distance traversed by each was the same.
Find ^." Brahmagupta gave the correct answer, namely
X = inhjim + 2). In the question as enunciated originally
/i = 100, 7)1 = 2.
Brahmagupta finds solutions in integers of several indeter-
minate equations of the first degree, using the same method as
that now practised. He states one indeterminate equation of
the second degree, namely, nx'^ + 1 = y"^, and gives as its solution
X = 2^/(^2 _ ^-^ amj .y = ^^2 ^ n)l{t'^ - n). To obtain this general
form he proved that, if one solution either of that or of certain
allied equations could be guessed, the general solution could be
written down ; but he did not explain how one solution could be
obtained. Curiously enough this equation was sent by Fermat
as a challenge to Wall is and Lord Brouncker in the seventeenth
century, and the latter found the same solutions as Brahmagupta
had previously done. Brahmagupta also stated that the equation
y2 = ^-j.2 _ \ could not be satisfied by integral values of x and y
unless n could be expressed as the sum of the squares of two
integers. It is perhaps worth noticing that the early algebraists,
whether Greeks, Hindoos, Arabs, or Italians, drew no distinc-
tion between the problems which led to determinate and those
which led to indeterminate equations. It was only after the in-
troduction of syncopated algebra that attempts were made to give
general solutions of equations, and the difficulty of giving such
solutions of indeterminate equations other than those of the first
degree has led to their practical exclusion from elementary algebra.
In geometry Brahmagupta proved the Pythagorean property
of a right-angled triangle (Euc. i, 47). He gave expressions for
the area of a triangle and of a quadrilateral inscribable in a
circle in terms of their sides ; and shewed that the area of a
circle was equal to that of a rectangle whose sides were the
radius and semiperimeter. He was less successful in his
attempt to rectify a circle, and his result is equivalent to
150 THE MATHEMATICS OF THE ARABS [ch. ix
taking ^^10 for the value of tt. He also determined the
surface and volume of a pyramid and cone ; problems over
which Arya-Bhata had blundered badly. The next part of
his geometry is almost unintelligible, but it seems to be an
attempt to find expressions for several magnitudes connected
with a quadrilateral inscribed in a circle in terms of its sides :
much of this is wrong.
It must not be supposed that in the original work all the
propositions which deal with any one subject are collected
together, and it is only for convenience that I have tried to
arrange them in that way. It is impossible to say whether the
whole of Brahmagupta's results given above are original. He
knew of Arya-Bhata's work, for he reproduces the table of sines
there given ; it is likely also that some progress in mathematics
had been made by Arya-Bhata's immediate successors, and that
Brahmagupta was acquainted with their works ; but there seems
no reason to doubt that the bulk of Brahmagupta's algebra and
arithmetic is original, although perhaps influenced by Dio-
phantus's writings : the origin of the geometry is more doubt-
ful, probably some of it is derived from Hero's works, and maybe
some represents indigenous Hindoo work.
"^Bhaskara. To make this account of Hindoo mathematics
complete I may depart from the chronological arrangement and
say that the only remaining Indian mathematician of exceptional
eminetce of whose works we know anything was BJuiskara, who
was born in 1114. He is said to have been the lineal successor
of Brahmagupta as head of an astronomical observatory at Ujein.
He wrote an astronomy, of which four chapters have been trans-
lated. Of these one termed Lilavati is on arithmetic ; a second
termed Bija Ganita is on algebra ; the third and fourth are on
astronomy and the sphere;^ some of the other chapters also
involve mathematics. This work was, I believe, known to the
1 See the article Viga Ganita in the Penny Cyclopaedia, London, 1843 ;
and the translations of the Lilavati and the Bija Ganita issued by H. T. Cole-
brooke, London, 1817. The chapters on astronomy and the sphere were
edited by L, Wilkinson, Calcutta, 1842.
cH.ix] BHASKARA 151
Arabs almost as soon as it was written, and influenced their
subsequent writings, though they failed to utilize or extend
most of the discoveries contained in it. The results thus became
indirectly known in the West before the end of the twelfth
century, but the text itself was not introduced into Europe till
within recent times.
The treatise is in verse, but there are explanatory notes in
prose. It is not clear whether it is original or whether it is
merely an exposition of the results then known in India ; but in
any case it is most probable that Bhaskara was acquainted with
the Arab works which had been written in the tenth and eleventh
centuries, and with the results of Greek mathematics as trans-
mitted through Arabian sources. The algebra is syncopated and
almost symbolic, which marks a great advance over that of
Brahraagupta and of the Arabs. The geometry is also superior
to that of Brahmagupta, but apparently this is due to the
knowledge of various Greek works obtained through the Arabs.
The first book or Lilavati commences with a salutation to
the god of wisdom. The general arrangement of the work may
be gathered from the following table of contents. Systems of
weights and measures. Next decimal numeration, briefly de-
scribed. Then the eight operations of arithmetic, namely,
addition, subtraction, multiplication, division, square, cube,
square-root, and cube-root. Reduction of fractions to a common
denominator, fractions of fractions, mixed numbers, and the
eight rules applied to fractions. The "rules of cipher," namely,
a + 0 = a, 0^ = 0, ;^0 = 0, a -r- 0 = oo . The solution of some
simple equations which are treated as questions of arithmetic.
The rule of false assumption. Simultaneous equations of the
first degree with applications. Solution of a few quadratic
equations. Rule of three and comj^ound rule of three, with
various cases. Interest, discount, and partnership. Time of
filling a cistern by several fountains. Barter. Arithmetical
progressions, -and sums of squares and cubes. Geometrical
progressions. Problems on triangles and quadrilaterals. Ap-
proximate value of 17, Some trigonometrical formulae. Contents
152 THE MATHEMATICS OF THE ARABS [ch. ix
of solids. Indeterminate equations of tho first degree. Lastly,
the book ends with a few questions on combinations.
This is the earliest known work which contains a systematic
exposition of the decimal system of numeration. It is possible
that Arya-Bhata was acquainted with it, and it is most likely
that Brahmagupta was so, but in Bhaskara's arithmetic we meet
with the Arabic or Indian numerals and a sign for zero as part
of a well-recognised notation. It is impossible at present to
definitely trace these numerals farther back than the eighth
century, but there is no reason to doubt the assertion that they
were in use at the beginning of the seventh century. Their
origin is a difficult and disputed question. I mention below ^
the view which on the whole seems most probable, and perhaps is
now generally accepted, and I reproduce there some of the forms
used in early times.
To sum the matter up briefly, it may be said that the
Lilavati gives the rules now current for addition, subtraction,
multiplication, and division, as well as for the more common pro-
cesses in arithmetic ; while the greater part of the work is taken
up with the discussion of the rule of three, which is divided
into direct and inverse, simple and compound, and is used to
solve numerous questions chiefly on interest and exchange — the
numerical questions being expressed in the decimal system of
notation with which we are familiar.
Bhaskara was celebrated as an astrologer no less than as a
mathematician. He learnt by this art that the event of his
daughter Lilavati marrying would be fatal to himself. He
therefore declined to allow her to leave his presence, but by
way of consolation he not only called the first book of his work
by her name, but propounded many of his problems in the form
of questions addressed to her. For example, " Lovely and dear
Lilavati, whose eyes are like a fawn's, tell me what are the
numbers resulting from 135 multiplied by 12. If thou be
skilled in multiplication, whether by whole or by parts, whether
by division or by separation of digits, tell me, auspicious damsel,
^ See below, page 184.
CH. ix] BHASKARA 153
what is the quotient of the product when divided by the same
multiplier."
I may add here that the problems in the Indian works give
a great deal of interesting information about the social and
economic condition of the country in which they were written.
Thus Bhaskara discusses «ome questions on the price of slaves,
and incidentally remarks that a female slave was generally
supposed to be most valuable when 16 years old, and subse-
quently to decrease in value in inverse proportion to the age ;
for instance, if when 16 years old she were worth 32 nishkas,
her value when 20 would be represented by (16x32)-^20
nishkas. It would appear that, as a rough average, a female
slave of 16 was worth about 8 oxen which had worked for two
years. The interest charged for money in India varied from 3 J
to 5 per cent per month. Amongst other data thus given will
be found the prices of provisions and labour.
The chapter termed Bija Ganita commences with a sentence
so ingeniously framed that it can be read as the enunciation of a
religious, or a philosophical, or a mathematical truth. Bhaskara
after alluding to his Lilavati, or arithmetic, states that he intends
in this book to proceed to the general operations of analysis.
The idea of the notation is as follows. Abbreviations and
initials are used for symbols ; subtraction is indicated by a dot
placed above the coefficient of the quantity to be subtracted ;
addition by juxtaposition merely ; but no symbols are used for
multiplication, equality, or inequality, these being written at
length. A product is denoted by the first syllable of the word
subjoined to the factors, between which a dot is sometimes
placed. In a quotient or fraction the divisor is written under
the dividend without a line of separation. The two sides of an
equation are written one under the other, confusion being pre-
vented by the recital in words of all the steps which accompany
the operation Various symbols for the unknown quantity are
used, but most of them are the initials of names of colours, and
the word colour is often used as synonymous with unknown
quantity ; its Sanskrit equivalent also signifies a letter, and
154 THE MATHEMATICS OF THE ARABS [ch. ix
letters are sometimes used either from the alphabet or from the
initial syllables of subjects of the problem. In one or two cases
symbols are used for the given as well as for the unknown
quantities. The initials of the words square and solid denote
the second and third powers, and the initial syllable of square
root marks a surd. Polynomials are» arranged in powers, the
absolute quantity being always placed last and distinguished by
an initial syllable denoting known quantity. Most of the
equations have numerical coefficients, and the coefficient is
always written after the unknown quantity. Positive or
negative terms are indiscriminately allowed to come first ; and
every power is repeated on both sides of an equation, with a
zero for the coefficient when the term is absent. After explain-
ing his notation, Bhaskara goes on to give the rules for addition,
subtraction, multiplication, division, squaring, and extracting
the square root of algebraical expressions ; he then gives the
rules of cipher as in the Lilavati ; solves a few equations ; and
lastly concludes with some operations on surds. Many of the
problems are given in a poetical setting with allusions to fair
damsels and gallant warriors.
Fragments of other chapters, involving algebra, trigonometry,
and geometrical applications, have been translated by Cole-
brooke. Amongst the trigonometrical formulae is one which is
equivalent to the equation cZ (sin ^) = cos 0 d9.
I have departed from the chronological order in treating here
of Bhaskara, but I thought it better to mention him at the same
time as I was discussing his compatriots. It must be remem-
bered, however, that he flourished subsequently to all the Arab
mathematicians considered in the next section. The works with
which the Arabs first became acquainted were those of Arya.
Bhata and Brahmagupta, and perhaps of their successors Sridhara
and Padmanabha ; it is doubtful if they ever made much use of
the great treatise of Bhaskara.
It is probable that the attention of the Arabs was called to
the works of the first two of these writers by the fact that the
Arabs adopted the Indian system of arithmetic, and were thus
CH.ix] THE MATHEMATICS OF THE ARABS 155
led to look at the mathematical text-books of the Hindoos.
The Arabs had always had considerable commerce with India,
and with the establishment of their empire the amount of trade
naturally increased ; at that time, about the year 700, they
found the Hindoo merchants beginning to use the system of
numeration with which we are familiar, and adopted it at once.
This immediate acceptance of it was made the easier, as they
had no works of science or literature in which another system
was used, and it is doubtful whether they then possessed any
but the most primitive system of notation for expressing
numbers. The Arabs, like the Hindoos, seem also to have
made little or no use of the abacus, and therefore must have
found Greek and Roman methods of calculation extremely
laborious. The earliest definite date assigned for the use in
Arabia of the decimal system of numeration is 773. In that
year some Indian astronomical tables were brought to Bagdad,
and it is almost certain that in these Indian numerals (including
a zero) were employed.
The Development of Mathematics in Arabia}
In the preceding sections of this chapter I have indicated
the two sources from which the Arabs derived their knowledge
of mathematics, and have sketched out roughly the amount of
knowledge obtained from each. We may sum the matter up
by saying that before the end of the eighth century the Arabs
were in possession of a good numerical notation and of
Brahmagupta's work on arithmetic and algebra; while before
the end of the ninth century they were acquainted with the
masterpieces of Greek " mathematics in geometry, mechanics,
and astronomy. I have now to explain what use they made of
these materials.
Alkarismi. The first and in some respects the most illus-
^ A work by B. Baldi on the lives of several of the Arab mathematicians
was printed in Boncompagni's Bnlletino di bibliograjiu, 1872, vol. v, pp. 427-
534.
156 THE MATHEMATICS OF THE ARABS [ch. ix
trious of the Arabian mathematicians was Mohammed ibn Musa
Abu Djefai^ Al-Khwdrizmi. There is no common agreement as
to which of these names is the one by which he is to be known :
the last of them refers to the place where he was born, or in
connection with which he was best known, and I am told that
it is the one by which he would have been usually known
among his contemporaries. I shall therefore refer to him by
that name ; and shall also generally adopt the corresponding
titles to designate the other Arabian mathematicians. Until
recently, this was almost always written in the corrupt form
Alkarismij and, though this way of spelling it is incorrect, it
has been sanctioned by so many writers that I shall make use
of it.
We know nothing of Alkarismi's life except that he was a
native of Khorassan and librarian of the caliph Al Mamun ; and
that he accompanied a mission to Afghanistan, and possibly
came back through India. On his return, about 830, he wrote
an algebra, 1 which is founded on that of Brahmagupta, but in
which some of the proofs rest on the Greek method of repre-
senting numbers by lines. He also wrote a treatise on arith-
metic : an anonymous tract termed Algoritmi De Numero
iThdorum^ which is in the university library at Cambridge, is
believed to be a Latin translation of this treatise. ^ Besides
these two works he compiled some astronomical tables, with
explanatory remarks ; these included results taken from both
Ptolemy and Brahmagupta.
The algebra of Alkarismi holds a most important place in the
history of mathematics, for we may say that the subsequent
Arab and the early medieval works on algebra were founded on
it, and also that through it the Arabic • or Indian system of
decimal numeration was introduced into the West. The work
is termed Al-gehr we' I mukahala : al-gebr, from which the word
algebra is derived, means the restoration, and refers to the fact
1 It was published by F. Rosen, with an English translation, Loudon,
1831.
2 It was published by B. Boncompagni, Rome, 1857.
CH. IX]
ALKARISMI
157
that any the same magnitude may be added to or subtracted
from both sides of an equation ; al mukahala means the process
of simplification, and is generally used in connection with the
combination of like terms into a single term. The unknown
quantity is termed either " the thing " or " the root " (that is,
of a plant), and from the latter phrase our use of the word root
as applied to the solution of an equation is derived. The
square of the unknown is called "the power." All the known
quantities are numbers.
The work is divided into five parts. In the first Alkarismi
gives rules for the solution of quadratic equations, divided
into five classes of the forms ax^ = bx^ ax^ = Cj ax^ + bx — c,
ax^ -{■c = bxj and ax^ = bx-hc, where a, ^, c are positive numbers,
and in all the applications a=l. He considers only real and
positive roots, but he recognises the existence of two roots,
which as far as we know was never done by the Greeks. It is
somewhat curious that when both roots are positive he generally
takes only that root which is derived from the negative value of
the radical.
He next gives geometrical proofs of these rules in a
manner analogous to that of Euclid ii, 4. For example, to
solve the equation ^2^io.r = 39, or any equation of the form
x^+px = q, he gives two methods of which one is as follows.
Let AB represent the value of x, and construct on it the
square ABC J) (see figure below). Produce DA to . H and
g —
DC to F so that AH=CF==5 (or ^p) ; and complete the
figure as drawn below. Then the areas AC, HB, and BF
158 THE MATHEMATICS OF THE AEABS [ch. ix
represent the magnitudes x^, 6x, and Ox. Thus the left-hand
side of the equation is represented by the sum of the areas AC,
BB, and £F, that is, by the gnomon IICG. To both sides of
the equation add the square KG, the area of which is 25 (or
Ip'^), and we shall get a new square whose area is by hypothesis
equal to 39 + 25, that is, to 64 (or q + jp^) and whose side
therefore is 8. The side of this square DII, which is equal to
8, will exceed AH, which is equal to 5, by the value of the
unknown required, which, therefore, is 3.
In the third part of the book Alkarismi considers the
product of {x±a) and {x±h). In the fourth part he states
the rules for addition and subtraction of expressions which
involve the unknown, its square, or its square root ; gives rules
for the calculation of square roots ; and concludes with the
theorems that a Jb= Ja% and Jajb= Jab. In the fifth
and last part he gives some problems, such, for example, as to
find two numbers whose sum is 10 and the difference of whose
squares is 40.
In all these early works there is no clear distinction between
arithmetic and algebra, and we find the account and explana-
tion of arithmetical processes mixed -up with algebra and
treated as part of it. It was from this book then that
the Italians first obtained not only the ideas of algebra, but
also of an arithmetic founded on the decimal system.
This arithmetic was long known as algorism, or the art of
Alkarismi, which served to distinguish it from the arithmetic
of Boethiusj this name remained in use till the eighteenth
century.
Tabit ibn Eorra. The work commenced by Alkarismi
was carried on by Tabit ibn Korra, born at Harran in 836, and
died in 901, who was one of the most brilliant and accom-
plished scholars produced by the Arabs, As I have already
stated, he issued translations of the chief works of Euclid,
ApoUonius, Archimedes, and Ptolemy. He also wrote several
original works, all of which are lost with the exception of a
fragment on algebra, consisting of one chapter on cubic equa-
CH. ix] ALKAYAML ALKARKI. 159
tions, which are solved by the aid of geometry in somewhat
the same way as that given later. ^
Algebra continued to develop very rapidly, but it remained
entirely rhetorical. The problems with which the Arabs were
chiefly concerned were solution of equations, problems leading
to equations, or properties of numbers. The tw^o most prominent
algebraists of a later date were Alkayami and Alkarki, both
of whom flourished at the beginning of the eleventh century.
Alkayami. The first of these, Omar Alkayami^ is notice-
able for his geometrical treatment of cubic equations by which
he obtained a root as the abscissa of a point of intersection
of a conic and a circle.^ The equations he considers are of
the following forms, where a and c stand for positive integers,
(i) x^ + IP-x = h\ whose root he says is the abscissa of a point
of intersection of x^ = hy and y^ = x{c - x) ; (ii) x^ + ax^ = c^,
whose root he says is the abscissa of a point of intersection
of xy = c^ and y'^ = c(x + a) ; (iii) x^±ax^ + b^x = b^c, whose
root he says is the abscissa of a point of intersection of
y'^ = (x±a) (c-x) and x{b±y) = bc. He gives one biquadratic,
namely, (100 -a;^) (10-ar)2 = 8100, the root of which is deter-
mined by the point of intersection of {10-x)y = 90 and
x'^ + y^=100. It is sometimes said that he stated that it was
impossible to solve the equation x^ + y^ = z^ in positive integers,
or in other words that the sum of two cubes can never be a
cube ; though whether he gave an accurate proof, or whether,
as is more likely, the proposition (if enunciated at all) was the
result of a wide induction, it is now impossible to say; but
the fact that such a theorem is attributed to him will serve to
illustrate the extraordinary progress the Arabs had made in
algebra.
Alkarki. The other mathematician of this time (circ. 1000)
whom I mentioned was Alkarki.^ He gave expressions for the
1 See below, page 224.
^ His treatise on algebra was published by Ft. Woepcke, Paris, 1851.
^ His algebra was published by Fr. Woepcke, 1853, and his arithmetic was
translated into German by Ad. Hochheim, Halle, 1878.
160 THE MATHEMATICS OF THE ARABS [ch. ix
sums of the first, second, and third powers of the first n natural
numbers ; solved various equations, including some of the forms
ax^-'TP±.hdc'P±c^^ ; and discussed surds, shewing, for example,
that V8 + 718= V^O.
Even where the methods of Arab algebra are quite general
the applications are confined in all cases to numerical problems,
and the algebra is so arithmetical that it is difficult to treat the
subjects apart. From their books on arithmetic and from the
observations scattered through various works on algebra, we may-
say that the methods used by the Arabs for the four funda-
mental processes were analogous to, though more cumbrous
than, those now in use ; but the problems to which the subject
was applied were similar to those given in modern books, and
were solved by similar methods, such as rule of three, &c.
Some minor improvements in notation were introduced, such,
for instance, as the introduction of a line to separate the nume-
rator from the denominator of a fraction; and hence a line
between two symbols came to be used as a sjonbol of division.^
Alhossein (980-1037) used a rule for testing the correctness of
the results of addition and multiplication by "casting out the
nines." Various forms of this rule have been given, but they
all depend on the proposition that, if each number in the ques-
tion be replaced by the remainder when it is divided by 9, and
if these remainders be added or multiplied as directed in the
question, then this result when divided by 9 will leave the same
remainder as the answer whose correctness it is desired to test
when divided by 9 : if these remainders differ, there is an error.
The selection of 9 as a divisor was due to the fact that the
remainder when a number is divided by 9 can be obtained by
adding the digits of the number and dividing the sum
by 9.
I am not concerned with the views of Arab writers on
astronomy or the value of their observations, but I may remark
in passing that they accepted the theory as laid down by Hippar-
chus and Ptolemy, and did not materially alter or advance it.
^ See below, page 241.
CH. ix] THE MATHEMATICS OF THE AEABS 161
I may, however, add that Al Mamun caused the length of a
degree of latitude to be measured, and he, as well as the two
mathematicians to be next named, determined the obliquity of
the ecliptic.
Albategni. Albuzjani. Like the Greeks, the Arabs rarely,
if ever, employed trigonometry except in connection with
astronomy ; but in effect they used the trigonometrical ratios
which are now current, and worked out the plane trigonometry
of a single angle. They are also acquainted with the elements
of spherical trigonometry. Alhategni^ born at Batan in Meso-
potamia, in 877, and died at Bagdad in 929, was among the
earliest of the many distinguished Arabian astronomers. He
wrote the Science of the Stars} which is worthy of note from
its containing a mention of the motion of the sun's apogee.
In this work angles are determined by " the semi-chord of twice
the angle," that is, by the sine of the angle (taking the radius
vector as unity). It is doubtful whether he was acquainted
with the previous introduction of sines by Arya-Bhata and
Brahmagupta ; Hipparchus and Ptolemy, it will be remembered,
had used the chord. Albategni was also acquainted with
the fundamental formula in spherical trigonometry, giving
the side of a triangle in terms of the other sides and the
angle included by them. Shortly after the death of Albategni,
Albuzjani^ who is also known as Ahnl-Wafa^ born in 940,
and died in 998, introduced certain trigonometrical func-
tions, and constructed tables of tangents and cotan-
gents. He was celebrated as a geometrician as well as an
astronomer.
Alhazen. Abd-al-gehl. The Arabs were at first content to
take the works of Euclid and Apollonius for their text-books
in geometry without attempting to comment on them, but
Alhazen, born at Bassora in 987 and died at Cairo in 1038,
issued in 1036 a collection ^ of problems something like the Data
of Euclid. Besides commentaries on the definitions of Euclid
1 It was edited by Regiomontanus, Nuremberg, 1537.
2 It was translated by L. A. Sedillot, and published at Paris in 1836.
M
162 THE MATHEMATICS OF THE ARABS [ch. ix
and on tlie Almagest^ Alhazen also wrote a work on optics,^ which
includes the earliest scientific account of atmospheric refraction.
It also contains some ingenious geometry, amongst other things,
a geometrical solution of the problem to find at what point of a
concave mirror a ray from a given point must be incident so as
to be reflected to another given point. Another geometrician
of a slightly later date wa.s Abd-al-(/ehl (circ. 1100), who wrote on
conic sections, and was also the author of three small geometri-
cal tracts.
It was shortly after the last of the mathematicians mentioned
above that Bhaskara, the third great Hindoo mathematician,
flourished ; there is every reason to believe that he was
familiar with the works of the Arab school as described
above, and also that his writings were at once known in
Arabia.
The Arab schools continued to flourish until the fifteenth
century. But they produced no other mathematician of any
exceptional genius, nor was there any great advance on the
methods indicated above, and it is unnecessary for me to
crowd my pages with the names of a number of writers
who did not materially affect the progress of the science in
Europe.
From this rapid sketch it will be seen that the work of the
Arabs (including therein writers who wrote in Arabia and
lived under Eastern Mohammedan rule) in arithmetic, algebra,
and trigonometry was of a high order of excellence. They
appreciated geometry and the applications of geometry to
astronomy, but they did not extend the bounds of the science.
It may be also added that they made no special progress in
statics, or optics, or hydrostatics ; though there is abundant
evidence that they had a thorough knowledge of practical
hydraulics.
The general impression left is that the Arabs were quick
to appreciate the work of others — notably of the Greek masters
and of the Hindoo mathematicians — but, like the ancient
1 It was published at Bale in 1572.
CH.IX] THE MATHEMATICS OF THE ARABS 163
Chinese and Egyptians, they did not systematically develop
a subject to any considerable extent. Their schools may be
taken to have lasted in all for about 650 years, and if the
work produced be compared with that of Greek or modern
European writers it is, as a whole, second-rate both in quantity
and qualitA
164
CHAPTEE X.
THE INTRODUCTION OF AEAB WOEKS INTO EUROPE.
CIEC. 1150-1450.
In the last chapter but one I discussed the development of
European mathematics to a date which corresponds roughly
with the end of the "dark ages"; and in the last chapter
I traced the history of the mathematics of the Indians and
Arabs to the same date. The mathematics of the two or
three centuries that follow and are treated in this chapter are
characterised by the introduction of the Arab mathematical
text-books and of Greek books derived from Arab sources, and
the assimilation of the new ideas thus presented.
It was, however, from Spain, and not from Arabia, that
a knowledge of eastern mathematics first came into western
Europe. The Moors had established their rule in Spain in 747,
and by the tenth or eleventh century had attained a high
degree of civilisation. Though their political relations with the
caliphs at Bagdad were somewhat unfriendly, they gave a
ready welcome to the works of the great Arab mathematicians.
In this way the Arab translations of the writings of Euclid,
,' Archimedes, Apollonius, Ptolemy, and perhaps of other Greek
, )j/a,uthors, together with the works of the Arabian algebraists,
f^/^ '<^were read and commented on at the three great Moorish schools of
vV^, (Granada, Cordova, and Seville. It seems probable that these
^orks indicate the full extent of Moorish learning, but, as
CH. x] ELEVENTH AND TWELFTH CENTURIES 165
all knowledge was jealously guarded from Christians, it is
impossible to speak with certainty either on this point or
on that of the time when the Arab books were first introduced
into Spain.
The eleventh century. The earliest Moorish writer of
distinction of whom I find mention is Geber ibn Aphla, who
was born at Seville and died towards the latter part of the
eleventh century at Cordova. He wrote on astronomy and
trigonometry, and was acquainted with the theorem that the
sines of the angles of a spherical triangle are proportional to the
sines of the opposite sides. ^
Arzachel.2 Another Arab of about the same date was
Arzachel, who was living at Toledo in 1080. He suggested
that the planets moved in ellipses, but his contemporaries with
scientific intolerance declined to argue about a statement which
was contrary to Ptolemy's conclusions in the Almagest.
The twelfth century. During the course of the twelfth
century copies of the books used in Spain were obtained in
western Christendom. The first step towards procuring a
knowledge of Arab and Moorish science was taken by an
English monk, Adelhard of Bath,^ who, under the disguise of
a Mohammedan student, attended some lectures at Cordova
about 1120 and obtained a copy of Euclid's Elements. This
copy, translated into Latin, was the foundation of all the
editions known in Europe till 1533, when the Greek text
was recovered. / How rapidly a knowledge of the work spread
we may judge when we recollect that before the end of the
thirteenth century Roger Bacon was familiar with it, while
before the close of the fourteenth century the first five books
formed part of the regular curriculum at many universities.
The enunciations of Euclid seem to have been known before
^ Geber's works were translated into Latin by Gerard, and published at
Nuremberg in 1533.
2 See a memoir by M. Steinscbneider in Boncompagni's Bulletino di
Bihliografia, 1887, vol xx.
^ On the influence of Adelhard and Ben Ezra, see the ** Abhandlungen
zur Geschichte der Mathematik " in the ZeitschriftfiXr Mathematik, vol. xxv,
1880.
166 INTRODUCTION OF ARAB WORKS [ch.x
Adelhard's time, and possibly as early as the year 1000, though
copies were rare. Adelhard also issued a text-book on the use
of the abacus.
Ben Ezra.^ During the same century other translations of
the Arab text-books or commentaries on them were obtained.
Amongst those who were most influential in introducing
Moorish learning into Europe I may mention Abraham Ben
Ezra. Ben Ezra was born at Toledo in 1097, and died at
Rome in 1167. He was one of the most distinguished Jewish
rabbis who had settled in Spain, where it must be recollected
that they were tolerated and even protected by the Moors
on account of their medical skill. Besides some astronomical
tables and an astrology, Ben Ezra wrote an arithmetic ; ^ in
this he explains the Arab system of numeration with nine
symbols and a zero, gives the fundamental processes of
arithmetic, and explains the rule of three.
Gerard.^ Another European who was induced by the
reputation of the Arab schools to go to Toledo was Gerard,
who was born at Cremona in 1114 and died in 1187. He
translated the Arab edition of the AlTfiagest, the works of
Alhazen, and the works of Alfarabius, whose name is other-
wise unknown to us : it is believed that the Arabic numerals
were used in this translation, made in 1136, of Ptolemy's work.
Gerard also wrote a short treatise on algorism which exists in
manuscript in the Bodleian Library at Oxford. He was
acquainted with one of the Arab editions of Euclid's Elements,
which he translated into Latin.
John Hispalensis. Among the contemporaries of Gerard
was John Hispalensis of Seville, originally a rabbi, but converted
to Christianity and baptized under the name given above. He
made translations of several Arab and Moorish works, and also
wrote an algorism which contains the earliest examples of the
^ See footnote 3 on p. 165.
2 An analysis of it was published by 0. Terquem in Liouville's Journal
for 1841.
^ See Boncorapagni's Delia vita e dclle opcre di Ohcrardo Qremonese,
Rome, 1851.
CH. x] LEONARDO 167
extraction of the square roots of numbers by the aid of the
decimal notation.
The thirteenth century. During the thirteenth century
there was a revival of learning throughout Europe, but the new
learning was, I believe, confined to a very limited class. The
early years of this century are memorable for the development
of several universities, and for the appearance of three remark-
able mathematicians — Leonardo of Pisa, Jordanus, and Roger
^^cQn, the Franciscan monk of Oxford. Henceforward it is
to Europeans that we have to look for the development of
mathematics, but until the invention of printing the knowledge
was confined to a very limited class.
Leonardo.^ Leonardo Fibonacci {i.e. filius Bonaccii) gener-
ally known as LeoTiardo of Pisa, was born at Pisa about 1175.
His father Bonacci was a merchant, and was sent by his fellow-
townsmen to control the custom-house at Bugia in Barbary;
there Leonardo was educated, and he thus became acquainted
with the Arabic or decimal system of numeration, as also with
x^lkarismi's work on Algebra, which was described in the last
chapter. It would seem that Leonardo was entrusted with some
duties, in connection with the custom-house, which required him
to travel. He returned to Italy about 1200, and in 1202
published a work called Algebra et almucJiabala (the title being
taken from Alkarismi's work), but generally known as the Liber
Abaci. He there explains the Arabic system of numeration, and
remarks on its great advantages over the Roman system. He
then gives an account of algebra, and points out the convenience
of using geometry to get rigid demonstrations of algebraical
formulae. He shews how to solve simple equations, solves a few
quadratic equations, and states some methods for the solution of
indeterminate equations ; these rules are illustrated by problems
on numbers. The algebra is rhetorical, but in one case letters
^ See the Lehen und Schriften Leonardos da Pisa, by J. Giesing, Dobeln,
1886 ; Cantor, cliaps. xli, xlii ; and an article by V. Lazzarini in the
Bollettino di Bihliografia e Storia, Rome, 1904, vol. vii. Most of Leonardo's
writings were edited and published by B. Boncompagni, Rome, vol. i, 1857,
and vol. ii, 1862.
168 INTRODUCTION OF ARAB WORKS [ch. x
are employed a« algebraical symbols. This work had a wide
circulation, and for at least two centuries remained a standard
authority from which numerous writers drew their inspiration.
The Liber Abaci is especially interesting in the history of
arithmetic, since practically it introduced the use of the Arabic
numerals into Christian Europe. The language of Leonardo
implies that they were previously unknown to his countrymen ;
he says that having had to spend some years in Barbary he there
learnt the Arabic system, which he found much more convenient
than that used in Europe ; he therefore published it "in order
that the Latin ^ race might no longer be deficient in that
knowledge." Now Leonardo had read very widely, and had
travelled in Greece, Sicily, and Italy ; there is therefore every
presumption that the system was not then commonly employed
in Europe.
Though Leonardo introduced the use of Arabic numerals
into commercial affairs, it is probable that a knowledge of them
as current in the East was previously not uncommon among
travellers and merchants, for the intercourse between Christians
and Mohammedans was sufficiently close for each to learn
something of the language and common practices of the other. We
can also hardly suppose that the Italian merchants were ignorant
of the method of keeping accounts used by some of their best
customers ; and we must recollect, too, that there were numerous
Christians w^ho had escaped or been ransomed after serving the
Mohammedans as slaves. It was, however, Leonardo who
brought the Arabic system into general use, and by the middle
of the thirteenth century a large proportion of the Italian
merchants employed it by the side of the old system.
The majority of mathematicians must have already known
of the system from the works of Ben Ezra, Gerard, and John
Hispalensis. But shortly after the appearance of Leonardo's
book Alfonso of Castile (in 1252) published some astronomical
^ Dean Peacock says that the earliest known application of the word
Italians to describe the inhabitants of Italy occurs about the middle of the
tliirteenth century ; by the end of that century it was in common use.
CH. x] LEONARDO 169
tables, founded on observations made in Arabia, which were
computed by Arabs, and which, it is generally believed, were
expressed in Arabic notation. Alfonso's tables had a wide
circulation among men of science, and probably were largely
instrumental in bringing these numerals into universal use
among mathematicians. By the end of the thirteenth century
it was generally assumed that all scientific men would be
acquainted with the system : thus Roger Bacon writing in that
century recommends algorism (that is, the arithmetic founded
on the Arab notation) as a necessary study for theologians who
ought, he says, "to abound in the power of numbering." -We
may then consider that by the year 1300, or at the latest 1350,
these numerals were familiar both to mathematicians and to
Italian merchants.
So great was Leonardo's reputation that the Emperor
Frederick II. stopped at Pisa in 1225 in order to hold a sort
of mathematical tournament to test Leonardo's skill, of which
he had heard such marvellous accounts. The competitors were
informed beforehand of the questions to be asked, some or all
of which were composed by John of Palermo, who was one of
Frederick's suite. This is the first time that we meet with an
instance of those challenges to solve particular problems which
were so common in the sixteenth and seventeenth centuries.
The first question propounded was to find a number of which
the square, when either increased or decreased by 5, would
remain a square. Leonardo gave an answer, which is correct,
namely 41/12. The next question was to find by the methods
used in the tenth book of Euclid a line whose length x
should satisfy the equation a;2 + 2a;2 + lOo; = 20. Leonardo
showed by geometry that the problem was impossible, but he
gave an approximate value of the root of this equation, namely,
1-22' 7" 42'" 33"" 4^ 40^, which is equal to 1-3688081075...,
and is correct to nine places of decimals.^ Another question
was as follows. Three men, A^ B, C, possess a sum of money u,
their shares being in the ratio 3:2:1. A takes away x, keeps
^ See Fr. Woepcke in Liouville's Journal for 1854, p. 401.
170 INTRODUCTION OF ARAB WORKS [ch. x
half of it, and deposits the remainder with D ; B takes away y,
keeps two-thirds of it, and deposits the remainder with D ; C
takes away all that is left, namely 2, keeps five-sixths of it, and
deposits the remainder with D. This deposit with D is found
to belong to A, B, and C in equal proportions. Find u, x, y,
and z. Leonardo showed that the problem was indeterminate,
and gave as one solution u = 47, ^ = 33, 3/ = 13, « = 1. The other
competitors failed to solve any of these questions.
The chief work of Leonardo is the Liber Abaci alluded to
above. This w^ork contains a proof of the well-known result
(a2 + b^) (c2 + cZ2) = (ac + bdf + {be - adf = {ad + bcf + {bd - acf.
He also WTote a geometry, termed Practica Geometriae^ which
was issued in 1220. This is a good compilation, and some
trigonometry is introduced ; among other propositions and
examples he finds the area of a triangle in terms of its sides.
Subsequently he published a Liber Quadratorum dealing with
problems similar to the first of the questions propounded at the
tournament.^ He also issued a tract dealing with determinate
algebraical problems : these are all solved by the rule of false
assumption in the manner explained above.
Frederick II. The Emperor Frederick LL.^ who was born
in 1194, succeeded to the throne in 1210, and died in 1250,
was not only interested in science, but did as much as any
other single man of the thirteenth century to disseminate a
knowledge of the works of the Arab mathematicians in western
Europe. The university of Naples remains as a monument
of his munificence. I have already mentioned that the presence
of the Jews had been tolerated in Spain on account of their
medical skill and scientific knowledge, and as a matter of fact
the titles of physician and algebraist ^ were for a long time
nearly synonymous ; thus the Jewish physicians were admirably
^ Fr. Woepcke in Liouville's Journal for 1855, p. 54, has given an analysis
of Leonardo's method of treating problems on square numbers.
'^ For instance the reader may recollect that in Don Qidxote (part ii,
ch. 15), when Samson Carasco is thrown by the knight from his horse and
has his ribs broken, an algehrista is summoned to bind up his wounds.
CH. x] FREDERICK II. JORDANUS 171
fitted both to get copies of the Arab works and to translate
them. Frederick II. made use of this fact to engage a staff of
learned Jews to translate the Arab works which he obtained,
though there is no doubt that he gave his patronage to them
the more readily because it was singularly offensive to the pope,
with whom he was then engaged in a quarrel. At any rate, by
the end of the thirteenth century copies of the works of Euclid,
Archimedes, Apollonius, Ptolemy, and of several Arab authors
were obtainable from this source, and by the end of the next
century were not uncommon. From this time, then, we may
say that the development of science in Europe was independent
of the aid of the Arabian schools.
Jordanus.^ Among Leonardo's contemporaries was a German
mathematician, whose works were until the last few years almost
unknown. This was Jordanus Nemorarius^ sometimes called
Jordanus de Saxo7iia or Teutonicus. Of the details of his life
we know but little, save that he was elected general of the
Dominican order in 1222. The works enumerated in the foot-
note 2 hereto are attributed to him, and if we assume that these
works have not been added to or improved by subsequent
annotators, we must esteem him one of the most eminent mathe-
maticians of the middle ages.
His knowledge of geometry is illustrated by his De Triangulis
and De Isoperimetris. The most * important of these is the
De Triangulis^ which is divided into four books. The first
book, besides a few definitions, contains thirteen propositions on
triangles which are based on Euclid's Elements. The second
^ See Cantor, chaps, xliii, xliv, where references to the authorities on
Jordanus are collected.
2 Prof. Curtze, who has made a special study of the subject, considers that
the following works are due to Jordanus. "Geometria vel de Triangulis,"
published by M. Curtze in 1887 in vol. vi of the Mitteilungen des Copernicus-
Vereins zu Thorn ; De Isoperimetris ; Arithmetica Demonstrata, published
by Faber Stapulensis at Paris in 1496, second edition, 1514 ; Algorithmus
Demonstratus, irablished by J. Schciner at Nuremberg in 1534 ; De Numeris
Datis, published by P. Treutleiu iu 1879 and edited in 1891 with comments
by M. Curtze in vol. xxxvi of the Zeitschrift f\lr Mathematik imd Physik ;
De Ponder ibus, published by P, Apian at Nuremberg in 1533, and reissued
at Venice in 1565 ; and, lastly, two or three tracts on Ptolemaic astronomy.
172 INTRODUCTION OF ARAB WORKS [ch. x
book contains nineteen propositions, mainly on the ratios of
straight lines and the comparison of the areas of triangles ; for
example, one problem is to find a point inside a triangle so that
the lines joining it to the angular points may divide the triangle
into three equal parts. The third book contains twelve proposi-
tions mainly concerning arcs and chords of circles. The fourth
book contains twenty -eight propositions, partly on regular
polygons and partly on miscellaneous questions such as the
duplication and tri section problems.
The Algorithmus Demonstratus contains practical rules for
the four fundamental processes, and Arabic numerals are
generally (but not always) used. It is divided into ten books
dealing with properties of numbers, primes, perfect numbers,
polygonal numbers, ratios, powers, and the progressions. It
would seem from it that Jordanus knew the general expres-
sion for the square of any algebraic multinomial.
The De Numeris Datis consists of four books containing
solutions of one hundred and fifteen problems. Some of these
lead to simple or quadratic equations involving more than one
unknown quantity. He shews a knowledge of proportion ; but
many of the demonstrations of his general propositions are only
numerical illustrations of them.
In several of the propositions of the Algorithmus and De
Numeris Datis letters are employed to denote both known and
unknown quantities, and they are used in the demonstrations of
the rules of arithmetic as well as of algebra. As an example
of this I quote the following proposition,^ the object of which is
to determine two quantities whose sum and product are known.
Daio numero per duo diuiso si, quod ex dudu unius in alterum pro-
ducitur, datum fuer it, et utrumque eorum datum esse necesse est.
Sit numerus datus ahc diuisus in ah et c, atque ex ah in c fiat d datus,
itemque ex ahc in se fiat e. Sumatur itaque quadruplnni d, qui fit /, quo
dempto de e remaueat g, et ipse erit quadratum differentiae ah ad c.
Extraliatur ergo radix ex g, et sit hy eritque h differentia ah ad c. cumque
sic h datum, erit et c et ah datum.
^ From the De Numeris Datis, book i, prop. 3.
CH.x] JORDANUS 173
Huius operatic facile constabit hoc modo. Yerbi gratia sit x diiiisus
in numeros duos, atque ex ductu unius eorum in alium fiat xxi ; cuius
quadruplum et ipsum est lxxxiiii, tollatur de quadrato x, hoc est c, et
remanent xvi, cuius radix extrahatur, quae erit quatuor, et ipse est
differentia. Ipsa tollatur de x et reliquum, quod est vi, dimidietur,
eritque medietas iii, et ipse est minor portio et maior vii.
It ^^iHr be noticed that Jordanus, like Diophantus and the
Hindoos, denotes addition by juxtaposition. Expressed in
modern notation his argument is as follows. Let the numbers
he a + b (which I will denote by 7) and c. Then y + c is
given ; hence (7 + c)^ is known ; denote it by e. Again yc is
given ; denote it hy d ; hence 4yc, which is equal to id, is
known ; denote it by /. Then (7 - c)^ is equal to e-f, which
i^ known ; denote it by g. Therefore 7 - c = J(/, which is
known ; denote it by h. Hence 7 + c and 7 - c are know^n,
and therefore 7 and c can be at once found. It is curious
that he should have taken a sum like a + b for one of his
unknowns. In his numerical illustration he takes the sum to
be 10 and the product 21.
Save for one instance in Leonardo's writings, the above
works are the earliest instances known in Eurojjean mathematics
of syncopated algebra in which letters are used for algebraical,
symbols. It is probable that the A Igorithmus was not generally
known until it was printed in 1534, and it is doubtful how far
the works of Jordanus exercised any considerable influence on
the development of algebra. In fact it constantly happens in
the history of mathematics that improvements in notation or
method are made long before they are generally adopted or
their advantages realized. Thus the same thing may be dis-
covered over and over again, and it is not until the general
standard of knowledge requires some such improvement, or it is
enforced by some one whose zeal or attainments compel atten-
tion, that it is adopted and becomes part of the science.
Jordanus in using letters or symbols to represent any quantities
which occur in analysis was far in advance of his contemporaries.
A similar notation was tentatively introduced by other and
174 INTRODUCTION OF ARAB WORKS [ch. x
later mathematicians, but it was not until it had been thus
independently discovered several times that it came into general
use.
It is not necessary to describe in detail the mechanics, optics,
or astronomy of Jordanus. The treatment of mechanics
throughout the middle ages was generally unintelligent.
No mathematicians of the same ability as Leonardo and
Jordanus appear in the history of the subject for over two
hundred years. Their individual achievements must not be
taken to imply the standard of knowledge then current, but
their works were accessible to students in the following two
centuries, though there were not many who seem to have
derived much benefit therefrom, or who attempted to extend the
bounds of arithmetic and algebra as there expounded.
During the thirteenth century the most famous centres of
learning in western Europe were Paris and Oxford, and I -must
now refer to the more eminent members of those schools.
Holywood.^ I will begin by mentioning John de Holy wood,
whose name is often written in the latinized form of Sacrobosco.
Holywood was born in Yorkshire and educated at Oxford ; but
after taking his master's degree he moved to Paris, and taught
there till his death in 1244 or 1246. His lectures on algorism
and algebra are the earliest of which I can find mention. His
work on arithmetic was for many years a standard authority ; it
contains rules, but no proofs; it was printed at Paris in 1496.
He also wrote a treatise on the sphere, which was made public
in 1256: 'this had a wide and long-continued circulation, and
indicates how rapidly a knowledge of mathematics was spreading.
Besides these, two pamphlets by him, entitled respectively De
Compiito Ecclesiastico and De Astrolabio, are still extant.
Roger Bacon. ^ Another contemporary of Leonardo and
^ See Cantor, chap. xlv.
2 See Roger Bacon, sa vie, ses ouvrages ... by E. Charles, Paris, 1861 ;
aud the memoir by J. S. Brewer, prefixed to the Opera Inedita, Rolls Series,
London, 1859 : a somewhat depreciatory criticLsiri of the former of these
works is given in Roger Bacon, cine Monographic, by L. Schneider, Augsburg,
1873.
CH. x] ROGER BACON 175
Jordanus was Roger Bacon, who for physical science did work
somewhat analogous to what they did for arithmetic and
algebra. Roger Bacon was born near Ilchester in 1214, and
died at Oxford on June 11, 1294. He was the son of royalists,
most of whose property had been confiscated at the end of the
civil -v^ara; at an early age he was entered as a student at
Oxford, and is said to have taken orders in 1233. In 1234
he removed to Paris, then the intellectual capital of western
Europe, where he lived for some years devoting himself especi-
ally to languages and physics; and there he spent on books
and experiments all that remained of his family property and
his savings. He returned to Oxford soon after 1240, and there
for the following ten or twelve years he laboured incessantly,
being chiefly occupied in teaching science. His lecture room
was crowded, but everything that he earned was spent in buying
manuscripts and instruments. He tells us that altogether at
Paris and Oxford he spent over £2000 in this way — a sum
which represents at least £20,000 nowadays.
Bacon strove hard to replace logic in the university curri-
culum by mathematical and linguistic studies, but the influences
of the age were too strong for him. His glowing eulogy on
" divine mathematics " which should form the foundation of a
liberal education, and which " alone can purge the intellect
and fit the student for the acquirement of all knowledge," fell
on deaf ears. We can judge how small was the amount of
geometry which was implied in the quadrivium, when he tells us
that in geometry few students at Oxford read beyond Euc. i, 5 ;
though we might perhaps have inferred as much from the
character of the work of Boethius.
At last worn out, neglected, and ruined. Bacon was per-
suaded by his friend Grosseteste, the great Bishop of Lincoln,
to renounce the world and take the Franciscan vows. The
society to which he now found himself confined was singularly
uncongenial to him, and he beguiled the time by writing on
scientific questions and perhaps lecturing. The superior of the
order heard of this, and in 1257 forbade him to lecture or
176 INTRODUCTION OF AEAB WORKS [ch. x
publish anything under penalty of the most severe punishments,
and at the same time directed him to take up his residence at
Paris, where he could be more closely watched.
Clement IV., when in England, had heard of Ration's abilities,
and in 1266 when he became Pope he invited Bacoft to write.
The Franciscan order reluctantly permitted him to do so, but
they refused him any assistance. With difficulty Bacon obtained
sufficient money to get paper and the loan of books, and in the
shott space of fifteen months he produced in 1267 his Opus
Majus with two supplements which summarized what was then
known in physical science, and laid down the principles on which
it, as well as philosophy and literature, should be studied. He
stated as the fundamental principle that the study of natural
science must rest solely on experiment ; and in the fourth part
he explained in detail how astronomy and physical sciences rest
ultimately on mathematics, and progress only when their funda-
mental principles are expressed in a mathematical form. Mathe-
matics, he says, should be regarded as the alphabet of all
philosophy.
The results that he arrived at in this and his other works
are nearly in accordance with modern ideas, but were too far
in advance of that age to be capable of appreciation or perhaps
even of comprehension, and it was left for later generations to
rediscover his works, and give him that credit which he never
experienced in his lifetime. In astronomy he laid down the
principles for a reform of the calendar, explained the pheno-
mena of shooting stars, and stated that the Ptolemaic system
was unscientific in so far as it rested on the assumption that
circular motion was the natural motion of a planet, while the
complexity of the explanations required made it improbable
that the theory was true. In optics he enunciated the laws of
reflexion and in a general way of refraction of light, and used
them to give a rough explanation of the rainbow and of magnify-
ing glasses. Most of his experiments in chemistry were directed
to the transmutation of metals, and led to no useful results. He
gave the composition of gunpowder, but there is no doubt that it
CH. x] ROGER BACON. CAMPANUS 177
was not his own invention, though it is the earliest European
mention of it. On the other hand, some of his statements
appear to be guesses which are more or less ingenious, while
some of them are certainly erroneous.
In the years immediately following the publication of his
Opus Jlajics he wrote numerous works which developed in
detail the principles there laid down. Most of these have now
been published, but I do not know of the existence of any
complete edition. They deal only with applied mathematics
and physics.
Clement took no notice of the great work for which he had
asked, except to obtain leave for Bacon to return to England.
On the death of Clement, the general of the Franciscan order
was elected Pope and took the title of Nicholas IV. Bacon's
investigations had never been approved of by his superiors,
and he was now ordered to return to Paris, where we are told
he was immediately accused of magic; he was condemned in
1280 to imprisonment for life, but was released about a year
before his death.
Campanus. The only other mathematician of this century
whom I need mention is Giovanni Campano, or in the latinized
form Campanus, a canon of Paris. A copy of Adelhard''s trans-
lation of Euclid's Elements fell into the hands of Campanus, who
added a commentary thereon in which he discussed the properties
of a regular re-entrant pentagon. ^ He also, besides some minor
works, wrote the Theory of the Planets, which was a free
translation of the Almagest.
The fourteenth century. The history of the fourteenth
century, like that of the one preceding it, is mostly concerned
with the assimilation of Arab mathematical text-books and of
Greek books derived from Arab sources.
Bradwardine.^ A mathematician of this time, who was
^ This edition of Euclid was printed by Ratdolt at Venice in 1482, and
was formerly believed to be due to Campanus. On this work see J. L.
Heiberg in the Zeitschrift filr Matheviatik, vol. xxxv, 1890.
'■^ See Cantor, vol. ii, p. 102 et seq.
N
178 INTRODUCTION OF ARAB WORKS [ch. x
perhaps sufficiently influential to justify a mention here, is
Thomas Bradwardine, Archbishop of Canterbury. BradWardine
was born at Chichester about 1290. He was educated at
Merton College, Oxford, and subsequently lectured in that
university. From 1335 to the time of his death he was chiefly
occupied with the politics of the church and state ; he took a
prominent part in the invasion of France, the capture of Calais,
and the victory of Cressy. He died at Lambeth in 1349. His
mathematical works, which were probably written when he was
at Oxford, are the Tractatus de ProportionibuSj jDrinted at Paris
in 1495 ; the Arithmetica Speculativa, printed at Paris in 1502 ;
the Geometria Speculativa, printed at Paris in 1511 ; and the
De Quadratura Circuli, printed at Paris in 1495. They prob-
ably give a fair idea of the nature of the mathematics then read
at an English university.
Oresmus..^ Nicholas Oresmus was another writer of the
fourteenth century. He was born at Caen in 1323, became the
confidential adviser of Charles V., by whom he was made tutor
to Charles VI., and subsequently was appointed bishop of
Lisieux, at which city he died on July 11, 1382. He wrote the
Algorismus Proportionum, in which the idea of fractional indices
is introduced. He also issued a treatise dealing with questions
of coinage and commercial exchange ; from the mathematical
point of view it is noticeable for the use of vulgar fractions and
the introduction of symbols for them.
By the middle of this century Euclidean geometry (as
expounded by Campanus) and algorism were fairly familiar to
all professed mathematicians, and the Ptolemaic astronomy was
also generally known. About this time the almanacks began to
add to the explanation of the Arabic symbols the rules of
addition, subtraction, multiplication, and division, "de algorismo."
The more important calendars and other treatises also inserted
a statement of the rules of proportion, illustrated by various
practical questions.
^ See Die mathematischen Schriften des Nicole Oresme, by M. Curtze,
Thorn, 1870.
cH.x] THE FOURTEENTH CENTURY 179
In the latter half of this century there was a general revolt
of the universities against the intellectual tyranny of the school-
men. This was largely due to Petrarch, who in his own genera-
tion was celebrated as a humanist rather than as a poet, and
who exerted all his power to destroy scholasticism and encourage
scholarship. The result of these influences on the study of
mathematics may be seen in the changes then introduced in
the study of the quadrivium. The stimulus came from the
university of Paris, where a statute to that efi'ect was passed
in 1366, and a year or two later similar regulations were
made at other universities ; unfortunately no text-books are
mentioned. We can, however, form a reasonable estimate of
the range of mathematical reading required, by looking at
the statutes of the universities of Prague, of Vienna, and of
Leipzig.
By the statutes of Prague, dated 1384, candidates for the
bachelor's degree were required to have read Holywood's treatise
on the sphere, and candidates for the master's degree to be
acquainted with the first six, books of Euclid, optics, hydrostatics,
the theory of the lever, and astronomy. Lectures were actually
delivered on arithmetic, the art of reckoning with the fingers,
and the algorism of integers ; on almanacks, which probably
meant elementary astrology ; and on the Almagest ^ that is, on
Ptolemaic astronomy. There is, however, some reason for
thinking that mathematics received far more attention here than
was then usual at other universities.
At Vienna, in 1389, a candidate for a master's degree was
required to have read five books of Euclid, common perspective,
proportional parts, the measurement of superficies, and the
Theory of the Planets. The book last named is the treatise by
Campanus which was founded on that by Ptolemy. This was a
fairly respectable mathematical standard, but I would remind
the reader that there was no such thing as "plucking" in a /
medieval university. The student had to keep an act or give
a lecture on certain subjects, but whether he did it well or
badly he got his degree, and it is probable that it was only the
180 INTRODUCTION OF ARAB WORKS [ch. x
few students whose interests were mathematical who really
mastered the subjects mentioned above.
The fifteenth century. A few facts gleaned from the history
of the fifteenth century tend to shew that the regulations about
the study of the quadrivium were not seriously enforced. The
lecture lists for the years 1437 and 1438 of the university of
Leipzig (founded in 1409, the statutes of which are almost
identical with those of Prague as quoted above) are extant, and
shew that the only lectures given there on mathematics in those
years were confined to astrology. The records of Bologna,
Padua, and Pisa seem to imply that there also astrology was
the only scientific subject taught in the fifteenth century, and
even as late as 1598 the professor of mathematics at Pisa was
required to lecture on the Quadripartitum^ an astrological work
purporting (probably falsely) to have been written by Ptolemy.
The only mathematical subjects mentioned in the registers of
the university of Oxford as read there between the years 1449
and 1463 were Ptolemy's astronomy, or some commentary on it,
and the first two books of Euclid. Whether most students got
as far as this is doubtful. It would seem, from an edition of
Euclid's Elements published at Paris in 1536, that after 1452
candidates for the master's degree at that university had to take
an oath that they had attended lectures on the first six books of
that work.
Beldomandi. The only writer of this time that I need
mention here is Prodocimo Beldomandi of Padua, born about
1380, who wrote an algoristic arithmetic, published in 1410,
which contains the summation of a geometrical series; and
some geometrical works. ^
By the middle of the fifteenth century printing had been
introduced, and the facilities it gave for disseminating knowledge
were so great as to revolutionize the progress of science. We
have now arrived at a time when the results of Arab and Greek
science were known in Europe ; and this perhaps, then, is as
^ For further details see Boncompagni's Bulletino di Mbliografia,
vols, xii, xviii.
CH.x] THE FIFTEENTH CENTURY 181
good a date as can be fixed for the close of this period, and the
commencement of that of the renaissance. The mathematical
history of the renaissance begins with the career of Regiomon-
tanus ; but before proceeding with the general history it will be
convenient to collect together the chief facts connected with the
development of arithmetic during the middle ages and the
renaissance. To this the next chapter is devoted.
182
CHAPTER XL
THE DEVELOPMENT OF ARITHMETIC.^
CIKC. 1300-1637.
We have seen in the last chapter that by the end of the
thirteenth century the Arabic arithmetic had been fairly intro-
duced into Europe and was practised by the side of the older
arithmetic which was founded on the work of Boethius. It will
be convenient to depart from the chronological arrangement and
briefly to sum up the subsequent history of arithmetic, but I
hope, by references in the next chapter to the inventions and
improvements in arithmetic here described, that I shall be able
to keep the order of events and discoveries clear.
The older arithmetic consisted of. two parts : practical arith-
metic or the art of calculation which was taught by means of
the abacus and possibly the multiplication table ; and theoretical
arithmetic, by which was meant the ratios and properties of
numbers taught according to Boethius — a knowledge of the
latter being confined to professed mathematicians. The theo-
retical part of this system continued to be taught till the middle
of the fifteenth century, and the practical part of it was used by
^ See the article on Arithmetic by G. Peacock in the Encyclopaedia
Metroj)olitana, vol. i, London, 1845 ; Arithmetical Books by A. De Morgan,
London, 1847 ; and an article by P. Trentlein of Karlsruhe, in the Zeitschnft
fur Mathematik, 1877, vol. xxii, supplement, pp. 1-100.
X
CH. xi] THE DEVELOPMENT OF ARITHMETIC 183
the smaller tradesmen in England/ Germany, and France till
the beginning of the seventeenth century.
The new Arabian arithmetic was called algorism or the art of
Alkarismi, to distinguish it from the old or Boethian arithmetic.
The text-books on algorism commenced with the Arabic system
of notation, and began by giving rules for addition, subtraction,
multiplication, and division ; the principles of proportion were
then applied to various practical problems, and the books usually
concluded with general rules for many of the common problems of
commerce. Algorism was in fact a mercantile arithmetic, though
at first it also included all that was then known as algebra.
Thus algebra has its origin in arithmetic ; and to most people
the term universal arithmetic, by which it was sometimes desig-
nated, conveys a more accurate impression of its objects and
methods than the more elaborate definitions of modern mathe-
maticians— certainly better than the definition of Sir William
Hamilton as the science of pure time, or that of De Morgan as
the calculus of succession. No doubt logically there is a marked
distinction between arithmetic and algebra, for the former is the
theory of discrete magnitude, while the latter is that of continu-
ous magnitude ; but a scientific distinction such as this is of
comparatively recent origin, and the idea of continuity was not
introduced into mathematics before the time of Kepler.
Of course the fundamental rules of this algorism were not at
first strictly proved — that is the work of advanced thought —
but until the middle of the seventeenth century there was some
discussion of the principles involved ; since then very few arith-
meticians have attempted to justify or prove the processes used,
or to do more than enunciate rules and illustrate their use by
numerical examples.
^ Sec, for instance, Chaucer, TJie Miller s Tale^ v, 22-25 ; Shakespeare,
The Winter s Tale, Act iv, Sc. 2 ; Othello, Act I, Sc. 1. There are similar
references in French and German literature ; notably by Montaigne and
Moliere. I believe that the Exchequer division of the High Court of Justice
derives its name from the table before which the judges and officers of the
court originally sat : this was covered with black cloth divided into squares
or chequers by white lines, and apparently was used as an abacus.
184 THE DEVELOPMENT OF ARITHMETIC [ch. xi
I have alluded frequently to the Arabic system of numerical
notation. I may therefore conveniently begin by a few notes on
the history of the symbols now current.
Their origin is obscure and has been much disputed.^ On
the whole it seems probable that the symbols for the numbers 4,
5, 6, 7, and 9 (and possibly 8 too) are derived from the initial
letters of the corresponding words in the Indo-Bactrian alphabet
in use in the north of India perhaps 150 years before Christ;
that the symbols for the numbers 2 and 3 are derived respectively
from two and three parallel penstrokes written cursively ; and
similarly that the symbol for the number 1 represents a single
penstroke. Numerals of this type were in use in India before
the end of the second century of our era. The origin of the
symbol for zero is unknown ; it is not impossible that it was
originally a dot inserted to indicate a blank space, or it may
represent a closed hand, but these are mere conjectures ; there
is reason to believe that it was introduced in India towards the
close of the fifth century of our era, but the earliest writing now
extant in which it occurs is assigned to the eighth century.
The numerals used in India in the eighth century and for a
long time afterwards are termed Devanagari numerals, and their
forms are shewn in the first line of the table given on the next
page. These forms were slightly modified by the eastern Arabs,
and the resulting symbols were again slightly modified by the
western Arabs or Moors. It is perhaps probable that at first
the Spanish Arabs discarded the use of the symbol for zero, and
only reinserted it when they found how inconvenient the omission
proved. The symbols ultimately adopted by the Arabs are
termed Gobar numerals, and an idea of the forms most commonly
used may be gathered from those printed in the second' line of
the table given on next page. From Spain or Barbary the Gobar
numerals passed into western Europe, and they occur on a
Sicilian coin as early as 1138. The further evolution of
^ See A. L'Esprit, Histoire des chiffres, Paris, 1893 ; A. P. Pihan, Signes
de numeration, Paris, 1860 ; Fr. Woepcke, La propagation des chiffres
Jndiens, Paris, 1863 ; A. C. Burnell, South Indian Palaeography, Mangalore,
1874 ; Is. Taylor, The Alphabet, London, 1883 ; and Cantor.
CH. xi] HISTORY OF THE AEABIC SYMBOLS 185
the forms of the symbols to those with which we are familiar is
indicated below by facsimiles ^ of the numerals used at different
times. All the sets of numerals here represented are written
from left to right and in the order 1, 2, 3, 4, 5, 6, 7, 8, 9, 10.
Devanagari (Indian) nu-
merals, circ. 950.
Gobar Arabic numerals,
circ. 1100 (?).
From a missal, circ. 1385, \ ^'y'7(^J/^^KOr\^^
of German origin. / 'i ^..J^,^ ,C^ , O, A ,i>,j; , l\r
European(probablyltalian)
numerals, circ. 1400.
Mirrour of ttie'\ .
printed by Cax- \ 1,1,), 4, U,^, ^, 8, 9, I O
480. J
From the Mirrour of
World,
ton in 1480.
From a Scotch calendar "j
for 1482, probably of [ t, Z, 5, 9-» ^. (3, A, 8 , Q, I O
French origin. J
From 1500 onwards the symbols employed are practically the
same as those now in use.^
The further evolution in the East of the Gobar numerals
proceeded almost independently of European influence. There
are minute differences in the forms used by various writers, and
in some cases alternative forms ; without, however, entering into
these details we may say that the numerals they commonly
employed finally took the form shewn above, but the symbol
^ The first, second, and fourth examples are taken from Is. Taylor's
Alphabet, Loudon, 1883, vol. 11, p. 266 ; the others are takeu from Leslie's
Philosophy of Arithmetic, 2nd ed., Edinburgh, 1820, pp. 114, 115.
^ See, for example, Tonstall's De Arte Supputandi, Loudon, 1522 ; or
Record's Grounde of Artes, Londou, 1540, and Whetstone of Witfe, London,
1557.
186 THE DEVELOPMENT OF AEITHMETIC [ch. xi
there given for 4 is at the present time generally written
cursively.
Leaving now the history of the symbols I proceed to discuss
their introduction into general use and the development of
algoristic arithmetic. I have already explained how men of
science, and particularly astronomers, had become acquainted
with the Arabic system by the middle of the thirteenth century.
The trade of Europe during the thirteenth and fourteenth
centuries was mostly in Italian hands, and the obvious ad-
vantages of the algoristic system led to its general adoption
in Italy for mercantile purposes. This change was not effected,
however, without considerable opposition ; thus, an edict was
issued at Florence in 1299 forbidding bankers to use Arabic
numerals, and in 1348 the authorities of the university of Padua
directed that a list should be kept of books for sale with the
prices marked "non per cifras sed per literas claras."
The rapid spread of the use of Arabic numerals and arithmetic
through the rest of Europe seems to have been as largely due to
the makers of almanacks and calendars as to merchants and
men of science. These calendars had a wide circulation in
medieval times. Some of them were composed with special
reference to ecclesiastical purposes, and contained the dates of
the different festivals and fasts of the church for a period of
some seven or eight years in advance, as well as notes on church
ritual. Nearly every monastery and church of any pretensions
possessed one of these. Others were written specially for the
use of astrologers and physicians, and some of them contained
notes on various scientific subjects, especially medicine and astro-
nomy. Such almanacks were not then uncommon, but, since it
was only rarely that they found their way into any corporate
library, specimens are now rather scarce. It was the fashion to
use the Arabic symbols in ecclesiastical works; while their
occurrence in all astronomical tables and their Oriental origin
(which savoured of magic) secured their use in calendars intended
for scientific purposes. Thus the symbols were generally em-
ployed in both kinds of almanacks, and there are but few specimens
CH.XI] THE DEVELOPMENT OF ARITHMETIC 187
of calendars issued after the year 1 300 in which an explanation
of the Arabic numerals is not included. Towards the middle of
the fourteenth century the rules of arithmetic de algorismo were
also sometimes added, and by the year 1400 we may consider
that the Arabic symbols were generally known throughout
Europe, and were used in most scientific and astronomical
works.
Outside Italy most merchants continued, however, to keep
their accounts in Roman numerals till about 1550, and
monasteries and colleges till about 1650; though in both
cases it is probable that in and after the fifteenth century the
processes of arithmetic were performed in the algoristic manner.
Arabic numerals are used in the pagination of some books issued
at Venice in 1471 and 1482. No instance of a date or number
being written in Arabic numerals is known to occur in any
English parish register or the court rolls of any English
manor before the sixteenth century; but in the rent-roll of
the St Andrews Chapter, Scotland, the Arabic numerals
were used in 1490. The Arabic numerals were used in
Constantinople by Planudes ^ in the fourteenth century.
The history of modern mercantile arithmetic in Europe
begins then with its use by Italian merchants, and it is
especially to the Florentine traders and writers that we owe
its early development and improvement. It was they who
invented the system of book-keeping by double entry. In this
system every transaction is entered on the credit side in one
ledger, and on the debtor side in another ; thus, if cloth be sold
to A^ A^s account is debited with the price, and the stock-book,
containing the transactions in cloth, is credited with the amount
sold. It was they, too, who arranged the problems to which
arithmetic could be applied in different classes, such as rule of
three, interest, profit and loss, <fcc. They also reduced the
fundamental operations of arithmetic "to seven, in reverence,"
says Pacioli, " of the seven gifts of the Holy Spirit : namely,
numeration, addition, subtraction, multiplication, division,
^ See above, p. 117.
L
188 THE DEVELOPMENT OF ARITHMETIC [ch. xi
raisiDg to powers, and extraction of roots." Brahmagupta
had enumerated twenty processes, besides eight subsidiary ones,
and had stated that *' a distinct and several knowledge of these "
was " essential to all who wished to be calculators " ; and,
whatever may be thought of Pacioli's reason for the alteration,
the consequent simplification of the elementary processes was
satisfactory. It may be added that arithmetical schools were
founded in various parts of Germany, especially in and after the
fourteenth century, and did much towards familiarizing traders
in northern and western Europe with commercial algoristic
arithmetic.
The operations of algoristic arithmetic were at first very
cumbersoii^e. The chief improvements subsequently introduced
into the early Italian algorism were (i) the simplification of the
four fundamental processes ; (ii) the introduction of signs for
addition, subtraction, equality, and (though not so important)
for multiplication and division; (iii) the invention of
logarithms; and (iv) the use of decimals. I will consider
these in succession.
(i) In addition and subtraction the Arabs usually worked
from left to right. The modern plan of working from right to
left is said to have been introduced by an Englishman named
Garth, of whose life I can find no account. The old plan con-
tinued in partial use till about 1600; even now it would be
more convenient in approximations where it is necessary to keep
only a certain number of places of decimals.
The Indians and Arabs had several systems of multiplication.
These were all somewhat laborious, and were made the more so
as multiplication tables, if not unknown, were at any rate used
but rarely. The operation was regarded as one of considerable
difficulty, and the test of the accuracy of the result by " casting
out the nines " was invented as a check on the correctness of the
work. Various other systems of multiplication were subse-
quently employed in Italy, of which several examples are
given by Pacioli and Tartaglia; and the use of the multipli-
cation table — at least as far as 5 x 5 — became common. From
CH. xi] MULTIPLICATION 189
this limited table the-resulting product of the multiplication of
all numbers up to 10 x 10 can be deduced by what was termed
the regula ignavi. This is a statement of the identity
(5 + a) (5 + ^) = (5 - a) {^-b) + I0{a + h). The rule was usually
^undated in the following form. Let the number five be
represented by the open hand ; the number six by the hand with
one finger closed; the number seven by the hand with two
fingers closed ; the number eight by the hand with three fingers
closed; and the number nine by the hand with four fingers
closed. To multiply one number by another let the multiplier be
represented by one hand, and the number multiplied by the
other, according to the above convention. Then the required
answer is the product of the number of fingers (counting the
thumb as a finger) open in the one hand by the number of
fingers open in the other together with ten times the total
number of fingers closed. The system of multiplication now
in use seems to have been first introduced at Florence.
The difficulty which all but professed mathematicians
experienced in the multiplication of large numbers led to the
invention of several mechanical ways of effecting the process.
Of these the most celebrated is that of Napier's rods invented in
1617. In principle it is the same as a method which had been
long in use both in India and Persia, and which has been
described in the diaries of several travellers, and notably m
the Travels of Sir John Chardin in Persia, London, 1686.
To use the method a number of rectangular slips of bone, wood,
metal, or cardboard are prepared, and each of them divided by
cross lines into nine little squares, a slip being generally about
three inches long and a third of an inch across. In the top
square one of the digits is engraved, and the results of multiplying
it by 2, 3, 4, 5, 6, 7, 8, and 9 are respectively entered in the
eight lower squares ; where the result is a number of two digits,
the ten-digit is wTitten above and to the left of the unit-digit
and separated from it by a diagonal line. The slips are usually
arranged in a box. Figure 1 on the next page represents nine
such slips side by side ; figure 2 shews the seventh slip, which
190 THE DEVELOPMENT OF ARITHMETIC [ch. xi
is supposed to be taken out of the box and put hy itself.
Suppose we wish to multiply 2985 by 317. The process as
1
2
3
4
5
6
7
8
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Figure 1-
Figure 2.
Figure 3.
effected by the use of these slips is as follows. The slips headed
2, 9, 8, and 5 are taken out of the box and put side by side as
shewn in figure 3 above. The result of multiplying 2985 by 7
may be written thus —
2985
7
35
56
63
14
20895
Now if the reader will look at the seventh line in figure 3,
he will see that the upper and lower rows of figures are respec-
tively 1653 and 4365 ; moreover, these are arranged by the
diagonals so that roughly the 4 is under the 6, the 3 under the
5, and the 6 under the 3 ; thus
5
The addition of these two numbers gives the required result.
CH.xi] MULTIPLICATION 191
Hence the result of multiplying by 7, 1, and 3 can be succes-
sively determined in this way, and the required answer (namely,
the product of 2985 and 317) is then obtained by addition.
The whole process was written as follows : —
2985
20895 / 7
2985 / 1
8955 /3
946245
The modification introduced by Napier in his Rahdologia^
published in 1617, consisted merely in replacing each slip by a
prism with square ends, which he called " a rod," each lateral
face being divided and marked in the same way as one of the
slips above described. These rods not only economized space,
but were easier to handle, and were arranged in such a way as
to facilitate the operations required.
If multiplication was considered difficult, division was at first
regarded as a feat which could be performed only by skilled
mathematicians. The method commonly employed by the
Arabs and Persians for the division of one number by another
will be sufficiently illustrated by a concrete instance. Suppose
we require to divide 17978 by 472. A sheet of paper is divided
into as many vertical columns as there are figures in the number
to be divided. The number to be divided is written at the top
and the divisor at the bottom ; the first digit of each number
being placed at the left-hand side of the paper. Then, taking
the left-hand column, 4 will go into 1 no times, hence the first
figure in the dividend is 0, which is written under the last figure
of the divisor. This is represented in figure 1 on the next page.
Next (see figure 2) rewrite the 472 immediately above its
former position, but shifted one place to the right, and cancel
the old figures. Then 4 will go into 17 four times; but, as
on trial it is found that 4 is too big for the first digit of the
dividend, 3 is selected ; 3 is therefore written below the last
digit of the divisor and next to the digit of the dividend last
192 THE DEVELOPMENT OF ARITHMETIC [ch. xi
found. The process of multiplying the divisor by 3 and sub-
tracting from the number to be divided is indicated in figure
2, and shews that the remainder is 3818. A similar process is
1
1
7
9
7
8
r
-
1
1
7
2
9
7
8
1
1
7
2
9
7
8
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1
7
7
L
I
1
7
X
5
2
9
1
7
8
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7
8
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8
7
6
8
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8
7
6
8
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3
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4
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7
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4
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7
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7
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3
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8
Figure 1.
Figure 2.
Figure 3.
then repeated, that is, 472 is divided into 3818, shewing that
the quotient is 38 and the remainder 42. This is represented in
figure 3, which shews the whole operation.
The method described above never found much favour in
Italy. The present system was in use there as early as the
beginning of the fourteenth century, but the method generally
employed was that known as the galley or scratch system. The
following example from Tartaglia, in which it is required to
0 7
49
05 90
1 3 30( 15
844
8
CH.xi] . V DIVISION 193
divide 1330 by 84, will serve to illustrate this method : the
arithmetic given by Tartaglia is shewn above, where numbers in
Wn tyjje are supposed to be scratched out in the course of the
workr
The process is as follows. First write the 84 beneath the
1330, as indicated below, then 84 will go into 133 once, hence
the first figure in the quotient is 1. Now 1 x 8 = 8, which sub-
tracted from 13 leaves 5. Write this above the 13, and cancel
the 13 and the 8, and we have as the result of the first step
5
1330(1
84
Next, 1 X 4 = 4, which subtracted from 53 leaves 49. Insert
the 49, and cancel the 53 and the 4, and we have as the next
step
4
5 9
13 3 0(1
8 4
which shews a remainder 490.
We have now to divide 490 by 84. Hence the next figure
in the quotient will be 5, and re-writing the divisor we have
4
5 9
1 3 3 0 ( 15
8 44
8
Then 5 x 8 = 40, and this subtracted from 49 leaves 9. Insert
the 9, and cancel the 49 and the 8, and we have the following
result
4 9
5 9
1 3 3 0 ( 15
8 44
8
o
194 THE DEVELOPMENT OF ARITHMETIC [ch. xi
Next 5 X 4 = 20, and this subtracted from 90 leaves 70. Insert
the 70, and cancel the 90 and the 4, and the final result, shewing
a remainder 70, is
7
4 9
5 9 0
1 3 3 0( 15
8 44
The three extra zeros inserted in Tartaglia's work are unneces-
sary, but they do not affect the result, as it is evident that a
figure in the dividend may be shifted one or more places up in
the same vertical column if it be convenient to do so.
The medieval writers were acquainted with the method now
in use, but considered the scratch method more simple. In
some cases the latter is very clumsy, as may be illustrated by the
following example taken from Pacioli. The object is to divide
23400 by 100. The result is obtained thus
0
04 0
0 3 4 0 0
2 3 4 0 0 ( 234
10000
^ 100
1
The galley method was used in India, and the Italians may
have derived it thence. In Italy it became obsolete somewhere
about 1600 ; but it continued in partial use for at least another
century in other countries. I should add that Napier's rods can
be, and sometimes were used to obtain the result of dividing
one number by another.
(ii) The signs + and - to indicate addition and subtraction ^
occur in Widman's arithmetic published in 1489, but were first
brought into general notice, at any rate as symbols of opera-
tion, by Stifel in 1544. They occur, however, in a work by
1 See below, pp. 206, 207, 214, 216.
DIVISION 195
G. V. Hoecke, published at Antwerp in 1514. I believe I am
in saying that Vieta in 1591 was the first well-known
writer who used these signs consistently throughout his work,
and that it was not until the beginning of the seventeenth
century that they became recognized as well-known symbols.
The sign = to denote equality ^ was introduced by Record in
1557.
(iii) The invention of logarithms,^ without which many of
the numerical calculations which have constantly to be made
would be practically impossible, was due to Napier of Merchis-
ton. The first public announcement of the discovery was
made in his Mirijici Logarithmorum Can/mis Description pub-
lished in 1614, and of which an English translation was issued
in the following year; but he had privately communicated a
summary of his results to Tycho Brahe as early as 1594. In
this work Napier explains the nature of logarithms by a com-
parison between corresponding terms of an arithmetical and
geometrical progression. He illustrates their use, and gives
tables of the logarithms of the sines and tangents of all angles
in the first quadrant, for differences of every minute, calculated
to seven places of decimals. His definition of the logarithm of a
quantity n was what we should now express by lO^logg (W/n).
This work is the more interesting to us as it is the first valuable
contribution to the progress of mathematics which was made by
any British writer. The method by which the logarithms were
calculated was explained in the Constriictio, a posthumous work
issued in 1619 : it seems to have been very laborious, and
depended either on direct involution and evolution, or on the
formation of geometrical means. The method by finding the
approximate value of a convergent series was introduced by
Newton, Cotes, and Eukr. Napier had determined to change
the base to one which was a power of 10, but died before he
could effect it.
^ See below, p. 214.
'^ See the article on Logarithms iu the Encyclopaedia Britannica, ninth
edition ; see also below, pp. 236, 237.
196 THE DEVELOPMENT OF ARITHMETIC [ch. xi
The rajDid recognition tliroughout Europe of the advantages
of using logarithms in practical calculations was mainly due to
Briggs, who was one of the earliest to recognize the value of
Napier's invention. Briggs at once realized that the base to
which Napier's logarithms were calculated was inconvenient;
he accordingly visited Napier in 1616, and urged the change
to a decimal base, which was recognized by Napier as an im-
provement. On his return Briggs immediately set to work to
calculate tables to a decimal base, and in 1617 he brought out
a table of logarithms of the numbers from 1 to 1000 calculated
to fourteen places of decimals.
It would seem that J. Biirgi, independently of Napier, had
constructed before 1611 a table of antilogarithms of a series of
natural numbers: this was published in 1620. In the same
year a table of the logarithms, to seven places of decimals, of
the sines and tangents of angles in the first quadrant was
brought out by Edmund Gunter, one of the Gresham lecturers.
Four years later the latter mathematician introduced a " line of
numbers," which provided a mechanical method for finding the
product of two numbers : this was the precursor of the slide-
rule, first described by Oughtred in 1632. In 1624, Briggs pub-
lished tales of the logarithms of some additional numbers and of
various trigonometrical functions. His logarithms of the natural
numbers are equal to those to the base 10 when multiplied by
10^, and of the sines of angles to those to the base 10 when
multiplied by 10^^. The calculation of the logarithms of
70,000 numbers which had been omitted by Briggs from his
tables of 1624 was performed by Adrian Vlacq and published
in 1628: with this addition the table gave the logarithms of
all numbers from 1 to 101,000.
The Arithmetica Logarithmica of Briggs and Vlacq are sub-
stantially the same as the existing tables : parts have at different
times been recalculated, but no tables of an equal range and
fulness entirely founded on fresh computations have been pub-
lished since. These tables were supplemented by Briggs's
Trigonometrica Britannica^ which contains tables not only of
CH. xi] THE DEVELOPMENT OF ARITHMETIC 197
e logaritlims of the trigonometrical functions, but also of their
natural values: it was published posthumously in 1633. A
table of logarithms to the base e of the numbers from 1 to 1000
and of the sines, tangents, and secants of angles in the first
quadrant was published by John Speidell at London as early
as 1619, but of course these were not so useful in practical
calculations as those to the base 10. By 1630 tables of
logarithms were in general use.
(iv) The introduction of the decimal notation for fractions
is also (in my opinion) due to Briggs. Stevinus had in 1585
used a somewhat similar notation, for he wrote a number
such as 25*379 either in the form 25, 3' 7" 9'", or in the form
25 ©307090; Napier in 1617 in his essay on rods had
adopted the former notation ; and Rudolff had used a somewhat
similar notation. Biirgi also employed decimal fractions, writing
141*4 as Q . But the above-mentioned writers had employed
the notation only as a concise way of stating results, and made
no use of it as an operative form. The same notation occurs, how-
ever, in the tables published by Briggs in 1617, and would
seem to have been adopted by him in all his works ; and, though
it is difficult to' speak with absolute certainty, I have myself but
little doubt that he there employed the symbol as an operative
form. In Napier's posthumous Constriictio, published in 1619,
it is defined and used systematically as an operative form, and
as this work was written after consultation with Briggs, about
1615-6, and probably was revised by the latter before it was issued,
I think it confirms the view that the invention is due to Briggs
and was communicated by him to Napier. At any rate it was
not employed as an operative form by Napier in 1617, and, if
Napier were then acquainted with it, it must be supposed that
he regarded its use as unsuitable in ordinary arithmetic. Before
the sixteenth century fractions were commonly written in the
sexagesimal notation.^
In Napier's work of 1619 the point is written in the form now
^ For examples, see above, pp. 97, 101, 169.
198 THE DEVELOPMENT OF ARITHMETIC [ch. xi
adopted in England. Witt in 1613 and Napier in 1617 used
a solidus to separate the integral from the fractional part.
Briggs underlined the decimal figures, and would have printed a
number such as 25*379 in the form 25379. Subsequent writers
added another line, and would have written it as 25|379 ; nor
was it till the beginning of the eighteenth century that the
current notation was generally employed. Even now the
notation varies slightly in different countries : thus the fraction
J would in the decimal notation be written in England as 0-25,
in America as 0.25, and in Germany and France as 0,25. A
knowledge of the decimal notation became general among
practical men with the introduction of the French decimal
standards.
1%
CHAPTER XII.
THE MATHEMATICS OF THE RENAISSANCE.^
CIRC. 1450-1637.
The last chapter is a digression from the chronological arrange-
ment to which, as far as possible, I have throughout adhered,
but I trust by references in this chapter to keep the order of
events and discoveries clear. I return now to the general
history of mathematics in western Europe. Mathematicians
had barfily_assiniilated the knowledge obtained from the Arabs,
Jncliiding their translations of Greek writerSj when the refugees
whoe§ca^e(iJrom Constantinople after the fall of the eastern
empire brought the original works and the traditions of Greek
science into Italy. Thus by the middle of the fifteenth century
the chief results of Greek and Arabian mathematics were
accessible to European students.
The invention of printing about that time rendered the
dissemination of discoveries comparatively easy. It is almost a
truism to remark that until printing was introduced a writer
appealed to a very limited class of readers, but w^e are perhaps
apt to forget that when a medieval writer " published " a work
the results were known to only a few of his contemporaries.
^ Where no other references are given, see parts xii, xiii, xiv, and the early
chapters of part xv of Cantor's Vorlesungen ; on the Italian mathematicians
of this period see also G. Libri, Histoire des sciences mathhnatiques en Italic,
4 vols., Paris, 1838-1841.
200 MATHEMATICS OF THE EENAISSANCE [ch. xii
This had not been the case in classical times, for then and
until the fourth century of our era Alexandria was the recog-
nized centre for the reception and dissemination of new works
and discoveries. In medieval Europe, on the other hand, there
was no common centre through which men of science could
communicate wdth one another, and to this cause the slow
and fitful development of medieval mathematics may be partly
ascribed.
The introduction of printing marks the beginning of the
modern world in science as in politics; for it was contempo-
raneous with the assimilation by the indigenous European
school (which was born from scholasticism, and whose history
was traced in chapter viii) of the results of the Indian and
Arabian schools (whose history and influence were traced in
chapters ix and x), and of the Greek schools (whose history
was traced in chapters ii to v).
The last two centuries of this period of our history, which
may be described as the renaissance, were distinguished by
great mental activity in all branches of learning. The creation
of a fresh group of universities (including those in Scotland),
of a somewhat less complex type than the medieval universities
above described, testify to the general desire for knowledge.
The discovery of America in 1492 and the discussions that
preceded the Reformation flooded Europe with new ideas which,
by the invention of printing, were widely disseminated ; butjhe^
advance in mathematics was at least as well marked as that in^
literature and that in politics.
During the first part of this time the attention of mathe-
maticians was to a large extent concentrated on syncopated
algebra and trigonometry; the treatment of these subjects is
discussed in the first section of this chapter, but the relative
importance of the mathematicians of this period is not very
easy to determine. The middle years of the renaissance were
distinguished by the development of symbolic algebra : this is
treated in the second section of this chapter. The close of the
sixteenth century saw the creation of the science of dynamics :
REGIOMONTANUS 201
this forms the subject of the first section of chapter xiii.
About the same time and in the early years of the seventeenth
century considerable attention was paid to pure geometry : this
forms the subject of the second section of chapter xiii.
The development of syncopated algebra and trigonometry.
Eegiomontanus.^ Amongst the many distinguished writers
of this time Johann Regiomontanus was the earliest and one
of the most able. He was born at Konigsberg on June 6,
1436, and died at Rome on July 6, 1476. His real name
was Johannes Miiller^ but, following the custom of that time,
he issued his publications under a Latin pseudonym which in
his case was taken from his birthplace. To his friends, his
neighbours, and his tradespeople he may have been Johannes
Miiller, but the literary and scientific world knew him as
Regiomontanus, just as they knew Zepernik as Copernicus,
and Schwarzerd as Melanchthon. It seems as pedantic as it is
confusing to refer to an author by his actual name when he
is universally recognized under another : I shall therefore in all
cases as far as possible use that title only, whether latinized or
not, by which a writer is generally known.
Regiomontanus studied mathematics at the university of
Vienna, then one of the chief centres of mathematical studies
in Europe, under Purbach who was professor there. His
first work, done in conjunction with Purbach, consisted of an
analysis of the Almagest. In this the trigonometrical functions
sine and cosine were used and a table of natural sines was
introduced. Purbach died before the book was finished : it
was finally published at Venice, but not till 1496. As soon as
this was completed Regiomontanus wTote a work on astrology,
^ His life was written by P. Gassendi, The Hague, second edition, 1655.
His letters, which afford much valuable information on the mathematics of his
time, were collected and edited by C. G. von Murr, Nuremberg, 1786. An
account of his works will be found in Regiomontanus, ein geistiger Vorldufer
des Copernicus, by A. Ziegler, Dresden, 1874 ; see also Cantor, chap. Iv.
202 MATHEMATICS OF THE RENAISSANCE [ch. xii
which contains some astronomical tables and a table of natural
tangents : this was published in 1490.
Leaving Vienna in 1462, Regiomontanus travelled for some
time in Italy and Germany; and at last in 1471 settled for
a few years at Nuremberg, where he established an observatory,
opened a printing-press, and probably lectured. Three tracts on
astronomy by' him were written here. A mechanical eagle, which
flapped its wings and saluted the Emperor Maximilian I. on his
entry into the city, bears witness to his mechanical ingenuity,
and was reckoned among the marvels of the age. Thence
Regiomontanus moved to Rome on an invitation from Sixtus IV.
who wished him to reform the calendar. He was assassinated,
shortly after his arrival, at the age of 40^.
Regiomontanus was among the first to take advantage of
the recovery of the original texts of the Greek mathematical
works in order to make himself acquainted with the methods of
reasoning and results there used ; the earliest notice in modern
Europe of the algebra of Diophantus is a remark of his that he
had seen a copy of it at the Vatican. He was also well read in
the works of the Arab mathematicians.
The fruit of his study was shewn in his De Triangulis
written in 1464. This is the earliest modern systematic
exposition of trigonometry, plane and spherical, though the
only trigonometrical functions introduced are those of the sine
and cosine. It is divided into five books. The first four are
given up to plane trigonometry, and in particular to deter-
mining triangles from three given conditions. The fifth book
is devoted to spherical trigonometry. The work was printed
at Nuremberg in 1533, nearly a century after the death of
Regiomontanus.
As an example of the mathematics of this time I quote one
of his propositions at length. It is required to determine a
triangle when the difference of two sides, the perpendicular on
the base, and the diflference between the segments into which
the base is thus divided are given [book ii, prop. 23]. The
following is the solution given by Regiomontanus.
CH. XIl]
EEGIOMONTANUS
203
Sit talis triangiilus ABG, cujas duo latera AB ei AG differentia
habeant nota HG, ductaque perpendiculari AD duorum casuum BD et
DG, differentia sit EG : hae duae differentiae sint datae, et ipsa perpen-
dicularis AD data. Dico quod omnia latera trianguli nota concludentur.
Per artem rei et census hoc problenia absolvemus. Detur ergo differentia
laterum ut 3, differentia casuum 12, et perpendicularis 10. Pono pro
basi unam rem, et pro aggregato laterum 4 res, nae proportio basis ad
congeriem laterum est ut HG ad GE, scilicet unius ad 4. Erit ergo BD
i rei minus 6, sed AB erit 2 res demptis f . Duco AB in se, producuntur
4 census et 2| demptis 6 rebus. Item BD in se facit J census et 36
minus 6 rebus : huic addo quadratum de 10 qui est 100. Colliguntur ^
census et 136 minus 6 rebus aequales videlicet 4 censibus et 2^ demptis
6 rebus. Restaurando itaque defectus et auferendo utrobique aequalia,
quemadmodum ars ipsa praecipit, habemus census aliquot aequales
numero, unde cognitio rei patebit, et inde tria latera trianguli more suo
innotescet.
To explain the language of the proof I should add that
Regiomontanus calls the unknown quantity res, and its square
census or zensus ; but though he uses these technical terms he
writes the words in full. He commences by saying that he will
solve the problem by means of a quadratic equation (per artem
rei et census) ; and that he will suppose the difference of the
sides of the triangle to be 3, the difference of the segments of
the base to be 12, and the altitude of the triangle to be 10.
He then takes for his unknown quantity (unam rem or x) the
base of the triangle, and therefore the sum of the sides will be
ix. Therefore BD will be equal to i^ - 6 (i rei minus 6), and
AB will be equal to 2j7 - | (2 res demptis f ) ; hence AB- {AB
204 MATHEMATICS OF THE RENAISSANCE [ch. xii
in se) will be 4^^ + 2 J - 6^ (4 census et 2^ demptis 6 rebus),
and BD'^ will be Ix'- + 36 - 6^. To BD'^ he adds AD"^ (quad-
ratum de 10) which is 100, and states that the sum of the two
is equal to AB"^. This he says will give the value of x'^ (census),
whence a knowledge of x (cognitio rei) can be obtained, and the
triangle determined.
To express this in the language of modern algebra we have
AG^-DG^ = AB''--DB\
.'. AG'^-AB'^ = DG^-DB\
but by the given numerical conditions
AG - AB^^ = \ {DG - DB),
.'. AG + AB = i {DG + DB) = ix.
Therefore AB = 2x- 1, and BD = ^x-6.
Hence (2x - 1)2 = (^x - 6)2 + 100.
From which x can be found, and all the elements of the triangle
determined.
It is worth noticing that Regiomontanus merely aimed at
giving a general method, and the numbers are not chosen with
any special reference to the particular problem. Thus in his
diagram he does not attempt to make GU anything like four
times as long as Gil, and, since x is ultimately found to be
equal to ^ ;^321, the point D really falls outside the base. The
order of the letters ABG, used to denote the triangle, is of
course derived from the Greek alphabet.
Some of the solutions which he gives are unnecessarily
complicated, but it must be remembered that algebra and
trigonometry were still only in the rhetorical stage of develop-
ment, and when every step of the argument is expressed in
words at full length it is by no means easy to realize all that is
contained in a formula.
It will be observed from the above example that Regiomon-
tanus did not hesitate to apply algebra to the solution of
geometrical problems. Another illustration of this is to be found
in his discussion of a question which appears in Brahmagupta's
CH.xii] REGIOMONTANUS. PURBACH. CUSA 205
Siddhanta. The problem was to construct a quadrilateral,
having its sides of given lengths, which should be inscribable in
a circle. The solution ^ given by Regiomontanus was effected
by means of algebra and trigonometry.
The Algorithmus Demonstratus of Jordanus, described above,
which was first printed in 1534, was formerly attributed to
Regiomontanus.
Regiomontanus was one of the most prominent mathema-
ticians of his generation, and I have dealt with his works in
some detail as typical of the most advanced mathematics of
the time. Of his contemporaries I shall do little more than
mention the names of a few of those who are best known ;
none were quite of the first rank, and I should sacrifice the
proportion of the parts of the subject were I to devote much
space to them.
Purbacli.^ I may begin by mentioning George Purbach,
first the tutor and then the friend of Regiomontanus, born
near Linz on May 30, 1423, and died at Vienna on April 8,
1461, who wrote a work on planetary motions which was
published in 1460; an arithmetic, published in 1511 ; a table
of eclipses, published in 1514; and a table of natural sines,
published in 1541.
Cusa.^ Next I may mention Nicolas de Cusa, who was
born in 1401 and died in 1464. Although the son of a poor
fisherman and without influence, he rose rapidly in the church,
and in spite of being "a reformer before the reformation"
became a cardinal. His mathematical writings deal with the
reform of the calendar and the quadrature of the circle; in
the latter problem his construction is equivalent to taking
|( ^3 + ;76) as the value of tt. He argued in favour of the
diurnal rotation of the earth.
Chuquet. I may also here notice a treatise on arithmetic,
^ It was published by C. G. von Murr at Nuremberg in 1786.
2 Purbach's life was written by P. Gassendi, The Hague, second edition,
1655.
^ Cusa's life was written by F. A. ScharpflF, Tubingen, 1871 ; and his
collected works, edited by H. Petri, were published at Bale in 1565.
206 MATHEMATICS OF THE RENAISSANCE [ch. xii
known as Le Tri'party} by Nicolas Chuquet, a bachelor of
medicine in the university of Paris, which was written in 1484.
This work indicates that the extent of mathematics then taught
was somewhat greater than was generally believed a few years
ago. It contains the earliest known use of the radical sign
with indices to mark the root taken, 2 for a square-root, 3 for
a cube-root, and so on ; and also a definite statement of the
rule of signs. The words plus and minus are denoted by the
contractions p, m. The work is in French.
Introduction ^ of signs + and - . In England and Germany
algorists were less fettered by precedent and tradition than in
Italy, and introduced some improvements in notation which
were hardly likely to occur to an Italian. Of these the most
prominent were the introduction, if not the invention, of the
current symbols for addition, subtraction, and equality.
The earliest instances of the regular use of the signs -l- and -
of which we have any knowledge occur in the fifteenth century.
Johannes Widman of Eger, born about 1460, matriculated at
Leipzig in 1480, and probably by profession a physician, wrote
a Mercantile Arithmetic^ published at Leipzig in 1489 (and
modelled on a work by Wagner printed some six or seven years
earlier) : in this book these signs are used merely as marks
signifying excess or deficiency ; the corresponding use of the
word surplus or overplus ^ was once common and is still
retained in commerce.
It is noticeable that the signs generally occur only in practical
mercantile questions : hence it has been conjectured that they
were originally' warehouse marks. Some kinds of goods were
sold in a sort of" wooden chest called a lagel, which when full
was apparently expected to weigh roughly either three or four
1 See an article by A. Marre ia Boncompagni's Bulletiiio di hihliografia
for 1880, vol. xiii, pp. 555-659.
2 See articles by P. Treutlein {Die. deutsche Coss) in the Ahhandlungen zur
Geschichte der Mathematik for 1879 ; by de Morgan in tlie Cambridge Philo-
sophical Transactions, 1871, vol. xi, pp. 203-212 ; ami by Boncompagni in
the Bulletino di bibliograjict for 1876, vol. ix, pp. 188-210.
^ ^ee passim Levit. xxv, verse 27, and 1 Maccab, x, verse 41.
CH. xii] INTRODUCTION OF SIGNS 207
centners ; if one of these cases were a little lighter, say 5 lbs.,
than four centners, Widman describes it as weighing 4c - 5 lbs. :
if it were 5 lbs. heavier than the normal weight it is described
as weighing Ac — | — 5 lbs. The symbols are used as if they
would be familiar to his readers ; and there are some slight
reasons for thinking that these marks were chalked on the
chests as they came into the warehouses. We infer that the
more usual case was for a chest to weigh a little less than
its reputed weight, and, as the sign - placed between two
numbers was a common symbol to signify some connection
between them, that seems to have been taken as the standard
case, while the vertical bar was originally a small mark super-
added on the sign - to distinguish the two symbols. It will
be observed that the vertical line in the symbol for excess,
printed above, is somewhat shorter than the horizontal line.
This is also the case with Stifel and most of the early writers
who used the symbol : some presses continued to print it in
this, its earliest form, till the end of the seventeenth century.
Xylander, on the other hand, in 1575 has the vertical bar
much longer than the horizontal line, and the symbol is some-
thing like -j-.
Another conjecture is that the symbol for plus is derived
from the Latin abbreviation (b for et ; while that for minus is
obtained from the bar which is often used in ancient manuscripts
to indicate an omission, or which is written over the contracted
form of a word to signify that certain letters have been left^out.
This view has been often supported on a priori grounds, but it
has recently found powerful advocates in Professors Zangmeister
and Le Paige who also consider that the introduction of these
symbols for plus and minus may be referred to the fourteenth
century.
These explanations of the origin of our symbols for phis and
minus are the most plausible that have been yet advanced, but
the question is difficult and cannot be said to be solved. Another
suggested derivation is that -f is a contraction of ^ the initial
letter in Old German of plus, while - is the limiting form of
208 MATHEMATICS OF THE RENAISSANCE [ch. xii
m (for minus) when written rapidly. De Morgan ^ proposed
yet another derivation : the Hindoos sometimes used a dot to
indicate subtraction, and this dot might, he thought, have been
elongated into a bar, and thus give the sign for minus ; while
the origin of the sign for plus was derived from it by a super-
added bar as explained above; but I take it that at a later
time he abandoned this theory for what has been called the
warehouse explanation.
I should perhaps here add that till the close of the sixteenth
century the sign + connecting two quantities like a and b was
also used in the sense that if a were taken as the answer to some
question one of the given conditions would b^ too little by h.
This was a relation which constantly occurred in solutions of
questions by the rule of false assumption.
Lastly, I would repeat again that these signs in Widman are
only abbreviations and not symbols of operation; he attached
little or no importance to them, and no doubt would have
been amazed if he had been told that their introduction was pre-
paring the way for a revolution of the processes used in algebra.
The Alfforithmus of Jordanus was not published till 1534;
Widman's work was hardly known outside Germany ; and it
is to Pacioli that we owe the introduction into general use
of syncopated algebra; that is, the use of abbreviations for
certain of the more common algebraical quantities and operations,
but where in using them the rules of syntax are observed.
ifacioli.^ Lucas Pacioli, sometimes known as Lucas di Burgo,
and sometimes, but more rarely, as Lucas Paciolus, was born at
Burgo in Tuscany about the middle of the fifteenth century.
We know little of his life except that he was a Franciscan
friar; that he lectured on mathematics at Rome, Pisa, Venice,
and Milan; and that at the last-named city he was the first
occupant, of a chair of mathematics founded by Sforza : he died
at Florence about the year 1510.
1 See his Arithmetical Books, London, 1847, p. 19.
2 See H. Staigmliller in the Zeitschrift filr Mathematik, 1889, vol. xxxiv ;
also Libri, vol. iii, pp. 133-145 ; and Canto;-, chap. Ivii.
CH. xii] PACIOLI 209
His chief work was printed at Venice in 1494 and is termed
Summa de arithmetica, geometria, proporzioni e proporzicmalita.
It is divided into two parts, the first dealing with arithmetic
and algebra, the second with geometry. This was the earliest
printed book on arithmetic^ and algebra, It is mainly based on
the writmgs of Leonardo of Pisa, and its importance in the
history of mathematics is largely due to its wide circulation.
In the arithmetic Pacioli fflves rules for the four simple
processes, and a method,.foiiextracting_g(][ug.re^ roots. He deals
pretty fully with all questions connected with mercantile
arithmetic, in which he works out numerous examples, and
in particular discusses at great length bills of___exchange _and
the thebry of book-keeping by double entry. This part was
the first systematic exposition of algoristic arithmetic, and has
been already alluded to in chapter xi. It and the similar
work by Tartaglia are the two standard authorities on the
subject.
Many of his problems are solved by "the^ method of false
assumption," which consists in assuming any number for the
unknownquantity, and if on trial the given conditions be
not satisfied, altering it by a simple proportion as in rule of
three. As an example of this take the problem to find the
original capital of a merchant who spent a quarter of it in
Pisa and a fifth of it in Venice, who received on these trans-
actions 180 ducats, and who has in hand 224 ducats. Suppose
that we assume that he had originally 100 ducats. Then if
he spent 25 + 20 ducats at Pisa and Venice, he would have
had 55 ducats left. But by the enunciation he then had
224-180, that is, 44 ducats. Hence the ratio of his original
capital to 100 ducats is* as 44 to 55. Thus his original capital
was 80 ducats.
The following example will serve as an illustration of the
kind of arithmetical problems discussed.
I buy for 1440 ducats at Venice 2400 sugar loaves, whose nett weight
is 7200 lire ; I pay as a fee to the agent 2 per cent. ; to the weighers and
porters on the whole, 2 ducats ; I afterwards spend in boxes, cords,
P
210 MATHEMATICS OF THE KENAISSANCE [ch. xii
canvas, and in fees to the ordinary packers in the whole, 8 ducats ; for
the tax or octroi duty on the first amount, 1 ducat per cent. ; afterwards
for duty and tax at the office of exports, 3 ducats per cent. ; for writing
directions on the boxes and booking their passage, 1 ducat ; for the bark
to Rimini, 13 ducats ; in compliments to the captains and in drink for
the crews of armed barks on several occasions, 2 ducats ; in expenses for
provisions for myself and servant for one month, 6 ducats ; for expenses
for several short journeys over land here and there, for barbers, for
washing of linen, and of boots for myself and servant, 1 ducat ; upon my
arrival at Rimini I pay to the captain of the port for port dues in the
money of that city, 3 lire ; for porters, disembarkation on land, and
carriage to the magazine, 5 lire ; as a tax upon entrance, 4 soldi a load
which are in number 32 (such being the custom) ; for a booth at the fair,
4 soldi per load ; I further find that the measures used at the fair are
different to those used at Venice, and that 140 lire of weight are there
equivalent to 100 at Venice, and that 4 lire of their silver coinage are
equal to a ducat of gold. I ask, therefore, at how much I must sell a
hundred lire Rimini in order that I may gain 10 per cent, upon my
whole adventure, and what is the sum which 1 must receive in Venetian
money ?
In the algebra he discusses in some detail simple and
quadratic equations, and problems on numbers which lead to
such equations. He mentions the Arabic classification of cubic
equations, but adds that their solution aj^pears to be as im-
possible as the quadrature of the circle. The following is the
rule he gives ^ for solving a quadratic equation of the form
x'^ + x = a: it is rhetorical, and not syncopated, and will serve
to illustrate the inconvenience of that method.
" Si res et census numero coaequantur, a rebus
dimidio sumpto censum producere debes,
addereque numero, cujus a radice totiens
tolle semis rerum, census latusque redibit."
He confines his attention to the positive roots of equations.
Though much of the matter described above is taken from
Leonardo's Liber Abaci, yet the notation in which it is expressed
is superior to that of Leonardo. Pacioli follows Leonardo and
1 Edition ofl 494, p. 145.
cH.xii] PACIOLI 211
the Arabs in calling the unknown qusintitjj^he^thinff, in Italian
cosa — hence algebra was sometimes known as the cossic art — or
in Latin res, and sometimes denotes it by co or B or BJ. He
calls the square of it census_OT zensus, and sometimes denotes
it by ce or Z ; similarly the cube of it, ox__cy^a, is sometimes
represented by cu ot C ; the fourth power, or censo di censo,
is written either at length or as ce di ce or as ce ce. It may
be noticed that all his equations are numerical, so that he did
not rise to the conception of representing known quantities by
letters as Jordanus had done and as is the case in modern
algebra ; but Libri gives two instances in which in a proportion
he represents a number by a letter. He indicates g^ditioii^by
p or p, the initial letter of the word plus, but he generally evades
the introduction of a symbol for minus by writing his quantities
on that side of the equation which makes them positive, though
in a few places he denotes it by m for minus or by de for demptus.
Similarly, equality is sometimes indicated by ae for aequalis.
This is a commencement of syncopated algebra.
There is nothing striking in the results he arrives at in the
second or geometrical part of the work ; nor in two other tracts
on geometry which he wrote and which were printed at Venice
in 1508 and 1509. It may be noticed, however, that, like
Regiomontanus, he applied algebra to aid him in investigating
the geometrical properties of figures.
The following problem will illustrate the kind of geometri-
cal questions he attacked. The radius of the inscribed circle
of a triangle is 4 inches, and the segments into which one side
is divided by the point of contact are 6 inches and 8 inches
respectively. Determine the other sides. To solve this it is
sufficient to remark that rs = A = Js (s-a) {s- b) (s - c) which
gives 4s = s/s X (s - 14) X 6 X 8, hence s = 21 ; therefore the
required sides are 21-6 and 21-8, that is, 15 and 13. But
Pacioli makes no use of these formulae (with which he was
acquainted), but gives an elaborate geometrical construction, and
then uses algebra to find the lengths of various segments of the
lines he wants. The work is too long for me to reproduce here,
212 MATHEMATICS OF THE RENAISSANCE [ch. xii
but the following analysis of it will afford sufficient materials
for its reproduction. Let ABC be the triangle, 2>, E^ F the
points of contact of the sides, and 0 the centre of the given
circle. Let H be the point of intersection of OB and DF, and
K that of OC and DE. Let L and M be the feet of the
perpendiculars drawn from E and F on BC. Draw EP
parallel to ^-6 and cutting BC in P. Then Pacioli determines
in succession the magnitudes of the following lines : (i) OB^
(ii) OC, (iii) FD, (iv) Fff, (v) ED, (vi) EK. He then
forms a quadratic equation, from the solution of which he
obtains the values of 3IB and 3ID. Similarly he finds the
values of ZC and LD. He now finds in succession the values
of EL, FM, EP, and LP ; and then by similar triangles obtains
the value of AB^ which is 13. This proof was, even sixty years
later, quoted by Cardan as " incomparably simple and excellent,
and the very crown of mathematics." I cite it as an illustration
of the involved and inelegant methods then current. The
problems enunciated are very similar to those in the De
Triangulis of Regiomontanus.
Leonardo da Vinci. The fame of Leonardo da Vinci as
an artist has overshadowed his claim to consideration as a
mathematician, but he may be said to have prepared the way
for a more accurate conception of mechanics and physics, while
his reputation and influence drew some attention to the subject ;
he was an intimate friend of Pacioli. Leonardo was the
illegitimate son of a lawyer of Vinci in Tuscany, was born in
1452, and died in France in 1519 while on a visit to Francis I.
Several manuscripts by him were seized by the French revolu-
tionary armies at the end of the last century, and Venturi, at
the request of the Institute, reported on those concerned with
physical or mathematical subjects.^
Leaving out of account Leonardo's numerous and important
artistic works, his mathematical writings are concerned chiefly
with mechanics, hydraulics, and optics — his conclusions being
^ Ussai sur les ouvrages physico-mathematig^iies de Leonard de Vinci, by
J.-B. Veuturi, Paris, 1797.
cH.xii] MATHEMATICS OF THE RENAISSANCE 213
usually based on experiments. His treatment of hydraulics
and optics involves but little mathematics. The mechanics
contain numerous and serious errors; the best portions are
those dealing with the equilibrium of a lever under any forces,
the laws of friction, the stability of a body as affected by the
position of its centre of gravity, the strength of beams, and the
orbit of a particle under a central force ; he also treated a few
easy problems by virtual moments. A knowledge of the triangle
of forces is occasionally attributed to him, but it is probable
that his views on the subject were somewhat indefinite.
Broadly speaking, we may say that his mathematical work
is unfinished, and consists largely of suggestions which he
did not discuss in detail and could not (or at any rate did
not) verify.
Diirer. Alhrecht Diirer^ was another artist of the same
time who was also known as a mathematician. He was born at
Nuremberg on May 21, 1471, and died there on April 6, 1528.
His chief mathematical work was issued in 1525, and contains a
discussion of perspective, some geometry, and certain graphical
solutions ; Latin translations of it were issued in 1532, 1555,
and 1605.
Copernicus. An account of Nicolaus Copernicus, born at
Thorn on Feb. 19, 1473, and died at Frauenberg on- May 7,
1543, and his conjecture that the earth and planets all revolved
round the sun, belong to astronomy rather than to mathematics.
I may, however, add that Copernicus wrote on trigonometry, his
results being published as a text-book at Wittenberg in 1542;
it is clear though it contains nothing new. It is evident from
this and his astronomy that he was well read in the literature
of mathematics, and was himself a mathematician of consider-
able power. I describe his statement as to the motion of the
earth as a conjecture, because he advocated it only on the
ground that it gave a simple explanation of natural phenomena.
Galileo in 163,2 was the first to try to supply a proof of this
hypothesis.
^ See Diirer aZs Mathematiker, by H. Staigmiiller, Stuttglft, 1891.
214 MATHEMATICS OF THE RENAISSANCE [ch. xii
By the beginning of the sixteenth century the printing-
press began to be active, and many of the works of the earlier
mathematicians became now for the first time accessible to all
students. This stimulated inquiry, and before the middle of
the century numerous works were issued which, though they
did not include any great discoveries, introduced a variety of
small improvements all tending to make algebra more analytical.
Record. The sign now used to denote equality was intro-
duced by Robert Record} Record was born at Tenby in
Pembrokeshire about 1510, and died at London in 1558. He
entered at Oxford, and obtained a fellowship at All Souls
College in 1531 ; thence he migrated to Cambridge, where he
took a degree in medicine in 1545. He then returned to
Oxford and lectured there, but finally settled in London and
became physician to Edward VI. and to Mary. His prosperity
must have been short-lived, for at the time of his death he was
confined in the King's Bench prison for debt.
In 1540 he published an arithmetic, termed the Grounde of
Artes, in which he employed the signs + to indicate excess
and - to indicate deficiency ; " -t- whyche betokeneth too
muche, as this line - plaine without a crosse line betokeneth
too little." In this book the equality of two ratios is indi-
cated by two equal and parallel lines whose opposite ends are
joined- diagonally, ex. gr. by Z. A few years later, in 1557, he
wrote an algebra under the title of the Whetstone of Witte,
This is interesting as it contains the earliest introduction of the
sign = for equality, and he says he selected that particular
symbol because ^ than two parallel straight lines " noe 2 thynges
can be moare equalle." M. Charles Henry has, however, asserted
that this sign is a recognized abbreviation for the word est
in medieval manuscripts; and, if this be established, it would
seem to indicate a more probable origin. In this work Record
shewed how the square root of an algebraic expression could be
^ See pp. 15-19 of my History of the Study of Matliematics at Carru
bridge, Cambridge, 1889.
2 See Whetstone of Witte, f. Ff, j. v.
CH. xii] RUDOLFF. RIESE. STIFEL 215
extracted. He also wrote an astronomy. These works give a
clear view of the knowledge of the time.
Rudolff. Riese. About the same time in Germany, Rudolff
and Riese took up the subjects of algebra and arithmetic. Their
investigations form the basis of StifeFs well - known work.
Christoff Rudolff^ published his algebra in 1525; it is entitled
Die Coss, and is founded on the writings of Pacioli, and perhaps
of Jordantis. Rudolff introduced the sign of J for the square
root, the symbol being a corruption of the initial letter of the
word radix, similarly J J J denoted the cube root, and sj a/
the fourth root. Adam Riese '^ was born near Bamberg, Bavaria,
in 1489, of humble parentage, and after working for some years
as a miner set up a school ; he died at Annaberg on March 30,
1559. He wrote a treatise on practical geometry, but his most
important book was his well-known arithmetic (which may be
described as algebraical), issued in 1536, and founded on Pacioli's
work. Riese used the symbols + and - .
Stifel.^ The methods used by Rudolff and Riese and their
results were brought into general notice through Stifel's work,
which had a mde circulation. Midmel Stifel, sometimes known
by the Latin name of Stiff elius, was born at Esslingen in 1486,
and died at Jena on April 19, 1567. He was originally an
Augustine monk, but he accepted the doctrines of Luther, of
whom he was a personal friend. He tells us in his algebra that
his conversion was finally determined by noticing that the pope
Leo X. was the beast mentioned in the Revelation. To shew
this, it was only necessary to add up the numbers represented by
the letters in Leo decimus (the m had to be rejected since it
*clearly stood for mysterium), and the result amounts to exactly
ten less than QQQ, thus distinctly implying that it was Leo the
tenth. Luther accepted his conversion, but frankly told him he
^ See E. Wappler, Geschichte der deutschen Algebra im xv. Jahrhunderte,
Zwickau, 1887.
2 See two works by B. Berlet, Ud)er Adam Riese, Annaberg, 1855 ; and
Die Coss von Adam Riese, Annaberg, 1860.
^ The authorities on Stifel are given by Cantor chap. Ixii.
216 MATHEMATICS OF THE RENAISSANCE [ch. xii
had better clear his mind of any nonsense about the number of
the beast.
Unluckily for himself Stifel did not act on this advice.
Believing that he had discovered the true way of interpreting
the biblical prophecies, he announced that the world would come
to an end on October 3, 1533. The peasants of Holzdorf, of which
place he was pastor, aware of his scientific reputation, accepted
his assurance on this point. Some gave themselves up to
religious exercises, others wasted their goods in dissipation, but
all abandoned their work. When the day foretold had passed,
many of the peasants found themselves ruined. Furious at having
been deceived, they seized the unfortunate prophet, and he was
lucky in finding a refuge in the prison at Wittenberg, from
which he was after some time released by the, personal interces-
sion of Luther.
Stifel wrote a small treatise on algebra, but his chief mathe-
matical work is his Arithmetica Integra, published at Nuremberg
in 1544, with a preface by Melanchthon.
The first two books of the Arithmetica Integra deal with
surds and incommensurables, and are Euclidean in form. The
third book is on algebra, and is noticeable for having called
general attention to the German practice of using the signs
+ and - to denote addition and subtraction. There are traces of
these signs being occasionally employed by Stifel as symbols of
operation and not only as abbreviations ; in this use of them he
seems to have followed G. V. Hoecke. He not only employed the
usual abbreviations for the Italian words which represent the
unknown quantity and its powers, but in at least one case when
there "wefe^everal unknown quantities lie represented them
respectively by the letters A, B, C, &c. ; thus re-introducing the
general algebraic notation which had fallen into disuse since the
time of Jordanus. It used to be said that Stifel was the real
inventor of logarithms, but it is now certain that this opinion
was due to a misapprehension of a passage in which he compares
geometrical and arithmetical progressions. Stifel is said to have
indicated a formula for writing down the coefficients of the
CH.XTi] STIFEL. TARTAGLIA . 217
various terms in the expansion x}£XL4-^4^^L those in the expan-
sTon ot (i + x)'^~'^ were known^ _
In 1553 Stifel brought out an edition of Rudolffs Die Coss, in
which he introduced an improvement in the algebraic notation
then current. The symbols at that time ordinarily used for the
unknown quantity and its powers were letters which stood for
abbreviations of the words. Among those frequently adopted
were B or BJ for radix or res (x), Z ov C for zensus or census
{x"^), C or K for cubus (aj^), &c. Thus x^ + bx-i would have
been written
1 ^ p. 5 i? m. 4 ;
where p stands for plus and m for minus. Other letters and
symbols were also employed : thus Xylander (1575) would have
denoted the above expression by
a notation similar to this was sometimes used by Yieta and even
by Fermat. The advance made by Stifel was that he introduced
the symbols lA, \AA, \AAA, for the unknown quantity, its
square, and its cube, which shewed at a glance the relation
between them.
Tartaglia. Niccolo Fontaim^ generally known as Nicholas
Tartaglia^ that is, Nicholas the stammerer, was born at Brescia
in 1500, and died at Venice on December 14, 1557. After the
capture of the town by the French in 1512, most of the in-
habitants took refuge in the cathedral, and were there massacred
by the soldiers. His father, who was a postal messenger at
Brescia, was amongst the killed. The boy himself had his skull
split through in three places, while his jaws and his palate were
cut open ; he was left for dead, but his mother got into the
cathedral, and finding him still alive managed to carry him off.
Deprived of all resources she recollected that dogs when wounded
always licked the injured place, and to that remedy he attributed
his ultimate recovery, but the injury to his palate produced an
impediment in his speech, from which he received his nickname.
218 MATHEMATICS OF THE RENAISSANCE [ch. xii
His mother managed to get sufficient money to pay for his
attendance at school for fifteen days, and he took advantage
of it to steal a copy-book from which he subsequently taught
himself how to read and write ; but so poor were they that he
tells us he could not afford to buy paper, and was obliged
to make use of the tombstones as slates on which to work his
exercises.
He commenced his public life by lecturing at Verona, but he
was appointed at some time before 1535 to a chair of mathe-
matics at Venice, where he was living, when he became famous
through his acceptance of a challenge from a certain Antonio del
Fiore (or Florido). Fiore had learnt from his master, one
Scipione Ferro (who died at Bologna in 1526), an empirical
solution of a cubic equation of the form x^-\-qx = r. This
solution was previously unknown in Europe, and it is possible
that Ferro had found the result in an Arab work. Tartaglia, in
answer to a request from Colla in 1530, stated that he could
effect the solution of a numerical equation of the form x^ -\-px^ -— r.
Fiore, believing that Tartaglia was an impostor, challenge^ him
to a contest. According to this challenge each of them was to
deposit a certain stake with a notary, and whoever could solve
the most problems out of a collection of thirty propounded by
the other was to get the stakes, thirty days being allowed for
the solution of the questions proposed. Tartaglia was aware
that his adversary was acquainted with the solution of a cubic
equation of some particular form, and suspecting that the
questions proposed to him would all depend on the solution of
such cubic equations, set himself the problem to find a general
solution, and certainly discovered how to obtain a solution of
some if not all cubic equations. His solution is believed to have
depended on a geometrical construction^^ but led to the formula
which is often, but unjustly, described as Cardan's.
When the contest took place, all the questions proposed to
Tartaglia were, as he had suspected, reducible to the solution
of a cubic equation, and he succeeded within two hours in
1 See below, p. 224.
CH. xii] TARTAGLIA 219
bringing them to particular cases of the equation oc^ + qx = r, of
which he knew the solution. His opponent failed to solve any
of the problems proposed to him, most of which were,, as a
matter of fact, reducible to numerical equations of the form
x^+px^ = r. Tartaglia was therefore the conqueror; he subse-
quently composed some verses commemorative of his victory.
The chief works of Tartaglia are as follows : (i) His J^ova
scienza, published in 1537 : in this he investigated the fall of
bodies under gravity ; and he determined the range of a pro-
jectile, stating that it was a maximum when the angle of
projection was 45°, but this seems to have been a lucky guess,
(ii) His Inventioni, published in 1546, and containing, inter
alia, his solution of cubic equations. (iii) His Trattato de
numeri e misuri, consisting of an arithmetic, published in 1556,
and a treatise on numbers, published in 1560 ; in this he shewed
how the coefficients_of_^ in the expansion of (1 +^^ could be
calculated, by the use of an arithmetical triangle,^ from those
in tEe expansion of (1 +x)^~'^ for the cases wlien" n is equaHto
2, 3, 4, 5, or 6. His works were collected into a single edition
and republished at Venice in 1606.
The treatise on arithmetic and numbers is one of the chief
authorities for our knowledge of the early Italian algorism. It
is verbose, but gives a clear account of the arithmetical methods
then in use, and has numerous historical notes which, as far as
we can judge, are reliable, and are ultimately the authorities for
many of the statements in the last chapter. It contains an
immense number of questions on every kind of problem which
would be likely to occur in mercantile arithmetic, and there
are several attempts to frame algebraical formulae suitable for
particular problems.
These problems give incidentally a good deal of information
as to the ordinary life and commercial customs of the time.
Thus we find that the interest demanded on first-class security
in Venice ranged from 5 to 12 per cent, a year; while the
interest on commercial transactions ranged from 20 per cent.
^ See below, pp. 284, 285.
220 MATHEMATICS OF THE RENAISSANCE [ch. xii
a year upwards. Tartaglia illustrates the evil effects of the
law forbidding usury by the manner in which it was evaded
in farming. Farmers who were in debt were forced by their
creditors to sell all their crops immediately after the harvest ;
the market being thus glutted, the price obtained was very low,
and the money-lenders purchased the corn in open market at an
extremely cheap rate. The farmers then had to borrow their
seed-corn on condition that they replaced an equal quantity, or
paid the then price of it, in the month of May, when corn was
dearest. Again, Tartaglia, who had been asked by the magis-
trates at Verona to frame for them a sliding scale by which the
price of bread would be fixed by that of corn, enters into a
discussion on the principles which it was then supposed should
regulate it. In another place he gives the rules at that time
current for preparing medicines.
Pacioli had given in his arithmetic some problems of an
amusing character, and Tartaglia imitated him by inserting a
large collection of mathematical puzzles. He half apologizes
for introducing them by saymg that it was not uncommon at
dessert to propose arithmetical questions to the company by way
of amusement, and he therefore adds some suitable problems.
He gives several questions on how to guess a number thought
of by one of the company, or the relationships caused by the
marriage of relatives, or difficulties arising from inconsistent
bequests. Other puzzles are similar to the following. " Three
beautiful ladies have for husbands three men, who are young,
handsome, and gallant, but also jealous. The party are travel-
ling, and find on the bank of a river, over which they have to
pass, a small boat which can hold no more than two persons.
How can they pass, it being agreed that, in order to avoid
scandal, no woman shall be left in the society of a man unless
her husband is present?" "A ship, carrying as passengers
fifteen Turks and fifteen Christians, encounters a storm ; and
the pilot declares that in order to save the ship and crew one-
half of the passengers must be thrown into the sea. To choose
the victims, the passengers are placed in a circle, and it is agreed
cii.xii] TARTAGLIA. CARDAN * 221
that every ninth man shall be cast overboard, reckoning from a
certain point. In what manner must they be arranged, so that
the lot may fall exclusively upon the Turks'?" "Three men
robbed a gentleman of a vase containing 24 ounces of balsam.
Whilst running away they met in a wood with a glass-seller
of whom in a great hurry they purchased three vessels. On
reaching a place of safety they wish to divide the booty, but
they find that their vessels contain 5, 11, and 13 ounces
respectively. How can they divide the l^alsam into equal
portions 1 "
These problems — some of which are of oriental origin — form
the basis of the collections of mathematical recreations by Bachet
de Meziriac, Ozanam, and Montucla.^
Cardan.^ The life of Tartaglia was embittered by a quarrel
with his contemporary Cardan, who published Tartaglia's solu-
tion of a cubic equation which he had obtained under a pledge
of secrecy. Girolamo Cardan was born at Pa via on September
24, 1501, and died at Rome on September 21, 1576. His
career is an account of the most extraordinary and inconsistent
acts. A gambler, if not a murderer, he was also an ardent
student of science, solving problems which had long baffled all
investigation ; at one time of his life he was devoted to intrigues
which were a scandal even in the sixteenth century, at another
he did nothing but rave on astrology, and yet at another he
^ Solutions of these aud other similar problems are given in my Mathe-
matical Recreations, chaps, i, ii. On Bachet, see below, p. 305. Jacques
Ozanam, born at Bouligneux in 1640, and died in 1717, leit numerous works
of which one, worth mentioning here, is his Recreations inathematlques et
physiques, two volumes, Paris, 1696. Jean Etienne Montucla, born at Lyons
in 1725, and died in Paris in 1799, edited and revised Ozanam's mathe-
matical recreations. His history of attempts to square the circle, 1754,
and history of mathematics to the end of the seventeenth century, in two
volumes, 1758, are interesting and valuable works.
^ There is an admirable account of Cardan's life in the Nonvelle hiograiiliie
generate, by V". Sardou. Cardan left an autobiography of which an analysis
by H. Morley was piiblished in two volumes in Ijondon in 1854. All
Cardan's printed works were collected by Sponius, and published in ten
volumes, Lyons, 1663 ; the works on arithmetic and geometry are contained
in the fourth volume. It is said that there are in the Vatican several
manuscript note-books of his which have not been yet edited.
222 MATHEMATICS OF THE KENAISSANCE [ch. xii
declared that philosophy was the only subject worthy of man's
attention. His was the genius that was closely allied to
madness.
He was the illegitimate son of a lawyer of Milan, and was
educated at the universities of Pa via and Padua. After taking
his degree he commenced life as a doctor, and practised his
profession at Sacco and Milan from 1524 to 1550 ; it was during
this period that he studied mathematics and published his chief
works. After spending a year or so in France, Scotland, and
England, he returned to Milan as professor of science, and shortly
afterwards was elected to a chair at Pa via. Here he divided his
time between debauchery, astrology, and mechanics. His two
sons were as wicked and passionate as himself : the elder was
in 1560 executed for poisoning his wife, and about the same
time Cardan in a fit of rage cut off the ears of the younger who
had committed some offence ; for this scandalous outrage he
suffered no punishment, as the Pope Gregory XIII. granted him
protection. In 1562 Cardan moved to Bologna, but the scandals
connected with his name were so great that the university took
steps to prevent his lecturing, and only gave way under pressure
from Rome. In 1570 he was imprisoned for heresy on account
of his having published the horoscope of Christ, and when
released he found himself so generally detested that he deter-
mined to resign his chair. At any rate he left Bologna in
1571, and shortly afterwards moved to Rome. Cardan was
the most distinguished astrologer of his time, and when he
settled at Rome he received a pension in order to secure his
services as astrologer to the papal court. This proved fatal to
him for, having foretold that he should die on a particular
day, he felt obliged to commit suicide in order to keep up his
^reputation — so at least the story runs.
The chief mathematical work of Cardan is the Ars Magna
published at Nuremberg in 1545. Cardan was much interested
in the contest between Tartaglia and Fiore, and as he had
already begun writing this book he asked Tartaglia to com-
municate his method of solving a cubic equation. Tartaglia
CH. xii] CARDAN 223
refused, whereupon Cardan abused him in the most violent
terms, but shortly afterwards wrote saying that a certain
Italian nobleman had heard of Tartaglia's fame and was most
anxious to meet him, and begged him to come to Milan at
once. Tartaglia came, and though he found no nobleman
awaiting him at the end of his journey, he yielded to Cardan's
importunity, and gave him the rule. Cardan on his side taking
a solemn oath that he would never reveal it. Cardan asserts
that he was given merely the result, and that he obtained
the proof himself, but this is doubtful. He seems to have
at once taught the method, and one of his pupils Ferrari
reduced the equation of the fourth degree to a cubic and so
solved it.
When the Ars Magna was published in 1545 the breach of
faith was made manifest.^ Tartaglia not unnaturally was very
angry, and after an acrimonious controversy he sent a challenge
to Cardan to take part in a mathematical duel. The pre-
liminaries were settled, and the place of meeting was to be a
certain church in Milan, but when the day arrived Cardan
failed to appear, and sent Ferrari in his stead. Both sides
claimed the victory, though I gather that Tartaglia was the
more successful ; at any rate ' his opponents broke up the
meeting, and he deemed himself fortunate in escaping with his
life. Not only did Cardan succeed in his fraud, but modern
writers have often attributed the solution to him, so that
Tartaglia has not even that posthumous reputation which at
least is his due.
The Ars Magna is a great advance on any algebra pre-
viously published. Hitherto algebraists had confined_their
attention to those rnnt« nf_ Puliation ^ whi^h wf"ff>- p^Hitiv^ -
Cardan discussed negative and even complex roots, and
proved that the latter would always occur in pairs, though he
declined to commit himself to any explanation as to the
[ ^ The history of the subject and of the doings of Fiore, Tartaglia, and
Cardan are given in an Appendix to the 2nd edition of the French translation
'u of my Mathematical Recreations, Paris, 1908, vol. ii, p. 322 et seq.
224 MATHEMATICS OF THE KENAISSANCE [ch. xii
meaning of these "sophistic" quantities which he said were
ingenious though useless. Most of his analysis of cubic equa-
tions seems to have been original ; he shewed that jf the three
roots were j;eal, Tartag^lja's solution gave them in aT form
which involved imaginary quantities. Except for the somewhat
similar researches of Bombelli a few years later, the theory
of imaginary quantities received little further attention from
mathematicians until John Bernoulli and Euler took up the
matter after the lapse of nearly two centuries. Gaussfirst_piit
the subject on a^s^stemati£_and^_sdeijti£c^iasi% introducedjthe-
notation of complex variables, and_iisedjbhe_sy7iiho1 ?', which had
beenln^rtJaucecl by Euler in 1777, to denote the square root of
( - 1) : the modern theory is chiefly based on his researches.
Cardan established the relations connecting the roots with
the coefficients of an equation. He was also aware of the
principle that underlies Descartes's "rule of signs," but as he
followed the custom, then general, of writing his equations as
the equality of two expressions in each of which all the terms
were positive he was unable to express the rule concisely. HeS
gave a method of approximating to the root of a numerical 1
equation, founded on the fact that, if a function have opposite I
signs when two numbers are substituted in it, the equation
obtained by equating the function to zero will have a rootj
between these two numbers.
Cardan's solution of a quadratic equation is geometrical
and substantially the same as that given by Alkarismi. His
solution of a cubic equation is also geometrical, and may be .
illustrated by the following case which he gives in chapter xi.
To solve the equation x^ + 6x== 20 (or any equation of the form
x^ + qx = r), take two cubes such that the rectangle under their
respective edges is 2 (or ^q) and the difference of their
volumes is 20 (or r). Then x will be equal to the difference
between the edges of the cubes.. To verify this he first gives a
geometrical lemma to shew that, if from a line AC Si portion
CB be cut off, then the cube on AB will be less than the
difference between the cubes on AC and BC by three times
CH.xii] CARDAN. FERRAKI 225
the right parallelepiped whose edges are respectively equal to
AC, BC\ and AB — this statement is equivalent to the alge-
braical identity (a - b)^ = a^-b^- 3ab{a - b) — and the fact
that X satisfies the equation is then obvious. To obtain the
lengths of the edges of the two cubes he has only to solve
a quadratic equation for which the geometrical solution pre-
viously given sufiiced.
Like all previous mathematicians he gives separate proofs
of his rule for the different forms of equations which can fall
under it. Thus he proves the rule independently for equa-
tions of the form x^+px = q, x^=px + q, x^+px + q==0, and
x^ + q =px. It will be noticed that with geometrical proofs
this was the natural course, but it does not appear that he was
aware that the resulting formulae were general. The equations
he considers are numerical.
Shortly after Cardan came a number of mathematicians
who did good work in developing the subject, but who are
hardly of sufficient importance to require detailed mention here.
Of these the most celebrated are perhaps Ferrari and Rheticus.
Ferrari. Ludovico Ferraro, usually known as Ferrari,
whose name I have already mentioned in connection with the
solution of a biquadjg;tic equation, was born at Bologna on
Feb. 2, 1522, and died on Oct. 5, 1565. His parents were
poor and he was taken into Cardan's service as an errand boy,
but was allowed to attend his master's lectures, and sub-
sequently became his most ceMbrated pupil. He is described as
"a neat rosy little fellow, with a bland voice, a cheerful face,
and an agreeable short nose, fond of pleasure, of great natural
powers," but " with the temper of a fiend." His manners and
numerous accomplishments procured him a place in the service
of the Cardinal Ferrando Gonzago, where he managed to make
a fortune. His dissipations told on his health, and he retired
in 1565 to Bologna where he began to lecture on mathematics.
He was poisoned the same year either by his sister, who seems
to have been the only person for whom he had any affection,
or by her paramour.
Q
^
226 MATHEMATICS OF THE RENAISSANCE [ch. xii
Such work as Ferrari produced is incorporated in Cardan's
Ars Magna or Bombelli's Algebra, but nothing can be defi-
nitely assigned to him except the solution of a biquadratic
equation. CoUa had proposed the solution of the equation
a?* + 6x^ + 36 = QOx as a challenge to mathematicians : this par-
ticular equation had probably been found in some Arabic
work. Nothing is known about the history of this problem
except that Ferrari succeeded where Tartaglia and Cardan
had failed.
Rheticus. Georg Joachim Rheticus, born at Feldkirch on
Feb. 15, 1514, and died at Kaschau on Dec. 4, 1576, was
professor at Wittenberg, and subsequently studied under
Copernicus whose works were produced under the direction of
Rheticus. Rheticus constructed various trigonometrical tables,
some of which were published by his pupil Otho in 1596.
These were subsequently completed and extended by Vieta
and Pitiscus, and are the basis of those still in use. Rheticus
also found the values of sin 20 and sin 3^ in terms of sin 0
and cos 0, and was aware that trigonometrical ratios might be
defined by means of the ratios of the sides of a right-angled
triangle without introducing a circle.
I add here the names of some other celebrated mathema-
ticians of about the same time, though their works are now
of little value to any save antiquarians. Franciscus
Maurolycus, born at Messina of Greek parents in 1494, and
died in 1575, translated numerous Latin and Greek mathe-
matical works, and discussed the conies regarded as sections of
a cone: his works were published at Venice in 1575. Jean
Borrel, born in 1492 and died at Grenoble in 1572, wrote an
algebra, founded on that of Stifel; and a history of the
quadrature of the circle : his works were published at Lyons
in 1559. Wilhelm Xylander, born at Augsburg on Dec. 26,
1532, and died on Feb. 10, 1576, at Heidelberg, where since
1558 he had been professor, brought out an edition of the
works of Psellus in 1556 ; an edition of Euclid's Elements in
1562; an edition of the Arithmetic of Diophantus in 1575;
CH. xii] MATHEMATICS OF THE RENAISSANCE 227
and some minor works which were collected and published in
1577. Frederigo Commandino, born at Urbino in 1509,
and died there on Sept. 3, 1575, published a translation of the
works of Archimedes in 1558 ; selections from Apollonius and
Pappus in 1566; an edition of Euclid's Elements in 1572; and
selections from Aristarchus, Ptolemy, Hero, and Pappus in
1574 : all being accompanied by commentaries. Jacques
Peletier, born at le Mans on July 25, 1517, and died at Paris
in July 1582, wrote text-books on algebra and geometry :
most of the results of Stifel and Cardan are included in the
former. Adrian Romanus, born at Lou vain on Sept. 29,
1561, and died on May 4, 1625, professor of mathematics and
medicine at the university of Louvain, was the first to prove
the usual formula for sin {A + B), And lastly, Bartholomaus
Pitiscus, born on Aug. 24, 1561, and died at Heidelberg,
where he was professor of mathematics, on July 2, 1613,
published his Trigonometry in 1599 : this contains the expres-
sions for sin {A ± B) and cos {A ± B) in terms of the trigono-
metrical ratios of A and B.
About this time also several tot-books were produced
which if they did not extend the boundaries of the subject
systematized it. In particular I may mention those by Ramus
and Bombelli.
Ramus. ^ Peter Ramus was born at Cuth in Picardy in
1515, and was killed at Paris in the massacre of St. Bartho-
lomew on Aug. 24, 1572. He was educated at the university
of Paris, and on taking his degree he astonished and charmed
the university with the brilliant declamation he delivered on
the thesis that everything Aristotle had taught was false. He
lectured — for it will be remembered that in early days there
were no professors — first at le Mans, and afterwards at Pariy ;
at the latter he founded the first chair of mathematics.
Besides some works on philosophy he wrote treatises on
arithmetic, algebra, geometry (founded on Euclid), astronomy
1 See the mpuograplis by Ch. Waddington, Paris, 1855 ; and by
C. Desmaze, Paris, 1864.
228 MATHEMATICS OF THE RENAISSANCE [ch.xii
(founded on the works of Copernicus), and physics, which were
long regarded on the Continent as the standard text-books in
these subjects. They are collected in an edition of his works
published at Bale in 1569.
Bombelli. Closely following the publication of Cardan's
great work, Rafaello Bombelli published in 1572 an algebra
which is a systematic exposition of the knowledge then current
on the subject. In the preface he traces the history of the
subject, and alludes to Diophantus who, in spite of the notice
of Eegiomontanus, was still unknown in Europe. He discusses
radicals, real and complex. He also treats the theory of
equations, and shews that in the irreducible case of a cubic
equation the roots are all real ; and he remarks that the
problem to trisect a given angle is the same as that of the
solution of a cubic equation. Finally he gave a large collection
of problems.
Bombelli's work is noticeable for his use of symbols which
indicate an approach to index notation. Following in the
steps of Stifel, he introduced a symbol (^ for the unknown
quantity^ vi^ for its square, ^^ for its cube, and so on, and
therefore wrote x^-\-^x- i as
1 va; p. 5 (^ m. 4.
Stevinus in 1586 employed 0, ©, 0, ... in a similar way;
and suggested, though he did not use, a corresponding notation
for fractional indices. He would have written the above
expression as »-
10 + 50-40.
But whether the symbols were more or less convenient they
were still only abbreviations for words, and were subject to
all the rules of syntax. They merely afforded a sort of short-
hand by which the various steps and results could be expressed
concisely. The next advance was the creation of symbolic
algebra, and the chief credit of that is due to Vieta.
CH. xn] THE DEVELOPMENT OF ALGEBRA 229
The development of symbolic algebra.
We have now readied a point beyond which any con-
siderable development of algebra, so long as it was strictly
syncopated, could hardly proceed. It is evident that Stifel
and Bombelli and other writers of the sixteenth century had
introduced or were on the point of introducing some of the
ideas of symbolic algebra. But so far as the credit of in-
venting_symbolic algebra can be put down to any one man
we may perhaps assign it to Vieta, while we may say that
Harriot and Descartes did more than any other writers to
bring it into general use. It must be remembered, however,
that it took time before all these innovations became generally
known, and they were not familiar to mathematicians until the
lapse of some years after they had been published.
Vieta. ^ Franciscus Vieta {Frangois Viete) was born in
1540 at Fontenay near la Rochelle, and died in Paris in 1603.
He was brought up as a lawyer and practised for some time
at the Parisian bar; he then became a member of the pro-
vincial parliament in Brittany; and finally in 1580, through
the influence of the Duke de Rohan, he was made master of
requests, an office attached to the parliament at Paris; the
rest of his life was spent in the public service. He was a
firm believer in the right divine of kings, and probably a
zealous catholic. After 1580 he gave up most of his leisure
to mathematics, though his great work. In Artem Analyticam
Isagoge, in which he explained how algebra could be applied
to the solution of geometrical problems, was not published till
1591.
His mathematical reputation was already considerable,
when one day the ambassador from the Low Countries re-
marked to Henry IV. that France did not possess any
geometricians capable of solving a problem which had been
propounded in 1593 by his countryman Adrian Romanus to
1 The best accouni of Vieta's life and works is that by A. De Morgan in
the E^iglish Cyclopaedia, London, vol. vi, 1858.
230 MATHEMATICS OF THE EENAISSANCE [ch. xii
all the mathematicians of the world, and which required the
solution of an equation of the 45th degree. The king there-
upon summoned Vieta, and informed him of the challenge.
Vieta saw that the equation was satisfied by the chord of a
circle (of unit radius) which subtends an angle 27r/45 at the
centre, and in a few minutes he gave back to the king two
solutions of the problem written in pencil. In explanation of
this feat I should add that Vieta had previously discovered
how to form the equation connecting sin nd with sin 9 and
cos 6. Vieta in his turn asked Romanus to give a geometrical
construction to describe a circle which should touch three
given circles. This was the problem which Apollonius had
treated in his D^ Tactionibus, a lost book which Vieta at
a later time conjecturally restored. Romanus solved the
problem by the use of conic sections, but failed to do it by
Euclidean geometry. Vieta gave a Euclidean solution which
so impressed Romanus that he travelled to Fontenay, where
the French court was then settled, to make Vieta's acquaint-
ance— an acquaintanceship which rapidly ripened into warm
friendship.
Henry was much struck with the ability shown by Vieta in
this matter. The Spaniards had at that time a cipher contain-
ing nearly 600 characters, which was periodically changed, and
which they believed it was impossible to decipher. A despatch
having been intercepted, the king gave it to Vieta, and asked
him to try to read it and find the key to the system. Vieta
succeeded, and for two years the French used it, greatly to
their profit, in the war which was then raging. So convinced
was Philip 11. that the cipher could not be discovered, that when
he found his plans known he complained to the Pope that the
French were using sorcery against him, " contrary to the practice
of the Christian faith."
Vieta wrote numerous works on algebra and geometry.
The most important are the In Artem Analyticam Isagoge,
Tours, 1591 ; the Supplementum Geometriae^ and a collection
of geometrical problems, Tours^ 1593; and the De Numerosa
CH. xii] VIETA 231
Potestatum Resolutione, Paris, 1600. All of these were printed
for private circulation only, but they were collected by F. van
Schooten and published in one volume at Ley den in 1646.
Vieta also wrote the De Aequationum Recognitione et Emenda-
tione, which was published after his death in 1615 by Alexander
Anderson.
The In Artem is the earliest work on symbolic algebra. It ' ^
also introduced the use of letters for both known and unknown
(positive) quantities, a notation for the powers of quantities,
and enforced the advantage of working with homogeneous
equations. To this an appendix called Logistice Speciosa was
added on addition and multiplication of algebraical quantities,
and on the powers of a binomial up to the sixth. Vieta
implies that he knew how to form the coefficients of these six
expansions by means of the arithmetical triangle as Tartaglia
had previously done, but Pascal gave the general rule for
forming it for any order, and Stifel had already indicated the
method in the expansion of (1 +^)" if those in the expansion
of (l+^)^~i were known; Newton was the first to give the
general expression for the coefficient of xP in the expansion of 1
(1 ■\-x)'^. Another appendix known as Zetetica on the solution
of equations was subsequently added to the In Artem.
The In Artem is memorable for two improvements in
algebraic notation which were introduced here, though it is
probable that Vieta took the idea of both from other authors.
One of these improvements was that he denoted the known
quantities by the consonants j5, C, Z>, &c., and the unknown
quantities by the vowels A, E, /, &c. Thus in any problem
he was able to use a number of unknown quantities. In this
particular point he seems to have been forestalled by Jordanus
and by Stifel. The present custom of using the letters at the
beginning of the alphabet a, 5, c, &c., to represent known
quantities and those towards the end^. x, y, 2, &c., to represent
the unknown quantities was introduced by Descartes in 1637. <
The other improvement was this. Till this time it had been
generally the custom to introduce new symbols to represent the
232 MATHEMATICS OF THE RENAISSANCE [ch. xii
square, cube, &c., of quantities which had already occurred in
the equations ; thus, if ^ or JV stood for or, Z or C oi Q stood
for x^, and C or K for x^, &c. So long as this was the case the
chief advantage of algebra was that it afforded a concise state-
ment of results every statement of which was reasoned out.
But when Vieta used A to denote the unknown quantity x, he
sometimes employed A quadratus, A cubus, ... to represent x^,
x^, ..., which at once showed the connection between the
different powers ; and later the successive powers of A were
commonly denoted by the abbreviations Aq, Ac, Aqq, &c. Thus
Vieta would have written the equation
3BA^-DA+A^ = Z,
as ^ 3 in A quad. - D piano in A -{• A cubo aequatur Z solido.
It will be observed that the dimensions of the constants {B, i>,
and Z) are chosen so as to make the equation homogeneous :
this is characteristic of all his work. It will be also noticed
that he does not use a sign for equality ; and in fact the parti-
cular sign = which we use to denote equality was employed by
him to represent " the difference between." Vieta's notation is
not so convenient as that previously used by Stifel, Bombelli,
and Stevinus, but it was more generally adopted.
These two steps were almost essential to any further progress
in algebra. In both of them Vieta had been forestalled, but it
was his good luck in emphasising their importance to be the
means of making them generally known at a time when opinion
was ripe for such an advance.
The De Aeqnationum Recognitione et Emendatione is mostly
on the theory of equations. It was not published till twelve
years after Vieta's death, and it is possible that the editor made
additions to it. Vieta here indicated how from a given equation
another could be obtained whose roots were equal to those of
the original increase by a given quantity, or multiplied by a
given quantity; he used this method to get rid of the co-
efficient of 0? in a quadratic equation and of the coefficient of
07- in a cubic equation, and was thus enabled to give the general
CH. xii] VIETA 233
algebraic solution of both. It would seem that he knew that
the first member of an algebraical equation (^ (a?) = 0 could be
resolved into linear factors, and that the coefficients of x could
be expressed as functions of the roots ; perhaps the discovery
of both these theorems should be attributed to him.
His solution of a cubic equation is as follows. First reduce
the equation to the form x^ + ^a?x = 26^. Next let x = a^jy - y,
and we get y'° + ^h^y^ = a^, which is a quadratic in y^. Hence y
can be found, and therefore x can be determined.
His solution of a biquadratic is similar to that known as
Ferrari's, and essentially as follows. He first got rid of the
term involving x^^ thus reducing the equation to the form
x^ + a^x'^ + h^x — c*. He then took the forms involving x^ and x
to the right-hand side of the equation and added a^y + 1^4 ^q
each side, so that the equation became
(^2 + 1^2)2 = ^2 (^2 _ ^2) _ 53^ + 1^4 + ^K
He then chose y so that the right-hand side of this equality is
a perfect square. Substituting this value of y, he was able to
take the square root of both sides, and thus obtain two quadratic
equations for x, each of which can be solved.
The De Numerosa Fofestatum Resolutione deals with nume-
rical equations. In this a method for approximating to the
values of positive roots is given, but it is prolix and of little
use, though the principle (which is similar to that of Newton's
rule) is correct. Negative roots are uniformly rejected. This
work is hardly worthy of^Vieta's reputation.
Vieta's trigonometrical^ researches are included in various
tracts which are collected in Van Schooten's edition. Besides
some trigonometrical tables he gave the general expression for
the sine (or chord) of an angle in terms of the sine and cosine
of its submultiples. Delambre considers this as the completion
of the Arab system of trigonometry. We may take it then
that from this time the results of elementary trigonometry were
familiar to mathematicians. Vieta also elaborated the theory
of right-angled spherical triangles.
234 MATHEMATICS OF THE RENAISSANCE [ch. xii
Among Vieta's miscellaneous tracts will be found a proof
that each of the famous geometrical problems of the trisection
of an angle and the duplication of the cube depends on the
solution of a cubic equation. There are also some papers
connected with an angry controversy with Clavius, in 1594,
on the subject of the reformed calendar, in which Vieta was
not well advised.
Vieta's works on geometry are good, but they contain
nothing which requires mention here. He applied algebra
and trigonometry to help him in investigating the properties
of figures. He also, as I have already said, laid great stress
on the desirability of always working with homogeneous
equations, so that if a square or a cube were given it should
be denoted by expressions like a^ or 5^, and not by terms like
m or n which do not indicate the dimensions of the quantities
they represent. He had a lively dispute with Scaliger on the
latter publishing a solution of the quadrature of the circle,
and Vieta succeeded in showing the mistake into which his
rival had fallen. He gave a solution of his own which as far
as it goes is correct, and stated that the area of a square is to
that of the circumscribing circle as
This is one of the earliest attempts to find the value of tt by
means of an infinite series. He was well acquainted with the
extant writings of the Greek geometricians, and introduced the
curious custom, which during the seventeenth and eighteenth
centuries became fashionable, of restoring lost classical works.
He himself produced a conjectural restoration of the De 2uc-
tionibus of Apollonius.
Girard. Vieta's results in trigonometry and the theory of
equations were extended by Albert Girard^ a Dutch mathe-
matician, who was born in Lorraine in 1595, and died on
December 9, 1632.
In 1626 Girard published at the Hague a short treatise on
trigonometry, to which were appended tables of the values of
CH.xii] GIRARD. NAPIER 235
the trigonometrical functions. This work contains the earliest
use of the abbreviations sin, tan, sec for sine, tangent, and
secant. The supplemental triangles in spherical trigonometry
are also discussed ; their properties seem to have been discovered
by Girard and Snell at about the same time. Girard also gave
the expression for the area of a spherical triangle in terms of
the spherical excess — this was discovered independently by
Cavalieri. In 1627 Girard brought out an edition of Marolois's
Geometry with considerable additions.
Girard's algebraical investigations are contained in his Inven-
tion nouvelle en Valgebre, published at Amsterdam in 1629.^ This
contains the earliest use of brackets ; a geometrical interpre-
tation of the negative sign ; the statement that the number of
roots of an algebraical question is equal to its degree ; the
distinct recognition of imaginary roots ; the theorem, known as
Newton's rule, for finding the sum of like powers of the roots
of an equation ; and (in the opinion of some writers) implies
also a knowledge that the first member of an algebraical equa-
tion <fi(x) = 0 could be resolved into linear factors. Giiard's
investigations were unknown to most of his contemporaries,
and exercised no appreciable influence on the development of
mathematics.
The invention of logarithms by Napier of Merchiston in
1614, and their introduction into England by Briggs and others,
have been already mentioned in chapter xi. A few words on
these mathematicians may be here added.
Napier. 2 John ]}^apier was born at Merchiston in 1550,
and died on April 4, 1617. He spent most of his time on the
family estate near Edinburgh, and took an active part in the
political and religious controversies of the day ; the business of
his life was to show that the Pope was Antichrist, but his
favourite amusement was the study of mathematics and science.
^ It was re-issued by B. de Haan at Leyden in 1884.
- See the Memoirs of Kapier by Mark Napier, Edinburgh, 1834. An
edition of all his works was issued at Edinburgh in 1839. A bibliography
of his writings is appended to a translation of the Gonstructio by W. R.
Macdonald, Edinburgh, 1889.
236 MATHEMATICS OF THE RENAISSANCE [ch. xii
As soon as the use of exponents became common in algebra
the introduction of logarithms would naturally follow, but
Napier reasoned out the result without the use of any symbolic
notation to assist him, and the invention of logarithms was the
result of the efforts of many years with a view to abbreviate
the processes of multiplication and division. It is likely that
Napier's attention may have been partly directed to the
desirability of facilitating computations by the stupendous
arithmetical efforts of some of his contemporaries, who seem
to have taken a keen pleasure in surpassing one another in
the extent to which they carried multiplications and divisions.
The trigonometrical tables by Rheticus, which were published
in 1596 and 1613, were calculated in a most laborious way:
Vieta himself delighted in arithmetical calculations which must
have taken days of hard w^ork, and of which the results often
served no useful purpose : L. van Ceulen (1539-1610) practically
devoted his life to finding a numerical approximation to the
value of TT, finally in 1610 obtaining it correct to 35 places of
decimals : while, to cite one more instance, P. A. Cataldi (1548-
1626), who is chiefly known for his invention in 1613 of the
form of continued fractions, must have spent years in numerical
calculations.
In regard to Napier's other work I may again mention that
in his Rahdologia^ published in 1617, he introduced an im-
proved form of rod by the use of which the product of two
numbers can be found in a mechanical way, or the quotient of
one number by another. He also invented two other rods
called "virgulae," by which square and cube roots can be
extracted. I should add that in spherical trigonometry he
discovered certain formulae known as Napier's analogies, and
enunciated the "rule of circular parts" for the solution of
right-angled spherical triangles.
Briggs. The name of Briggs is inseparably associated with
the history of logarithms. Henry Briggs'^ was born near
^ See pp. 27-30 of my History of the Stvdy of Matliematics at Caonbridge,
Cambridge, 1889.
CH. xii] BRIGGS. HARRIOT 237
Halifax in 1561 : he was educated at St. John's College,
Cambridge, took his degree in 1581, and obtained a fellowship
in 1588 : he was elected to the Gresham professorship of
geometry in 1596, and in 1619 or 1620 became Savilian
professor at Oxford, a chair which he held until his death on
January 26, 1631. It may be interesting to add that the
chair of geometry founded by Sir Thomas Gresham was the
earliest professorship of mathematics established in Great
Britain. Some twenty years earlier Sir Henry Savile had
given at Oxford open lectures on Greek geometry and geo-
metricians, and in 1619 he endowed the chairs of geometry
and astronomy in that university which are still associated
with his name. Both in London and at Oxford Briggs was
the first occupant of the chair of geometry. He began his
lectures at Oxford with the ninth proposition of the first book
of Euclid — that being the furthest point to which Savile had
been able to carry his audiences. At Cambridge the Lucasian
chair was established in 1663, the earliest occupants being
Barrow and Newton.
The almost immediate adoption throughout Europe of
logarithms for astronomical and other calculations was mainly
the work of Briggs, who undertook the tedious work of calculat-
ing and preparing tables of logarithms. Amongst others he
convinced Kepler of the advantages of Napier's discovery, and
the spread of the use of logarithms was rendered more rapid by
the zeal and reputation of Kepler, who by his tables of 1625
and 1629 brought them into vogue in Germany, while Cavalieri
in 1624 and Edmund Wingate in 1626 did a similar service for
Italian and French mathematicians respectively. Briggs also
was instrumental in bringing into common use the method of
long division now generally employed.
Harriot. Thomas Harriot, who was born at Oxford in
1560, and died in London on July 2, 1621, did a great deal to
extend and codify the theory of equations. The early part of
his life was spent in America with Sir Walter Raleigh ; while
there he made the first survey of Virginia and North Carolina,
238 MATHEMATICS OF THE RENAISSANCE [ch. xii
the maps of these being subsequently presented to Queen
Elizabeth. On his return to England he settled in London,
and gave up most of his time to mathematical studies.
The majority of the propositions I have assigned to Vieta
are to be found in Harriot's writings, but it is uncertain
whether they were discovered by him independently of Vieta
or not. In any case it is probable that Vieta had not fully
realised all that was contained in the propositions he had
enunciated. Some of the consequences of these, with exten-
sions and a systematic exposition of the theory of equations,
were given by Harriot in his Artis Analyticae Praxis, which
was first printed in 1631. The Praxis is more analytical than
any algebra that preceded it, and marks an advance both in
symbolism and notation, though negative and imaginary roots
are rejected. It was widely read, and proved one of the most
powerful instruments in bringing analytical methods into general
use. Harriot was the first to use the signs > and < to repre-
sent greater than and less than. When he denoted the unknown
quantity by a he represented a^ by an, a^ by aaa, and so on.
This is a distinct improvement on Vieta's notation. The same
symbolism was used by Wallis as late as 1685, but concurrently
with the modern index notation which was introduced by
Descartes. I need not allude to the other investigations of
Harriot, as they are comparatively of small importance ; extracts
from some of them were published by S. P. Rigaud in 1833.
Oughtred. Among those who contributed to the general
adoption in England of these various improvements and
additions to algorism and algebra was William Oughtred,^ who
was born at Eton on March 5, 1575, and died at his vicarage
of Albury in Surrey on June 30, 1660 : it is sometimes said
that the cause of his death was the excitement and delight
which he experienced " at hearing the House of Commons [or
Convention] had voted the King's return " ; a recent critic adds
^ See pp. 30-31 of my History of the Study of Mathematics at Oavibridge,
Cambridge, 1889. A complete edition of Oughtred's works was published at
Oxford in 1677.
CH. xii] OUGHTRED 239
that it should be remembered "by way of excuse that he
[Oughtred] was then eighty-six years old," but perhaps the
story is sufficiently discredited by the date of his death.
Oughtred was educated at Eton and King's College, Cambridge,
of the latter of which colleges he was a fellow and for some time
mathematical lecturer.
His Clavis Mathematicae published in 1631 is a good system-
atic text-book on arithmetic, and it contains practically all that
was then known on the subject. In this work he introduced the
symbol x for multiplication. He also introduced the symbol
: : in proportion : previously to his time a proportion such as
a-.h — cd was usually written as a-b-c-d; he denoted it
hj a . h : : c . d. Wallis says that some found fault with the
book on account of the style, but that they only displayed their
own incompetence, for Oughtred's "words be always full but
not redundant." Pell makes a somewhat similar remark.
Oughtred also wrote a treatise on trigonometry published in
1657, in which abbreviations for sine, cosine, &c., were employed.
This was really an important advance, but the works of Girard
and Oughtred, in which they were used, were neglected and soon
forgotten, and it was not until Euler reintroduced contractions
for the trigonometrical functions that they were generally adopted.
In this work the colon {i.e. the symbol :) was used to denote a ratio.
We may say roughly that henceforth elementary arithmetic,
algebra, and trigonometry were treated in a manner which is
not substantially different from that now in use ; and that the
subsequent improvements introduced were additions to the
subjects as then known, and not a rearrangement of them
on new foundations.
The origin of the more common symbols in algebra.
It may be convenient if I collect here in one place the
scattered remarks I have made on the introduction of the
various symbols for the more common operations in algebra.^
^ See also two articles by C. Henry in the June and July numbers of the
Revue ArcMologique, 1879, vol. xxxvii, pp. 324-333, vol. xxxviii, pp. 1-10.
240 MATHEMATICS OF THE RENAISSANCE [ch. xii
The later Greeks, the Hindoos, and Jordanus indicated
addition by mere juxtaposition. It will be observed that this
is still the custom in arithmetic, where, for instance, 2| stands
for 2 + J. The Italian algebraists, when they gave up expressing
every operation in words at full length and introduced synco-
pated algebra, usually denoted jo^^ts by its initial lefter P or ^;,
a line being sometimes drawn through the letter to show that it
was a contraction, or a symbol of operation, and not a quantity.
The practice, however, was not uniform; Pacioli, for example,
sometimes denoted plus by p, and sometimes by e, and Tartaglia
commonly denoted it by <^. The German and English algebraists,
on the other hand, introduced the sign + almost as soon as they
used algorism, but they spoke of it as signum additorum and
employed it only to indicate excess ; they also used it with a
special meaning in solutions by the method of false assumption.
Widman used it as an abbreviation for excess in 1489 : by 1630
it was part of the recognised notation of algebra, and was
used as a symbol of operation.
Subtraction was indicated by Diophantus by an inverted and
truncated xj^. The Hindoos denoted it by a dpt. The Italian
algebraists when they introduced syncopated algebra generally
denoted minus by if or m, a line being sometimes drawn through
the letter; but the practice was not uniform — Pacioli, for ex-
ample, denoting it sometimes by m, and sometimes by de for
demptus. The German and English algebraists introduced the
present symbol which they described as signum subtractorum.
It is most likely that the vertical bar in the symbol for plus
was superimposed on the symbol for minus to distinguish the
two. It may be noticed that Pacioli and Tartaglia found the
sign - already used to denote a division, a ratio, or a proportion
indifferently. The present sign for minus was in general use by
about the year 1630, and was then employed as a symbol of
operation.
Vieta, Schooten, and others among their contemporaries
employed the sign = written between two quantities to denote
the difference between them ; thus a = 6 means with them what
CH.xii] ALGEBRAIC SYMBOLS 241
we denote hy a c\j h. On the other hand, Barrow wrote — : for
the same purpose. I am not aware when or by whom the current
symbol oj was first used with this signification.
Oughtred in 1631 used the sign x to indicdite multiplication ;
Harriot in 1631 denoted the operation by a dot; Descartes in
1637 indicated it by juxtaposition. I am not aware of any
symbols for it which were in previous use. Leibnitz in 1686
employed the sign ^^ to denote multiplication.
Division was ordinarily denoted by the Arab way of
writing the quantities in the form of a fraction by means of
a line drawn between them in any of the forms a-b, ajb, or
J. Oughtred in 1631 employed a dot to denote either division
or a ratio. Leibnitz in 1686 employed the sign ^ to denote
division. The colon (or symbol :), used to denote a ratio,
occurs on the last two pages of Oughtred's Canones Sinuum,
published in 1657. I believe that the current symbol for
division -^ is only a combination of the - and the symbol : for
a ratio ; it was used by Johann Heinrich Rahn at Zurich in
1659, and by John Pell in London in 1668. The symbol -H-
was used by Barrow and other writers of his time to indicate
continued proportion.
The current symbol for equality was introduced by Record
in 1557; Xylander in 1575 denoted it by two parallel vertical
lines ; but in general till the year 1 600 the word was written at
length ; and from then until the time of Newton, say about
1680, it was more frequently represented by oo or by oo than
by any other symbol. Either of these latter signs was used as
a contraction for the first two letters of the word aequalis.
The symbol : : to denote proportion, or the equality of two
ratios, was introduced by Oughtred in 1631, and was brought
into common use by Wallis in 1686. There is no object in
having a symbol to indicate the equality of two ratios which is
different from that used to indicate the equality of other things,
and it is better to replace it by the sign = .
The sign > for is greater than and the sign < for is less than
E
242 MATHEMATICS OF THE. KENAISSANCE [ch. xii
were introduced by Harriot in 1631, but Oughtred simultaneously
invented the symbols H and H for the same purpose ; and
these latter were frequently used till the beginning of the
eighteenth century, ex. gr. by Barrow.
The symbols 4= for is not equal to, ;}> is not greater than, and
<tr for is not less than, are, I believe, now rarely used outside
Great Britain ; they were employed, if not invented, by Euler.
The symbols > and < were introduced by P. Bouguer in 1734.
The vinculum was introduced by Vieta in 1591 ; and brackets
were first used by Girard in 1629.
The symbol J to denote the square root was introduced by
Rudolff in 1526 ; a similar notation had been used by Bhaskara
and by Chuquet.
The different methods of representing the power to which
a magnitude was raised have been already briefly alluded to.
The earliest known attempt to frame a symbolic notation was
made by Bombelli in 1572, when he represented the unknown
quantity by viy, its square by ,^, its cube by v^, &c. In
1586 Stevinus used 0, 0, 0, &c., in a similar way; and
suggested, though he did not use, a corresponding notation
for fractional indices. In 15'91 Yieta improved on this by
denoting the different powers of A hj A, A quad., A cub., &c.,
so that he could indicate the powers of different magnitudes ;
Harriot in 1631 further improved on Vieta's notation by
writing aa for a^, aaa for d^, &c., and this remained in use for
fifty years concurrently with the index notation. In 1634
P. Herigonus, in his Cursus mathematicus, published in five
volumes at Paris in 1634-1637, wrote a, a2, a3, ... for a, a^,
a^ ....
The idea of using exponents to mark the power to which a
quantity was raised was due to Descartes, and was introduced
by him in 1637; but he used only positive integral indices
a^, a^, a^, .... Wallis in 1659 explained the meaning of negative
and fractional indices in expressions such as a~^, ax^'"^, &c. ; the
latter conception having been foreshadowed by Oresmus and
perhaps by Stevinus. Finally the idea of an index unrestricted
CH.xii] ALGEBRAIC SYMBOLS 243
in magnitude, such as the n in the expression a'^, is, I believe,
due to Newton, and was introduced by him in connection with
the binomial theorem in the letters for Leibnitz written in
1676.
The symbol oo for infinity was first employed by Wallis in
1655 in his Arithmetica Injinitorum ; but does not occur again
until 1713, when it is used in James Bernoulli's Ars Con-
jectandi. This sign was sometimes employed by the Romans
to denote the number 1000, and it has been conjectured that
this led to its being applied to represent any very large
number.
There are but few special symbols in trigonometry ; I may,
however, add here the following note which contains all that I
have been able to learn on the subject. The current sexagesimal
division of angles is derived from the Babylonians through the
Greeks. The Babylonian unit angle was the angle of an
equilateral triangle; following their usual practice this was
divided into sixty equal parts or degrees, a degree was sub-
divided into sixty equal parts or minutes, and so on ; it is said
that 60 was assumed as the base of the system in order that the
number of degrees corresponding to the circumference of a circle
should be the same as the number of days in a year which it is
alleged was taken (at any rate in practice) to be 360.
The word sine was used by Regiomontanus and was derived
from the Arabs ; the terms secant and tangent were introduced
by Thomas Finck (born in Denmark in 1561 and died in 1646)
in his Gecmietriae Rotundi, Bale, 1583 ; the word cosecant
was (I believe) first used by Rheticus in his Opus Palatinum,
1596 ; the terms cosine and cotangent were first employed by
E. Gunter in his Canon Triangulorwn^ London, 1620. The
abbreviations sm, tan, sec were used in 1626 by Girard, and
those of cos and cot by Oughtred in 1657 ; but these contractions
did not come into general use till Euler reintroduced them in
1748. The idea of trigonometrical functions originated with
John Bernoulli, and this view of the subject was elaborated in
1748 by Euler in his Introductio in Analysin Infinitorum.
244
CHAPTER XIII.
THE CLOSE OF THE RENAISSANCE. ^
ciEC. 1586-1637.
The closing years of the renaissance were marked by a revival
of interest in nearly all branches of mathematics and science.
As far as pure mathematics is concerned we have already seen
that during the last half of the sixteenth century there had been
a great advance in algebra, theory of equations, and trigono-
metry ; and we shall shortly see (in the second section of this
chapter) that in the early part of the seventeenth century
some new processes in geometry were invented. If, however,
we turn to applied mathematics it is impossible not to be
struck by the fact that even as late as the middle or end of the
/sixteenth century no marked progress in the theory had been
' made from the time of Archimedes. Statics (of solids) and
hydrostatics remained in much the state in which he had left
them, while dynamics as a science did not exist. It was
Stevinus who gave the first impulse to the renewed study of
staticSj and Galileo who laid the foundation of dynamics ; and
to their works the first section of this chapter is devoted.
The development of niecJianics and experimental methods.
Stevinus.^ Simon Stevinus was born at Bruges in 1548,
^ See footnote to chapter xii.
^ All analysis of his works is given in the Histoire des sciences mathe-
CH.xm] STEVmUS ^45
and died at the Hague in 1620. We know very little of his life
save that he was originally a merchant's clerk at Antwerp, and at
a later period of his life was the friend of Prince Maurice of
Orange, by whom he was made quartermaster -general of the
Dutch army.
To his contemporaries he was best known for his works on
fortifications and military engineering, and the principles he
laid down are said to be in accordance with those which are now
usually accepted. To the general populace he was also well
known on account of his invention of a carriage which was
propelled by sails; this ran on the sea-shore, carried twenty-
eight people, and easily outstripped horses galloping by the
side; his model of it was destroyed in 1802 by the French
when they invaded Holland. It was chiefly owing to the
influence of Stevinus that the Dutch and French began a
proper system of book-keeping in the national accounts.
I have already alluded to the introduction in his Arithmetic^
published in 1585, of exponents to mark the power to which
quantities were raised ; for instance, he wrote S^r- - 5^ -t- 1
as 30-50 + 10. His notation for decimal fractions was
of a similar character. He further suggested the use of
fractional (but not negative) exponents. In the same book he
likewise suggested a decimal system of weights and measures.
He also published a geometry which is ingenious though it
does not contain many results which w^ere not previously
known ; in it some theorems on perspective are enunciated.
It is, however, on his Statics and Hydrostatics, published (in
Flemish) at Leyden in 1586, that his fame rests. In this
work he enunciates the triangle of forces — a theorem which
some think was first propounded by Leonardo da Vinci.
Stevinus regards this as the fundamental proposition of the
mafiques et physiques chez les Beiges, by L. A. J, Quetelet, Brussels, 1866,
pp. 144-168 ; see also Notice histoi-ique stir la vie et les ouvrages de Stevinus,
by J. V. Gothals, Brussels, 1841 ; and Les travaux de Stevinus, by
M. Steichen, Brussels, 1846. The works of Stevinus were collected by
Snell, translated into Latin, and published at Leyden in 1608 under the title
HypoTiineviata Mathetnatica.
246 THE CLOSE OF THE RENAISSANCE [ch. xiii
subject. Previous to tlie publication of his work the science of
statics had rested on the theory of the lever, but subsequently it
became usual to commence by proving the possibility of repre-
senting forces by straight lines, and thus reducing many
theorems to geometrical propositions, and in particular to
obtaining in this way a proof of the parallelogram (which is
equivalent to the triangle) of forces. Stevinus is not clear in
his arrangement of the various propositions or in their logical
sequence, and the new treatment of the subject was not definitely
established before the appearance in 1687 of Varignon's work
on mechanics. Stevinus also found the force which must be
I exerted along the line of greatest slope to support a given
weight on an inclined plane — a problem the solution of which
had been long in dispute. He further distinguishes between
stable and unstable equilibrium. In hydrostatics he discusses
the question of the pressure which a fluid can exercise, and
explains the so-called hydrostatic paradox.
His method ^ of finding the resolved part of a force in a
given direction, as illustrated by the case of a weight resting on
an inclined plane, is a good specimen of his work and is worth
quoting. /^
He takes a wedge ABC whose base ^.^^ is horizontal [and
whose sides BA, BG are in the ratio of 2 to 1]. A thread
connecting a number of small equal equidistant weights is placed
over the wedge as indicated in the figure on the next page (which
I reproduce from his demonstration) so that the number of these
weights on BA is to the number on BC in the same proportion
as BA is to BC. This is always possible if the dimensions of the
wedge be properly chosen, and he places four weights resting on
BA and two on BC ; but we may replace these weights by a
heavy uniform chain TSLVT without altering his argument.
He says in effect, that experience shews that such a chain will
remain at rest ; if not, we could obtain perpetual motion. Thus
the effect in the direction BA of the weight of the part TS of
the chain must balance the effect in the direction BC of the
' Tlypomnemata Mathcmalica, vol. iv, cle Statica, prop. 19.
CH.xiii] STEVINUS. GALILEO 247
weight of the part TV oi the chain. Of course BC may be
vertical, and if so the above statement is equivalent to saying
that the effect in the direction BA of the weight of the chain on
it is diminished in the proportion of BC to BA ; in other words,
if a weight W rests on an inclined plane of inclination a the
component of W down the line of greatest slope is W sin a.
Stevinus was somewhat dogmatic in his statements, and
allowed no one to differ from his conclusions, "and those," says
he, in one place, " who cannot see this, may the Author of nature
have pity upon their unfortunate eyes, for the fault is not
in the thing, but in the sight which we are unable to give them."
Galileo.^ Just as the modern treatment of statics originates
with Stevinus, so the foundation of the science ^>f_dynamics is
due to Galileo. Galileo Galilei was born at Pisa on February
18, 1564, and died near Florence on January 8, 1642. His
father, a poor descendant of an old and noble Florentine house,
was himself a fair mathematician and a good musician. Galileo
was educated at the monastery of Vallombrosa, where his
literary ability and mechanical ingenuity attracted considerable
^ See the biography of Galileo, by J. J. Fahie, London, 1903. An
edition of Galileo's works was issued in 16 volumes by E. Alberi, Florence,
1842-1856. A good many of his letters on various mathematical subjects
have been since discovered, and a new and complete edition is in process of
issue by the Italian Government, Florence ; vols, i to xix and a bibliography,
1890-1907.
248 THE CLOSE OF THE RENAISSANCE [ch. xiii
attention. 'He was persuaded to become A novitiate of the order
in 1579, but his father, who had other views, at once removed
him', and sent him in 1581 to the university of Pisa to study
medicine. It was there that he noticed that the great bronze
lamp, hanging from the roof of the cathedral, performed its
oscillations in equal times, and independently of whether the
oscillations were large or small — a fact which he verified by
counting his pulse. He had been hitherto kept in ignorance of
mathematics, but one day, by chance hearing a lecture on
geometry (by Ricci), he was so fascinated by the science that
thenceforward he devoted his leisure to its study, and finally
got leave to discontinue his medical studies. He left the
university in 1585, and almost immediately commenced his
original researches.
He published in 1586 an account of the hydrostatic balance,
and in 1588 an essay on the centre of gravity in solids ; these
were not printed till later. The fame of these works secured for
V him in 1589 the appointment to the mathematical chair at Pisa
— the stipend, as was then the case with most professorships,
being very small. During the next three years he carried on,
from the leaning tower, that series of experiments on falling
bodies which established the first principles of dynamics.
Unfortunately, the manner in which he promulgated his dis-
coveries, and the ridicule he threw on those who opposed him,
gave not unnatural ofi'ence, and in 1591 he was obliged to
resign his position.
At this time he seems to have been much hampered by want
of money. Influence was, however, exerted on his behalf with
the Venetian senate, and he was appointed professor at Padua, a
chair which he held for eighteen years, 1592-1610. His
lectures there seem to have been chiefly on mechanics and
hydrostatics, and the substance of them is contained in his
treatise on mechanics, which was published in 1612. In these
lectures he repeated his Pisan experiments, and demonstrated
that falling bodies did not (as was then commonly believed)
descend with velocities proportional, amongst other things, to
CH.XIII] GALILEO 249
their weights. He further shewed that, if it were assumed that
they descended with a uniformly accelerated motion, it was
possible to deduce the relations connecting velocity, space, and
time which did actually exist. At a later date, by observing the
times of descent of bodies sliding down inclined, planes, he
shewed that this hypothesis was true. He also proved that the
path of a projectile is a parabola, and in doing so implicitly
used the principles laid down in the first two laws of motion as
enunciated by Newton. He gave an accurate definition of
momentum which some writers have thought may be taken to
imply a recognition of the truth of the third law of motion.
The laws of motion are, however, nowhere enunciated in a pre-
cise and definite form, and Galileo must be regarded rather as
preparing the way for Newton than as being himself the creator
of the science of dynamics.
In statics he laid down the principle that in machines what^
' was gained in power was lost in speed, and in the same ratio. |
In the statics of solids he found the force which can support
a given weight on an inclined plane ; in hydrostatics he pro-
pounded the more elementary theorems on pressure and on
floating bodies ; while among hydrostatical instruments he used,
and perhaps invented, the thermometer, though in a somewhat
imperfect form.
It is, however, as an astronomer that most people regard
Galileo, and though, strictly speaking, his astronomical researches
lie outside the subject-matter of this book, it may be interest-
ing to give the leading facts. It was in the spring of 1609
that Galileo heard that a tube containing lenses had been made
by an optician, Hans Lippershey, of Middleburg, which served
to magnify objects seen through it. This gave him the clue,
and he constructed a telescope of that kind which still bears his
name, and of w^hich an ordinary opera-glass is an example.
Within a few months he had produced instruments which were
capable of magnifying thirty-two diameters, and within a year
he had made and published observations on the solar spots, the
lunar mountains, Jupiter's satellites, the phases of Venus, and
250 THE CLOSE OF THE RENAISSANCE [ch. xiii
Saturn's ring. The discovery of the microscope followed natu-
rally from that of the telescope. Honours and emoluments were
showered on him, and he was enabled in 1610 to give up his
professorship and retire to Florence. In 1611 he paid a tem-
porary visit to Rome, and exhibited in the gardens of the
Quirinal the new worlds revealed by the telescope.
It would seem that Galileo had always believed in the
Copernican system, but was afraid of promulgating it on
account of the ridicule it excited. The existence of Jupiter's
satellites seemed, however, to make its truth almost certain, and
he now boldly preached it. The orthodox party resented his
action, and in February 1616 the Inquisition declared that to
suppose the sun the centre of the solar system was false, and
opposed to Holy Scripture. The edict of March 5, 1616, which
carried this into effect, has never been. repealed, though it has
been long tacitly ignored. It is well known that towards the
middle of the seventeenth century the Jesuits evaded it by
treating the theory as an hypothesis from which, though false,
certain results would follow.
In January 1632 Galileo published his dialogues on the
system of the world, in which in clear and forcible language he
expounded the Copernican theory. In these, apparently through
jealousy of Kepler's fame, . he does not so much as mention
Kepler's laws (the first two of which had been published in
1609, and the third in 1619); he rejects Kepler's hypothesis
that the tides are caused by the attraction of the moon, and
tries to explain their existence (which he alleges is a confirma-
tion of the Copernican hypothesis) by the statement that
different parts of the earth rotate with different velocities. He
was more successful in showing that mechanical principles
would account for the fact that a stone thrown straight up
falls again to the place from which it was thrown — a fact
which previously had been one of the chief difficulties in the
way of any theory which supposed the earth to be in
motion.
The publication of this book was approved by the papal
cH.xiii] GALILEO 251
censor, but substantially was contrary to the edict of 1616.
Galileo was summoned to Rome, forced to recant, do penance,
and was released only on promise of obedience. The documents
recently printed show that he was threatened Avith the torture,
but probably there was no intention of carrying the threat into
effect.
When released he again took up his work on mechanics, and
by 1636 had finished a book which was published under the
title Discorsi intorno a due nuove scienze at Ley den in 1638.
In 1637 he lost his sight, but with the aid of pupils he con-
tinued his experiments on mechanics and hydrostatics, and in
particular on the possibility of using a pendulum to regulate a
clock, and on the theory of impact.
An anecdote of this time has been preserved which, though
probably not authentic, is sufficiently interesting to bear repeti-
tion. According to one version of the story, Galileo was
interviewed by some members of a Florentine guild who wanted
their pumps altered so as to raise water to a height which was
greater than thirty feet; and thereupon he remarked that it
might be desirable to first find out why the water rose at all.
A bystander intervened and said there was no difficulty about
that, because nature abhorred a vacuum. Yes, said Galileo, but
apparently it is only a vacuum which is less than thirty feet.
His favourite pupil Torricelli was present, and thus had his
attention directed to the question, which he subsequently
elucidated. ^
Galileo's work may, I think, be fairly summed up by saying
that his researches on mechanics are deserving of high praise,
and that they are memorable for clearly enunciating the fact
that science must be founded on laws obtained by experiment ;
his astronomi-cal observations and his deductions therefrom were
also excellent, and were expounded with a literary skill which
leaves nothing to be desired ; but though he produced some of
the evidence which placed the Copernican theory on a satis-
factory basis, he did not himself make any special advance in
the theory of astronomy. .
252 THE CLOSE OF THE RENAISSANCE [ch. xiii
Francis Bacon. ^ The necessity of an experimental founda-
tion for science was also advocated with considerable effect by
Galileo's contemporary Francis Bacon (Lord Yerulam), who was
born at London on Jan. 22, 1561, and died on April 9, 1626.
He was educated at Trinity College, Cambridge. His career in
politics and at the bar culminated in his becoming Lord Chan-
cellor, with the title of Lord Verulam. The story of his subse-
quent degradation for accepting bribes is well known.
His chief work is the Novum Organum, published in 1620,
in which he lays down the principles which should guide those
who are making experiments on which they propose to found
a theory of any branch of physics or applied mathematics. He
gave rules by which the results of induction could be tested,
hasty generalisations avoided, and experiments used to check
one another. The influence of this treatise in the eighteenth
century was great, but it is probable that during the preceding
century it was little read, and the remark repeated by several
French writers that Bacon and Descartes are the creators of
modern philosophy rests on a misapprehension of Bacon's
influence on his contemporaries ; any detailed account of this
book belongs, however, to the history of scientific ideas rather
tl*an to that of mathematics.
Before leaving the subject of applied mathematics I may
add a few words on the writings of Guldinus, Wright, and
Snell.
Guldinus. Hahakkuk Guldinus^ born at St. Gall on June
12, 1577, and died at Gratz on Nov. 3, 1643, was of Jewish
descent, but was brought up as a Protestant ; he was converted
to Roman Catholicism, and became a Jesuit, when he took the
Christian name of Paul, and it was to him that the Jesuit
colleges at Rome and Gratz owed their mathematical reputation.
The two theorems known by the name of Pappus (to which I
have alluded above) were published by Guldinus in the fourth
^ See his life by J. Spedding, London, 1872-74. The best edition of his
works is that by Ellis, Spedding, and Heath, in 7 volumes, London, second
edition, 1870.
CH.xiii] GULDINUS WRIGHT 253
book of his De Centra Gravitatis, Vienna, 1635-1642. Not^^,^ ^
only were the rules in question taken without acknowledgment
from Pappus, but (according to Montucla) the proof of them
given by Guldinus was faulty, though he was successful in
applying them to the determination of the volumes and surfaces
of certain solids. The theorems were, however, previously
unknown, and their enunciation excited considerable interest.
Wright.^ I may here also refer to Edward Wright, who is
worthy of mention for having put the art of navigation on a ^
scientific basis. Wright was born in Norfolk about 1560, and
died in 1615. He was educated at Caius College, Cambridge,
of which society he was subsequently a fellow. He seems to
have been a good sailor, and he had a special talent for the con-
struction of instruments. About 1600 he was elected lecturer
on mathematics by the East India Company ; he then settled in
London, and shortly afterwards was appointed mathematical
tutor to Henry, Prince of Wales, the son of James I. His
mechanical ability may be illustrated by an orrery of his con-
struction by which it was possible to predict eclipses; it was
shewn in the Tower as a curiosity as late as 1675.
In the maps in use before the time of Gerard Mercator a
degree, whether of latitude or longitude, had been represented
in all cases by the same length, and the course to be pursued
by a vessel was marked on the map by a straight line joining
the ports of arrival and departure. Mercator had seen that
this led to considerable errors, and had realised that to make
this method of tracing the course of a ship at all accurate the
space assigned on the map to a degree of latitude ought gradu-
ally to increase as the latitude increased. Using this principle,
he had empirically constructed some charts, which were published
about 1560 or 1570. Wright set himself the problem to deter-
mine the theory on which such maps should be drawn, and
succeeded in discovering the law of the scale of the maps,
though his rule is strictly correct for small arcs only. The
^ See pp. 25-27 of mv History of the Study of Mathematics at Gairibridge,
Cambridge, 1889.
254 THE CLOSE OF THE RENAISSANCE [ch. xiii
result was published in the second edition of Blundeville's
Exercises.
In 1599 Wright published his Certain Errors in Navigation
Detected and Corrected, in which he explained the theory and
inserted a table of meridional parts. The reasoning shews con-
siderable geometrical power. In the course of the work he gives
the declinations of thirty-two stars, explains the phenomena of
the dip, parallax, and refraction, and adds a table of magnetic
declinations; he assumes the earth to be stationary. In the
following year he published some maps constructed on his
principle. In these the northernmost point of Australia is
shewn; the latitude of London is taken to be 51° 32'.
Snell. A contemporary of Guldinus and Wright was
Willehrod Snell, whose name is still well known through his
f discovery in 1619 of the law of refraction in optics. Snell was
born at Leyden in 1581, occupied a chair of mathematics at the
university there, and died there on Oct. 30, 1626. He was one
of those infant prodigies who occasionally appear, and at the
age of twelve he is said to have been acquainted with the
standard mathematical works. I will here only add that in
geodesy he laid down the principles for determining the length
of the arc of a meridian from the measurement of any base line,
and in spherical trigonometry he discovered the properties of the
polar or supplemental triangle.
Revival of interest in pure geometry.
The close of the sixteenth century was marked not only by
the attempt to found a theory of dynamics based on laws derived
from experiment, but also by a revived interest in geometry.
This was largely due to the influence of Kepler.
Kepler.i Johann Kepler, one of the founders of modern
^ See Johann Kepplers Lehen und Wirken, by J. L. E. von Breitscliwert,
Stuttgart, 1831 ; and R. Wolf's Geschichte der Astronomie, Munich, 1877.
A complete edition of Kepler's works was'published by C. Frisch at Frankfort,
in 8 volumes, 1858-71 ; and an analysis of the mathematical part of his chief
work, the Hannonice Mundi, is given by Chasles in his Apergu historique.
See also Cantor, vol. ii, part xv.
CH. xiii] KEPLER 255
astronomy, was born of humble parents near Stuttgart on
Dec. 27, 1571, and died at Ratisbon on Nov. 15, 1630. He
was educated under Mastlin at Tubingen. In 1593 he was
appointed professor at Gratz, where he made the acquaintance
of a wealthy widow, whom he married, but found too late that
he had purchased his freedom from pecuniary troubles at the
expense of domestic happiness. In 1599 he accepted an ap-
pointment as assistant to Tycho Brahe, and in 1601 succeeded
his master as astronomer to the emperor Rudolph II. But his
career was dogged by bad luck : first his stipend was not paid ;
next his wife went mad and then died, and a second marriage in
1611 did not prove fortunate ; while, to complete his discomfort,
he was expelled from his chair, and narrowly escaped condemna-
tion for heterodoxy. During this time he depended for his
income on teUing fortunes and casting horoscopes, for, as he
says, " nature which has conferred upon every animal the means
of existence has designed astrology as an adjunct and aUy to
astronomy." He seems, however, to have had no scruple in
charging heavily for his services, and to the surprise of his con-
temporaries was found at his death to possess a considerable
hoard of money. He died while on a journey to try and
recover for the benefit of his children some of the arrears of his
stipend.
In describing Galileo's work I alluded briefly to the three
laws in astronomy that Kepler had discovered, and in connection
with which his name wiU be always associated. I may further
add that he suggested that the planets might be retained in
their orbits by magnetic vortices, but this was little more than
a crude conjecture. I have also already mentioned the prominent
part he took in bringing logarithms into general use on the con-
tinent. These are familiar facts ; but it is not known so generally
that Kepler was also a geometrician and algebraist of consider-
able power, and that he, Desargues, and perhaps Galileo, may
be considered as forming a connecting link between the mathe-
maticians of the renaissance and those of modern times.
Kepler's work in geometry consists rather in certain general
25« THE CLOSE OF THE RENAISSANCE [ch. xiii
principles enunciated, and illustrated by a few cases, than in any
systematic exposition of the subject. In a short chapter on
conies inserted in his Paraliponiena^ published in 1604, he lays
down what has been called the principle of continuity, and
gives as an example the statement that a parabola is at once the
limiting case of an ellipse and of a hyperbola ; he illustrates the
same doctrine by reference to the foci of conies (the word focus
was introduced by him) ; and he also explains that parallel lines
should be regarded as meeting at infinity. He introduced the
use of the eccentric angle in discussing properties of the ellipse.
In his Stereometriay which was published in 1615, he deter-
mines the volumes of certain vessels and the areas of certain
surfaces, by means of infinitesimals instead of by the long and
tedious method of exhaustions. These investigations as well
as those of 1604 arose from a dispute with a wine merchant as
to the proper way of gauging the contents of a cask. This
use of infinitesimals was objected to by Guldinus and other
writers as inaccurate, but though the methods of Kepler are
not altogether free from objection he was substantially correct,
and by applying the law of continuity to infinitesimals he pre-
pared the way for Cavalieri's method of indivisibles, and the
infinitesimal calculus of Newton and Leibnitz.
Kepler's work on astronomy lies outside the scope of this
book. I will mention only that it was founded on the observa-
tions of Tycho Brahe,^ whose assistant he was. His three laws
of planetary motion were the result of many and laborious
eff'orts to reduce the phenomena of the solar system to certain
simple rules. The first two were published in 1609, and stated
that the planets describe ellipses round the sun, the sun being
in a focus; and that the line joining the sun to any planet
sweeps over equal areas in equal times. The third was pub-
lished in 1619, and stated that the squares of the periodic times
of the planets are proportional to the cubes of the major axes of
their orbits. The laws were deduced from observations on the
1 For an account of Tycho Brahe, born at Knudstrup in 1546 and died at
Prague in 1601, see his life by J. L. E. Dreyer, Edinburgh, 1890.
CH.xiii] KEPLER. DESARGUES 257
motions of Mars and the earth, and were extended by analogy
to the other planets. I ought to add that he attempted to
explain why these motions took place by a hypothesis which is
not very different from Descartes's theory of vortices. He sug-
gested that the tides were caused by the attraction of the moon.
Kepler also devoted considerable time to the elucidation of the
theories of vision and refraction in optics.
While the conceptions of the geometry of the Greeks were
being extended by Kepler, a Frenchman, w^hose works until
recently were almost unknown, was inventing a new method of
investigating the subject — a method which is now known as
projective geometry. This was the discovery of Desargues,
whom I put (with some hesitation) at the close of this period,
and not among the mathematicians of modern times.
Desargues.^ Gerard Desargues, born at Lyons in 1593, and
died in 1662, was by profession an engineer and architect, but
he gave some courses of gratuitous lectures in Paris from 1626
to about 1630 which made a great impression upon his contem-
poraries. Both Descartes and Pascal had a high opinion of his
work and abilities, and both made considerable use of the
theorems he had enunciated.
In 1636 Desargues issued a work on perspective; but most
of his researches were embodied in his Brouillon proiect on
conies, published in 1639, a copy of which was discovered
by Chasles in 1845. I take the following summary of it from
C. Taylor's work on conies. Desargues commences with a
statement of the doctrine of continuity as laid down by
Kepler : thus the points at the opposite ends of a straight
line are regarded as coincident, parallel lines are treated as
meeting at a point at infinity, and parallel planes on a line at
infinity, while a straight line may be considered as a circle whose
centre is at infinity. The theory of involution of six points,
with its special cases, is laid down, and the projective property
of pencils in involution is established. The theory of polar lines
^ See CEuvres de Desargues, by M. Poudra, 2 vols., Paris, 1864 ; and a note
in the Bibliotheco, Matheviatica, 1885, p. 90.
258 THE CLOSE OF THE RENAISSANCE [ch. xiii
is expounded, and its analogue in space suggested. A tangent
is defined as the limiting case of a secant, and an asymptote as
a tangent at infinity. Desargues shows that the lines which join
four points in a plane determine three pairs of lines in involu-
tion on any transversal, and from any conic through the four
points another pair of lines can be obtained which are in
involution with any two of the former. He proves that the
points of intersection of the diagonals and the two pairs of
opposite sides of any quadrilateral inscribed in a conic are a
conjugate triad with respect to the conic, and when one of the
three points is at infinity its polar is a diameter ; but he fails to
explain the case in which the quadrilateral is a parallelogram,
although he had formed the conception of a straight line which
was wholly at infinity. The book, therefore, may be fairly said
to contain the fundamental theorems on involution, homology,
poles and polars, and perspective.
The influence exerted by the lectures of Desargues on
Descartes, Pascal, and the French geometricians of the
seventeenth century was considerable ; but the subject of
projective geometry soon fell into oblivion, chiefly because the
analytical geometry of Descartes was so much more powerful as
a method of proof or discovery.
The researches of Kepler and Desargues will serve to remind
us that as the geometry of the Greeks, was _not. capable of
much further extension, mathematicians were now beginning
to seek for new methods of investigation, and were extending
the conceptions of geometry. The invention of analytical
geometry and of\the infinitesimal calculus temporarily diverted
attention from pUre geometry, but at the beginning of the
last century there was a revival of interest in it, and since
then it has been a N^vourite subject of study with many
mathematicians. \
Mathematical knowledge at the close of the reinaissance.
Thus by the beginning of the seventeenth century we may
say that the fundamental principles of arithmetic, algebra,
-. )t ^^
CH. xiii] THE CLOSE OF THE RENAISSANCE 259
theory of equations, and trigonometry had been laid down, and
the outlines of the subjects as we know them had been traced.
It must be, however, remembered that there were no good
elementary text -books on these subjects ; and a knowledge of
them was therefore confined to those who could extract it from
the ponderous treatises in which it lay buried. Though much of
the modern algebraical and trigonometrical notation had been
introduced, it was not familiar to mathematicians, nor was it
even universally accepted ; and it was not until the end of the
seventeenth century that the language of these subjects was
definitely fixed. Considering the absence of good text -books,
I am inclined rather to admire the rapidity with which it came
into universal use, than to cavil at the hesitation to trust to it
alone which many writers showed.
If we turn to applied mathematics, we find, on the other
hand, that the science of statics had made but little advance
in the eighteen centuries that had elapsed since the time of
Archimedes, while the foundations of dynamics were laid by
Galileo only at the close of the sixteenth century. In fact, as
we shall see later, it was not until the time of Newton that the
science of mechanics was placed on a satisfactory basis. The
fundamental conceptions of mechanics are difficult, but the
ignorance of the principles of the subject shown by the
mathematicians of this time is greater than would have been
anticipated from their knowledge of pure mathematics.
With this exception, we may say that the principles of
analytical geometry and of the infinitesimal calculus were needed
before there was likely to be much further progress. The
former was employed by Descartes in 1637, the latter was
invented by Newton some thirty or forty years later, and
their introduction may be taken as marking the commencement
of the period of modern mathematics.
261
THIRD PERIOD.
JEotrern ^athtmatitz.
The history of modem mathematics begins with the inventimi
of analytical geometry and the infinitesimal calculus. The
mathemxitics is far more complex than that ^produced in either of
the preceding periods ; hut^ during the seventeenth and eighteenth
centuries, it may he generally described as cliaracterized by the
development of analysis, and its application to the phenomena
of nature.
I continue the chronological arrangement of the subject.
Chapter xv contains the history of the forty years from 1635
to 1675, and an account of the mathematical discoveries of
Descartes, Cavalieri, Pascal, Wallis, Fermat, and Huygens.
Chapter xvi is given up to a discussion of Newton's researches.
Chapter xvii contains an account of the works of Leibnitz and
his followers during the first half of the eighteenth century
(including D'Alembert), and of the contemporary English school
to the death of Maclaurin. The works of Euler, Lagrange,
Laplace, and their contemporaries form the subject-matter of
chapter xviii.
Lastly, in chapter xix I have added some notes on a few of
the mathematicians of recent times ; but I exclude all detailed
reference to living writers, and partly because of this, partly
for other reasons there given, the account of contemporary
mathematics does not profess to cover the subject.
263
CHAPTER XIV.
THE HISTORY OF MODERN MATHEMATICS.
The division between this period and that treated in the
last six chapters is by no means so well defined as that which
separates the history of Greek mathematics from the mathe-
matics of the middle ages. The methods of analysis used in
the seventeenth century and the kind of problems attacked
changed but gradually ; and the mathematicians at the begin-
ning of this period were in immediate relations with those at
the end of that last considered. For this reason some writers
have divided the history of mathematics into two parts only,
treating the schoolmen as the lineal successors of the Greek
mathematicians, and dating the creation of modern mathe-
matics from the introduction of the Arab text -books into
Europe. The division I have given is, I think, more con-
venient, for the introduction of analytical geometry and of the
infinitesimal calculus revolutionized the development of the sub-
ject, and therefore it seems preferable to take their invention as
marking the commencement of modern mathematics.
The time that has elapsed since these methods were in-
vented has been a period of incessant intellectual activity in
all departments of knowledge, and the progress made in mathe-
matics has been immense. The greatly extended range of
knowledge, the mass of materials to be mastered, the absence
264 HISTORY OF MODERN MATHEMATICS [ch. xiv
of perspective, and even the echoes of old controversies, com-
bine to increase the difficulties of an author. As, however, the
leading facts are generally known, and the works published
during this time are accessible to any student, I may deal more
concisely with the lives and writings of modern mathematicians
than with those of their predecessors, and confine myself more
strictly than before to those who have materially affected the
progress of the subject.
To give a sense of unity to a history of mathematics it is
necessary to treat it chronologically, but it is possible to do
this in two ways. We may discuss sej^arately the development
of different branches of mathematics during a certain period
(not too long), and deal with the works of each mathematician
under such heads as they may fall. Or we may describe in
succession the lives and writings of the mathematicians of a
certain period, and deal with the develoj)ment of different sub-
jects under the heads of those who studied them. Personally,
I prefer the latter course ; and not the least advantage of this,
from my point of view, is that it adds a human interest to the
narrative. No doubt as the subject becomes more complex
this course becomes more difficult, and it may be that when the
history of mathematics in the nineteenth century is written it
will be necessary to deal separately with the separate branches
of the subject, but, as far as I can, I continue to present the
history biographically.
Roughly speaking, we may say that five distinct stages in
the history of modern mathematics can be discerned.
First of all, there is the invention of analytical geometry by
Descartes in 1637; and almost at the same time the intro-
duction of the method of indivisibles, by the use of which
areas, volumes, and the positions of centres of mass can be
determined by summation in a manner analogous to that effected
nowadays by the aid of the Integral calculus. The method of
indivisibles was soon superseded by the integral calculus. Ana-
lytical geometry, however, maintains its position as part of the
necessary training, of every mathematician, and for all purposes
CH. xiv] HISTORY OF MODERN MATHEMATICS 265
of research is incomparably more potent than the geometry of
the ancients. The latter is still, no doubt, an admirable intel-
lectual training, and it frequently affords an elegant demonstra-
tion of some proposition the truth of which is already known,
but it requires a special procedure for every particular problem
attacked. The former, on the other hand, lays down a few simple
rules by which any property can be at once proved or disproved.
In the second place, we have the invention, some thirty
years later, of the fluxional or differential calculus. Wherever
a quantity changes according to some continuous law — and most
things in nature do so change — the differential calculus enables
us to measure its rate of increase or decrease ; and, from its rate
of increase or decrease, the integral calculus enables us to find
the original quantity. Formerly every separate function of x
such as (1-1-^;)^, log (1+a;), sin x^ tan"^^^, &c., could be ex-
panded in ascending powers of x only by means of such special
procedure as was suitable for that particular problem ; but, by
the aid of the calculus, the expansion of any function of x in
ascending powers of x is in general reducible to one rule which
covers all cases alike. So, again, the .theory of maxima and
minima, the determination of the lengths of curves and the
areas enclosed by them, the determination of surfaces, of volumes,
and of centres of mass, and many other problems, are each re-
ducible to a single rule. The theories of differential equations,
of the calculus of variations, of finite differences, &c., are the
developments of the ideas of the calculus.
These two subjects — analytical geometry and the calculus —
became the chief instruments of further progress in mathematics.
In both of them a sort of machine was constructed : to solve a
problem, it was only necessary to put in the particular function
dealt with, or the equation of the particular curve or surface
considered, and on performing certain simple operations the
result came out. The validity of the process was proved once
for all, and it was no longer requisite to invent some special
method for every separate function, curve, or surface.
In the third place, Huygens, following Galileo, laid the
266 HISTORY OF MODERN MATHEMATICS [ch. xiv
foundation of a satisfactory treatment of dynamics, and Newton
reduced it to an exact science. The latter mathematician pro-
ceeded to apply the new analytical methods not only to numerous
problems in the mechanics of solids and fluids on the earth,
but to the solar system ; the whole of mechanics terrestrial and
celestial was thus brought within the ___domain of mathematics.
There is no doubt that Newton used the calculus to obtain many
of his results, but he seems to have thought that, if his demon-
strations were established by the aid of a new science which was
at that time generally unknown, his critics (who would not
understand the fluxional calculus) would fail to realise the truth
and importance of his discoveries. He therefore determined to
give geometrical proofs of all his results. He accordingly cast
the Principia into a geometrical form, and thus presented it to
the world in a language which all men could then understand.
The theory of mechanics was extended, systematized, and put
in its modern form by Lagrange and Laplace towards the end
of the eighteenth century.
In the fourth place, we may say that during this period
the chief branches of physics have been brought within the
scope of mathematics. This extension of the ___domain of mathe-
matics was commenced by Huygens and 'Newton when they
propounded their theories of lights but it was not until the
beginning of the last century that sufficiently accurate observa-
tions were made in most physical subjects to enable mathematical
reasoning to be applied to them.
Numerous and far-reaching conclusions have been obtained
in physics by the application of mathematics to the results of
observations and experiments, but we now want some more
simple hypotheses from which we can deduce those laws which
at present form our starting-point. If, to take one example,
we could say in what electricity consisted, we might get some
simple laws or hypotheses from which by the aid of mathe-
matics all the observed phenomena could be deduced, in the
same way as Newton deduced all the results of physical astro-
nomy from the law of gravitation. All lines of research seem,
CH. xiv] HISTORY OF MODERN MATHEMATICS 267
moreover, to indicate that there is an intimate connection be-
tween the different branches of physics, e.g. between light, heat,
elasticity, electricity, and magnetism. The ultimate explan^ation
of this and of the leading facts in physics seems to demand a
study of molecular physics ', a knowledge of molecular physics
in its turn seems to require some theory as to the constitution
of matter ; it would further appear that the key to the constitu-
tion of matter is to be found in electricity or chemical physics.
So the matter stands at present ; the connection between the
different branches of physics, and the fundamental laws of those
branches (if there be any simple ones), are riddles which are yet /
unsolved. This history does not pretend to treat of problems
which are now the subject of investigation ; the fact also that
mathematical physics is mainly the creation of the nineteenth
century would exclude all detailed discussion of the subject.
Fifthly, this period has seen an immense extension of pure
mathematics. Much of this is the creation of comparatively
recent times, and I regard the details of it as outside the limits
of this book, though in chapter xix I have allowed myself to
mention some of the subjects discussed. The most striking
features of this extension are the critical discussion of
fundamental principles, the developments of higher geometry,
of higher arithmetic or the theory of numbers, of higher
algebra (including the theory of forms), and of the theory
of equations, also the discussion of functions of double and
multiple periodicity, and the creation of a theory of functions.
This hasty summary will indicate the subjects treated and
the limitations I have imposed on myself. The history of the
origin and growth of analysis and its application to the
material universe comes within my purview. The extensions
in the latter half of the nineteenth century of pure mathe-
matics and of the application of mathematics to physical
problems open a new period which lies beyond the limits of
this book; and I allude to these subjects only so far as they
may indicate the directions in which the future history of
mathematics appears to be developing.
a
268
CHAPTER XV.
HISTORY OF MATHEMATICS FROM DESCARTES TO HUYGENS.l
CIRC. 1635-1675.
I PROPOSE in this chapter to consider the history of mathematics
during the forty years in the middle of the seventeenth century.
I regard Descartes, Cavalieri, Pascal, Wallis, Fermat, and
Huygens as the leading mathematicians of this time. I shall
treat them in that order, and I shall conclude with a brief list of
the more eminent remaining mathematicians of the same date.
I have already stated that the mathematicians of this period
— and the remark applies more particularly to Descartes, Pascal,
and Fermat — were largely influenced by the teaching of Kepler
and Desargues, and I would repeat again that I regard these
latter and Galileo as forming a connecting link between the
writers of the renaissance and those of modern times. I should
also add that the mathematicians considered in this chapter were
contemporaries, and, although I have tried to place them roughly
in such an order that their chief works shall come in a chrono-
logical arrangement, it is essential to remember that they were
in relation one with the other, and in general were acquainted
with one another's researches as soon as these were published.
Descartes. 2 Subject to the above remarks, we may consider
^ See Cantor, part xv, vol. ii, pp. 599-844 ; other authorities for the
mathematicians of this period are mentioned in the footnotes.
'■^ See Descartes, by E. S. Haldane, London, 1905. A complete edition of
his works, edited by C. Adam and P. Tanner, is in process of issue by the
CH. xv] DESCARTES 269
Descartes as the first of the modern school of mathematics.
Rene Descartes was born near Tours on March 31, 1596, and
died at Stockholm on February 11, 1650; thus he was a con-
temporary of Galileo and Desargues. His father, who, as the
name implies, was of a good family, was accustomed to spend
half the year at Rennes when the local parliament, in which he
held a commission as councillor, was in session] and the rest of
the time on his family estate of Les Cartes at^La Haye. Rene,
the second of a family of two sons and one daughter, was sent
at the age of eight years to the Jesuit School at La Fleche, and
of the admirable discipline and education there given he speaks
most highly. On account of his delicate health he was per-
mitted to lie in bed till late in the mornings ; this was a custom
which he always followed, and when he visited Pascal in 1647
he told him that the only way to do good work in mathematics
and to preserve his health was never to allow any one to make
him get up in the morning before he felt inclined to do so ; an
opinion which I chronicle for the benefit of any schoolboy into
whose hands this work may fall.
On leaving school in 1612 Descartes went to Paris to be
introduced to the world of fashion. Here, through the medium
of the Jesuits, he made the acquaintance of Mydorge, and
renewed his schoolboy friendship with Mersenne, and together
with them he devoted the two years of 1615 and 1616 to the
study of mathematics. At that time a man of position usually
entered either the army or the church; Descartes chose the
former profession, and in 1617 joined the army of Prince
Maurice of Orange, then at Breda. Walking through the streets
there he saw a placard in Dutch which excited his curiosity,
and stopping the first passer, asked him to translate it into
either French or Latin. The stranger, who happened to be
Isaac Beeckman, the head of the Dutch College at Dort, offered
French Government ; vols, i-ix, 1897-1904. A tolei'ably complete account of
Descartes's mathematical and physical investigations is given in Ersch and
Griiber's Encyclopadie. The most complete edition of his works is that by
Victor Cousin in 11 vols., Paris, 1824-26. Some minor papers subsequently
discovered were printed by F. de Careil, Paris, 1859.
270 HISTORY OF MATHEMATICS [ch. xv
to do so if Descartes would answer it ; the placard being, in fact,
a challenge to all the world to solve a certain geometrical
problem.^ Descartes worked it out within a few hours, and a
warm friendship between him and Beeckman was the result. This
unexpected test of his mathematical attainments made the un-
congenial life of the army distasteful to him, and though, under
family influence and tradition, he remained a soldier, he con-
tinued to occupy his leisure with mathematical studies. He was
accustomed to date the first ideas of his new philosophy and of
his analytical geometry from three dreams which he experienced
on the night of November 10, 1619, at Neuberg, when campaign-
ing on the Danube, and he regarded this as the critical day of
his life, and one which determined his whole future.
He resigned his commission in the spring of 1621, and
spent the next five years in travel, during most of which time
he continued to study pure mathematics. In 1626 we find
him settled at Paris, " a little well-built figure, modestly clad in
green taffety, and only wearing sword and feather in token of
his quality as a gentleman." During the first two years there
he interested himself in general society, and spent his leisure in
the construction of optical instruments ; but these pursuits were
merely the relaxations of one who failed to find in philosophy
that theory of the universe which he was convinced finally
awaited him.
In 1628 Cardinal de Berulle, the founder of the Oratorians,
met Descartes, and was so much impressed by his conversation
that he urged on him the duty of devoting his life to the
examination of truth. Descartes agreed, and the better to
secure himself from interruption moved to Holland, then at the
height of its power. There for twenty years he lived, giving up
all his time to philosophy and mathematics. Science, he says,
may be compared to a tree ; metaphysics is the root, physics is
the trunk, and the three chief branches are mechanics, medicine,
^ Some doubt has been recently expressed as to whether the story is
well founded : see L Intermedixtire des Mathemaiiciens, Paris, 1909, vol. xvi,
pp. 12-13.
CH. xv] DESCARTES 271
and morals, these forming the three applications of our know-
ledge, namely, to the external world, to the human body, and
to the conduct of life.
He spent the first four years, 1629 to 1633, of his stay in
Holland in writing Le Monde^ which embodies an attempt to
give a physical theory of the universe ; but finding that its
publication was likely to bring on him the hostility of the
church, and having no desire to pose as a martyr, he abandoned
it : the incomplete manuscript was published in 1664. He
then devoted himself to composing a treatise on universal
science; this was published at Leyden in 1637 under the title
Discours de la methode pour Men conduire sa raison et chercher
la verite dans les sciences, and was accompanied with three
appendices (which possibly were not issued till 1638) entitled
La Dioptrique, Les Meteores, and La Geometrie ; it is from the
last of these that the invention of analytical geometry dates.
In 1641 he published a work called Meditationes, in which he
explained at some length his views of philosophy as sketched
out in the Discours. In 1644 he issued the Principia
Philosophiae, the greater part of which was devoted to physical
science, especially the laws of motion and the theory of vortices.
In 1647 he received a pension from the French court in honour
of his discoveries. He went to Sweden on the invitation of the
Queen in 1649, and died a few months later of inflammation of
the lungs.
/" In appearance, Descartes was a small man with large head,
{ projecting brow, prominent nose, and black hair coming down
I to his eyebrows. His voice was feeble. In disposition he was
I cold and selfish. Considering the range of his studies he was
by no means widely read, and he despised both learning and
art unless something tangible could be extracted therefrom.
He never married, and left no descendants, though he had one
illegitimate daughter, who died young.
As to his philosophical theories, it will be sufficient to say
that he discussed the same problems which have been debated
for the last two thousand years, and probably will be debated
272 HISTORY OF MATHEMATICS [ch. xv
with equal zeal two thousand years hence. It is hardly neces-
sary to say that the problems themselves are of importance
and interest, but from the nature of the case no solution ever
offered is capable either of rigid proof or of disproof; all
that can be effected is to make one explanation more probable
than another, and whenever a philosopher like Descartes
believes that he has at last finally settled a question it has
been possible for his successors to point out the fallacy in
his assumptions. I have read somewhere that philosophy has
always been chiefly engaged with the inter-relations of God,
Nature, and Man. The earliest philosophers were Greeks
who occupied themselves mainly with the relations between
God and Nature, and dealt with Man , separately. The
Christian Church was so absorbed in the relation of God to
Man as entirely to neglect Nature. Finally, modern philos-
ophers concern themselves chiefly with the relations between
Man and Nature. Whether this is a correct historical
generalization of the views which have been successively
prevalent I do not care to discuss here, but the statement as
to the scope of modern philosophy marks the limitations of
Descartes's writings.
Descartes's chief contributions to mathematics were his
analytical geometry and his theory of vortices, and it is on his
researches in connection with the former of these subjects that
his mathematical reputation rests.
Analytical geometry does not consist merely (as is sometimes
loosely said) in the application of algebra to geometry ; that had
been done by Archimedes and many others, and had become the
usual method of procedure in the works of the mathematicians
of the sixteenth century. The great advance made by Descartes
was that he saw that a point in a plane could be completely
determined if its distances, say x and y, from two fixed lines
drawn at right angles in the plane were given, with the convention
familiar to us as to the interpretation of positive and negative
values ; and that though an equation /(^, 3/) = 0 was indeter-
minate and could be satisfied by an infinite number of values of
CH. xv] DESCARTES 273
X and y, yet these values of x and y determined the co-ordinates
of a number of points which form a curve, of which the equation
f(^x, y) = 0 expresses some geometrical property, that is, a
property true of the curve at every point on it. Descartes
asserted that a point in space could be similarly determined by
three co-ordinates, but he confined his attention to plane
curves.
It was at once seen that in order to investigate the properties
of a curve it was sufficient to select, as a definition, any
characteristic geometrical property, and to express it by means
of an equation between the (current) co-ordinates of any point
on the curve, that is, to translate the definition into the
language of analytical geometry. The equation so obtained
contains implicitly every property of the curve, and any
particular property can be deduced from it by ordinary algebra
without troubling about the geometry of the figure. This
may have been dimly recognized or foreshadowed by earlier
writers, but Descartes went further and pointed out the very
important facts that two or more curves can be referred to one
and Ihe same system of co-ordinates, and that the points in
which two curves intersect can be determined by finding TRp.
roots common to their two equations. I need not go further
into details, for nearly everyone to whom the above is intelligible
will have read analytical geometry, and is able to appreciate the
value of its invention.
Descartes's Geometrie is divided into three books : the first
two of these treat of analytical geometry, and the third includes
an analysis of the algebra then current. It is somewhat difficult
to follow the reasoning, but the obscurity was intentional.
" Je n'ai rien omis," says he, " qu'a dessein . . . j'avois prevu
que certaines gens qui se van tent de s^avoir tout n'auroient
pas manque de dire que je n'avois rien ecrit qu'ils n'eussent
SQU auparavant, si je me fusse rendu assez intelligible pour
eux."
The first book commences with an explanation of the
principles of analytical geometry, and contains a discussion
274 HISTORY OF MATHEMATICS [ch.xv
of a certain problem which had been propounded by Pappus in
the seventh book of his Iwayoiyrj and of which some particular
cases had been considered by Euclid and Apollonius. The
general theorem had baffled previous geometricians, and it
was in the attempt to solve it that Descartes was led to the
invention of analytical geometry. The full enunciation of the
problem is rather involved, but the most important case is to
find the locus of a point such that the product of the
perpendiculars on m given straight lines shall be in a constant
ratio to the product of the perpendiculars on n other given
straight lines. The ancients had solved this geometrically
for the case m=l, n = \, and the case m = l, n = 2. Pappus
had further stated that, if m = w = 2, the locus is a conic,
but he gave no proof; Descartes also failed to prove this by
pure geometry, but he shewed that the curve is represented
by an equation of the second degree, that is, is a conic ;
subsequently Newton gave an elegant solution of the problem
by pure geometry.
In the second book Descartes divides curves into two
classes, namely, geometrical and mechanical curves. He
defines geometrical curves as those which can be generated
by the intersection of two lines each moving parallel to one
co-ordinate axis with " commensurable " velocities ; by which
terms he means that dyjdx is an algebraical function, as, for
example, is the case in the ellipse and the cissoid. He calls a
curve mechanical when the ratio of the velocities of these lines
is " incommensurable " ; by which term he means that dyjdx is
a transcendental function, as, for example, is the case in the
cycloid and the quadratrix. Descartes confined his discussion
to geometrical curves, and did not treat of the theory of
mechanical curves. The classification into algebraical and trans-
cendental curves now usual is due to Newton.^
Descartes also paid particular attention to the theory of the
tangents to curves — as perhaps might be inferred from his
system of classification just alluded to. The then current
^ See below, page 340.
CH. xv] DESCARTES 275
definition of a tangent at a point was a straight line through
the point such that between it and the curve no other straight
line could be drawn, that is, the straight line of closest contact.
Descartes proposed to substitute for this a statement equivalent
to the assertion that the tangent is the limiting position of the
secant; Fermat, and at a later date Maclaurin and Lagrange,
adopted this definition. Barrow, followed by Newton and
Leibnitz, considered a curve as the limit of an inscribed
polygon when the sides become indefinitely small, and stated
that a side of the polygon when produced became in the limit a
tangent to the curve. Roberval, on the other hand, defined a
tangent at a point as the direction of motion at that instant of a
point which was describing the curve. The results are the same
whichever definition is selected, but the controversy as to which
definition was the correct one was none the less lively. In his
letters Descartes illustrated his theory by giving the general
rule for drawing tangents and normals to a roulette.
The method used by Descartes to find the tangent or normal
at any point of a given curve was substantially as follows. He
determined the centre and radius of a circle which should cut
the curve in two consecutive points there. The tangent to the
circle at that point will be the required tangent to the curve.
In modern text-books it is usual to express the condition that
two of the points in which a straight line (such as y = mx + c) cuts
the curve shall coincide with the given point : this enables us to
determine m and c, and thus the equation of the tangent there
is determined. Descartes, however, did not venture to do this,
but selecting a circle as the simplest curve and one to which he
knew how to draw a tangent, he sa fixed his circle as to make it
touch the given curve at the point in question, and thus reduced
the problem to drawing a tangent to a circle. I should note
in passing that he only applied this method to curves which are
symmetrical about an axis, and he took the centre of the circle
on the axis.
The obscure style deliberately adopted by Descartes
diminished the circulation and immediate appreciation of these
276 HISTORY OF MATHEMATICS [ch. xv
books ; but a Latin translation of them, with explanatory-
notes, was prepared by F. de Beaune, and an edition of this,
with a commentary by F. van Schooten, issued in 1659, was
widely read.
The third book of the Geometrie contains an analysis of the
algebra then current, and it has affected the language of the
subject by fixing the custom of employing the letters at the
beginning of the alphabet to denote known quantities, and those
I at the end of the alphabet to denote unknown quantities.^
(^Descartes further introduced the system of indices now in use ;
very likely it was original on his part, but I would here remind
the reader that the suggestion had been made by previous
writers, though it had not been generally adopted. It is
doubtful whether or not Descartes recognised that his letters
might represent any quantities, positive or negative, and that it
was sufficient to prove a proposition for one general case. He
was the earliest writer to realize the advantage to be obtained
by taking all the terms of an equation to one side of it, though
Stifel and Harriot had sometimes employed that form by choice.
He realised the meaning of negative quantities and used them
freely. In this book he made use of the rule for finding a limit
tq_the number of positiveand_of negativejioots-of-an algebraical
equation^ which is still known by his name ; and introduced the
metiiod""ori!RfetCTniinate coelEcients tor the solution oFeq nations.
^Tenieb'eved that he had given a method by which algebraical
equations of any order could be solved, but in3his~he was
mistaken. It may bealso mentioned that Ee~enunciated the
theorem, commonly attributed to Euler, on the relation between
the numbers of faces, edges, and angles of a polyhedron : this is
in one of the papers published by Careil.
Of the two other appendices to the Discours one was devoted
to optics. The chief interest of this consists in the statement
given of the law of refraction. This appears to have been taken
^ On the origin of the custom of using x to represent an unknown
example, see a note by G. Euestrom in the BihUotheca Mathematica, 1885,
p. 43.
CH.xv] DESCARTES 277
from Snell's work, though, unfortunately, it is enunciated in a
way which might lead a reader to suppose that it is due to the
researches of Descartes. Descartes would seem to have repeated
Snell's experiments when in Paris in 1626 or 1627, and it is
possible that he subsequently forgot how much he owed to the
earlier investigations of Snell. A large part of the optics is
devoted to determining the best shape for the lenses of a
telescope, but the mechanical difficulties in grinding a surface of
glass to a required form are so great as to render these investi-
gations of little practical use. Descartes seems to have been
doubtful whether to regard the rays of light as proceeding from
the eye and so to speak touching the object, as the Greeks
had done, or as proceeding from the object, and so affecting the
eye ; but, since he considered the velocity of light to be infinite,
he did not deem the point particularly important.
The other appendix, on meteo7^s, contains an explanation of
numerous atmospheric phenomena, including the rainbow ; the
explanation of the latter is necessarily incomplete, since
Descartes was unacquainted with the fact that the refractive
index of a substance is different for lights of different colours.
Descartes's physical theory of the universe, embodying mo^t
of the results contained in his earlier and unpublished Le Monde,
is given in his Principia, 1644, and rests on a metaphysical
basis. He commences with a discussion on motion ; and then
lays down ten laws of nature, of which the first two are almost
identical with the first two laws of motion as given by Newton ;
the remaining eight laws are inaccurate. He next proceeds to
discuss the nature of matter which he regards as uniform in
kind though there are three forms of it. He assumes that the
matter of the universe must be in motion, and that the motion
must result in a number of vortices. He states that the sun is
the centre of an immense whirlpool of this matter, in which the
planets float and are swept round like straws in a whirlpool of
water. Each planet is supposed to be the centre of a secondary
whirlpool by which its satellites are carried : these secondary
whirlpools are supposed to produce variations of density in the
278 HISTORY OF MATHEMATICS [ch. xv
surrounding medium which constitute the primary whirlpool,
and so cause the planets to move in ellipses and not in circles.
All these assumptions are arbitrary and unsupported by any
investigation. It is not difficult to prove that on his hypothesis
the sun would be in the centre of these ellipses, and not at a
focus (as Kepler had shewn was the case), and that the weight
of a body at every place on the surface of the earth except the
equator would act in a direction which was not vertical ; but it
will be sufficient here to say that Newton in the second book of
his Principia^ 1687, considered the theory in detail, and shew^ed
that its consequences are not only inconsistent with each of
Kepler's laws and with the fundamental laws of mechanics, but
are also at variance with the laws of nature assumed by Descartes.
Still, in spite of its crudeness and its inherent defects, the
theory of vortices marks a fresh era in astronomy, for it was
an attempt to explain the phenomena of the whole universe by
the same mechanical laws which experiment shews to be true
on the earth.
Cavalieri.^ Almost contemporaneously with the publication
in 1637 of Descartes's geometry, the principles of the integral
calculus, so far as they are concerned with summation, were
being worked out in Italy. This was effected by what was
( called the principle of indivisibles, and was the invention of
Cavalieri. It was applied by him and his contemporaries to
numerous problems connected with the quadrature of curves and
surfaces, the determination of volumes, and the positions of
centres of mass. It served the same purpose as the tedious
method of exhaustions used by the Greeks; in principle the
methods are the same, but the notation of indivisibles is more
concise and convenient. It was, in its turn, superseded at the
beginning of the eighteenth century by the integral calculus.
BoTiaventura Cavalieri was born at Milan in 1598, and died
^ Cavalieri's life has been vn:itteii by P. Frisi, Milan, 1778 ; by F.
Predari, Milan, 1843 ; by Gabrio Piola, Milan, 1844 ; and by A. Favaro,
Bologna, 1888. An analysis of his works is given in Marie's Histoire des
Sciences, Paris, 1885-8, vol. iv, pp. 69-90.
CH. xv] CAVALIERI 279
at Bologna on November 27, 1647. He became a Jesuit at an
early age; on the recommendation of the Order he was in 1629
made professor of mathematics at Bologna ; and he continued
to occupy the chair there until his death. I have already
mentioned Cavalieri's name in connection with the introduc-
tion of the use of logarithms into Italy, and have alluded to
his discovery of the expression for the area of a spherical
triangle in terms of the spherical excess. He was one of the
most influential mathematicians of his time, but his subsequent
reputation rests mainly on his invention of the pxjnciple of
indivisibles.
The principle of indivisibles had been used by Kepler in
1604 and 1615 in a somewhat crude form. It was first stated
by Cavalieri in 1629, but he did not publish his results till
1635. In his early enunciation of the principle in 1635
Cavalieri asserted that a line was made up of an infinite
number of points (each without magnitude), a surface of an
infinite number of lines (each without breadth), and a volume
of an infinite number of surfaces (each without thickness). To
meet the objections of Guldinus and others, the statement
was recast, and in its final form as used by the mathematicians
of the seventeenth century it was published in Cavalieri's
Exercitationes Geonietricae in 1647; the third exercise is
devoted to a defence of the theory. This book contains the
earliest demonstration of the properties of Pappus.^ Cavalieri's
works on indivisibles were reissued with his later corrections in
1653.
The method of indivisibles rests, in effect, on the assumption
that any magnitude may be divided into an infinite number of
small quantities which can be made to bear any required ratios
{ex. gr. equality) one to the other. The analysis given by
Cavalieri is hardly worth quoting except as being one of the
first steps taken towards the formation of an infinitesimal \
calculus. One example mil suffice. Suppose it be required to
find the area of a right-angled triangle. Let the base be made
1 See above, pp. 101, 252.
/
280 HISTORY OF MATHEMATICS [ch. xv
up of, or contain n points (or indivisibles), and similarly let the
other side contain na points, then the ordinates at the successive
points of the base will contain a, 2a ... , na points. Therefore
the number of points in the area is a + 2a+ ... +7ia ; the sum
of which is ^n^a + \na. Since n is very large, we may neglect
^na, for it is inconsiderable compared with \7i^a. Hence the
area is equal to ^{na)nj that is, J x altitude x base. There is
no difficulty in criticizing such a proof, but, although the form
in which it is presented is indefensible, the substance of it is
correct.
It would be misleading to give the above as the only
specimen of the method of indivisibles, and I therefore quote
another example, taken from a later writer, which will fairly
illustrate the use of the method when modified and corrected by
the method of limits. Let it be required to find the area
outside a parabola APC and bounded by the curve, the tangent
at A, and a line DC parallel to AB the diameter at A. Com-
plete the parallelogram ABCD. Divide AD into ii equal parts,
let AM contain r of them, and let JfiVbe the (r+l)th part.
Draw MP and NQ parallel to AB, and draw PR parallel to AD.
Then when n becomes indefinitely large, the curvilinear area
APCD will be the limit of the sum of all parallelograms like PN.
Now
area PN : area BD = MP . MN : DC . AD,
cH.xv] CAVALIERI. PASCAL 281
But by the properties of the parabola
MP : DC = AM^ : AD^ = r^ : n\
and MN \AD = l \n.
Hence MF . MN -.DC . AD^r^ -.n^,
3
Therefore area FN : area BD = r^ : w
Therefore, ultimately,
area AFCD-.area, BD = r- + 2^ + ... +(n-lY :n^
= i?i (n-l) (27i-l) -.71^
which, in the limit, =1:3.
It is perhaps worth noticing that Cavalieri and his successors
always used the method to find the ratio of two areas, volumes,
or magnitudes of the same kind and dimensions, that is, they
never thought of an area as containing so many units of area.
The idea of comparing a magnitude with a unit of the same
kind seems to have been due to Wallis.
It is evident that in its direct form the method is ap-
plicable to only a few curves. Cavalieri proved that, if m be
a positive integer, then the limit, when n is infinite, of
(l*'* + 2''^+...+?i'«)/?i"^+Ms l/(m+l), which is equivalent to
saying that he found the integral to x of x''^ from x = 0 to
x=l ; he also discussed the quadrature of the hyperbola.
Pascal.^ Among the contemporaries of Descartes none
displayed greater natural genius than Pascal, but his mathe-
matical reputation rests more on what he might have done
than on what he actually effected, as during a considerable part
of his life he deemed it his duty to devote his whole time
to religious exercises.
Blaise Fascal was born at Clermont on June 19, 1623, and
died at Paris on Aug. 19, 1662. His father, a local judge at
^ See Pascal by J. Bertrand, Paris, 1891 ; and Pascal, sein Leben und
seine Kdmpfe, by J. G. Dreydorff, Leipzig, 1870. Pascal's life, written by
his sister Mme. Perier, was edited by A. P. Faugere, Paris, 184.5, and has
formed the basis for several works. An edition of his writings was published
in five volumes at the Hague in 1779, second edition, Paris, 1819 ; some
additional pamphlets and letters were published in three volumes at Paris
in 1858.
282 HISTORY OF MATHEMATICS [ch.xv
Clermont, and himself of some scientific reputation, moved to
Paris in 1631, partly to prosecute his own scientific studies,
partly to carry on the education of his only son, who had
already displayed exceptional ability. Pascal was kept at home
in order to ensure his not being overworked, and with the same
object it was directed that his education should be at first
confined to the study of languages, and should not include any
mathematics. This naturally excited the boy's curiosity, and
one day, being then twelve years old, he asked in what geometry
consisted. His tutor replied that it was the science of con-
structing exact figures and of determining the proportions
between their different parts. Pascal, stimulated no doubt by
the injunction against reading it, gave up his play-time to this
new study, and in a few weeks had discovered for himself many
properties of figures, and in particular the proposition that the
sum of the angles of a triangle is equal to two right angles. I
have read somewhere, but I cannot lay my hand on the authority,
that his proof merely consisted in turning the angular points of
a triangular piece of paper over so as to meet in the centre of
the inscribed circle : a similar demonstration can be got by
♦turning the angular points over so as to meet at the foot of the
perpendicular drawn from the biggest angle to the opposite side.
His father, struck by this display of ability, gave him a copy of
Euclid's Elements^ a book which Pascal read with avidity and
soon mastered.
At the age of fourteen he was admitted to the weekly
meetings of Roberval, Mersenne, Mydorge, and other French
geometricians ; from which, ultimately, the French Academy
sprung. At sixteen Pascal wrote an essay on conic sections;
and in 1641, at the age of eighteen, he constructed the first
arithmetical machine, an instrument which, eight years later, he
further improved. His correspondence with Fermat about this
time shews that he was then turning his attention to analytical
geometry and physics. He repeated Torricelli's experiments, by
which the pressure of the atmosphere could be estimated as a
weight, and he confirmed his theory of the cause of barometrical
CH. xv] PASCAL 283
variations by obtaining at the same instant readings at different
altitudes on the hill of Puy-de-D6me.
In 1650, when in the midst of these researches, Pascal
suddenly abandoned his favourite pursuits to study religion, or,
as he says in his Peiisees, " to contemplate the greatness and the
misery of man " ; and about the same time he persuaded the
younger of his two sisters to enter the Port Royal society.
In 1653 he had to administer his father's estate. He now
took up his old life again, and made several experiments on the
pressure exerted by gases and liquids ; it was also about this
period that he invented the arithmetical triangle, and together
with Fermat created the calculus of probabilities. He was
meditating marriage when an accident again turned the current
of his thoughts to a religious life. He was driving a four-in-
hand on November 23, 1654, when the horses ran away; the
two leaders dashed over the parapet of the. bridge at Neuilly,
and Pascal was saved only by the traces breaking. Always
somewhat of a mystic, he considered this a special summons to
abandon the world. He wrote an account of the accident on
a small piece of parchment, w^hich for the rest of his life he
wore next to his heart, to perpetually remind him of his
covenant ; and shortly moved to Port Royal, where he continued
to live until his death in 1662. Constitutionally delicate, he
had injured his health by his incessant study ; from the age of
seventeen or eighteen he suffered from insomnia and acute
dyspepsia, and at the time of his death was physically worn
out.
His famous Provincial Letters directed against the Jesuits,
and his Fensees, were written towards the close of his life, and
are the first example of that finished form which is characteristic
of the best French literature. The only mathematical work
that he produced after retiring to Port Royal was the essay on
the cycloid in 1658. He was suffering from sleeplessness and
toothache when the idea occurred to him, and to his surprise his
teeth immediately ceased to ache. Regarding this as a divine
intimation to proceed with the problem, he worked incessantly
284 HISTORY OF MATHEMATICS [ch. xv
for eight days at it, and completed a tolerably full account of
the geometry of the cycloid.
I now proceed to consider his mathematical works in rather
greater detail.
His early essay on the geometry of conies, written in 1639,
but not published till 1779, seems to have been founded on the
teaching of Desargues. Two of the results are important as
well as interesting. The first of these is the theorem known
now as "Pascal's theorem," namely, that if a hexagon be
inscribed in a conic, the points of intersection of the opposite
sides will lie in a straight line. The second, which is really due
to Desarg-ues, is that if a quadrilateral be inscribed in a conic,
and a straight line be drawn cutting the sides taken in order in
the points A, B, C, and D, and the conic in P and Q, then
PA . PC : PB.PD=^QA.QC'.QB. QD.
Pascal employed his arithmetical triangle in 1653, but no
account of his method was printed till 1665. The triangle is
constructed as in the figure below, each horizontal line
being formed from the one above it by making every number
in it equal to the sum of those above and to the left of it in the
row immediately above it; ex. gr. the fourth number in the
/
1
1
1
1/
/
2
3
V^
^b
..
3
6 /
^0
15
..
4/
^0
20
35
.
;y
^
15
35
70
.
fourth line, namely, 20, is equal to 1+3 + 6 + 10. The numbers
in each line are what are now called figurate numbers. Those
CH. XV] PASCAL 285
in* the first line are called numbers of the first order ; those in
the second line, natural numbers or numbers of the second
order ; those in the third line, numbers of the third order, and
so on. It is easily shewn that the with number in the nth. row
is(m + 7i-2)!/ {m-l)l {n-l)\
Pascal's arithmetical triangle, to any required order, is got
by drawing a diagonal downwards from right to left as in the
figure. The numbers in any diagonal give the coefiicients of the
expansion of a binomial; for example, the figures in the fifth
diagonal, namely, 1, 4, 6, 4, 1, are the coefficients in the
expansion (a -{-by. Pascal used the triangle partly for this
purpose, and partly to find the numbers of combinations of m
things taken n a.t a, time, which he stated, correctly, to be
(n+1) {n+'2) (n + S) ...ml(m- n) !
Perhaps as a mathematician Pascal is best known in connec-
tion with his correspondence with Fermat in 1654, in which he
laid down the principles of the theory of probabilities. This
correspondence arose from a problem proposed by a gamester,
the Chevalier de Mere, to Pascal, who communicated it to
Fermat. The problem was this. Two players of equal skill
want to leave the table before finishing their game. Their
scores and the number of points which constitute the game
being given, it is desired to find in what proportion they should
divide the stakes. Pascal and Fermat agreed on the answer,
but gave different proofs. The following is a translation of
Pascal's solution. That of Fermat is given later.
The following is my method for determining the share of each player
when, for example, two players play a game of three points and each
player has staked 32 pistoles.
Suppose that the first player has gained two points, and the second
player one point ; they have now to play for a point on this condition,
that, if the first player gain, he takes all the money Avhich is at stake,
namely, 64 pistoles ; while, if the second player gain, each player has two
points, so that they are on terms of equality, and, if they leave off play-
ing, each ought to take 32 pistoles. Thus, if the first player gain, then
64 pistoles belong to him, and, if he lose, then 32 pistoles belong to him.
If therefore the players do not wish to play this game, but to separate
286 HISTORY OF MATHEMATICS [ch.xv
without playing it, the first player would say to the second, "I am
certain of 32 pistoles even if I lose this game, and as for the other 32
pistoles perhaps I shall have them and perhaps you will have them ;
the chances are equal. Let us then divide these 32 pistoles equally,
and give me also the 32 pistoles of which I am certain." Thus the first
player will have 48 pistoles and the second 16 pistoles.
Next, suppose that the first player has gained two points and the
second player none, and that they are about to play for a point ; the
condition then is that, if the first player gain this point, he secures the
game and takes the 64 pistoles, and, if the second player gain this point,
then the players will be in the situation already examined, in which the
first player is entitled to 48 pistoles and the second to 16 pistoles.
Thus, if they do not wish to play, the first player would say to the second,
"If I gain the point I gain 64 pistoles ; if I lose it, I am entitled to
48 pistoles. Give me then the 48 pistoles of which I am certain, and
divide the other 16 equally, since our chances of gaining the point are
equal." Thus the first player will have 56 pistoles and the second player
8 pistoles.
Finally, suppose that the first player has gained one point and the
second player none. If they proceed to play for a point, the condition is
that, if the first player gain it, the players will be in the situation first
examined, in which the first player is entitled to 56 pistoles ; if the first
player lose the point, each player has then a point, and each is entitled
to 32 pistoles. Thus, if they do not wish to play, the first player would
say to the second, '* Give me the 32 pistoles of which I am certain, and
divide the remainder of the 56 pistoles equally, that is, divide 24 pistoles
equally." Thus the first player will have the sum of 32 and 12
pistoles, that is, 44 pistoles, and consequently the second will have 20
pistoles.
Pascal proceeds next to consider the similar problems when
the game is won by whoever first obtains m + n points, and
one player has m while the other has n points. The answer
is obtained by using the arithmetical triangle. The general
solution (in which the skill of the players is unequal) is given
in many modern text-books on algebra, and agrees with Pascal's
result, though of course the notation of the latter is different
and less convenient.
Pascal made an illegitimate use of the new theory in
the seventh chapter of his Pen^ees. In effect, he puts his
argument that, as the value of eternal happiness must be
,4 ^..p-
CH. xv] PASCAL V ^ ^
infinite, then, even if the probability of a religious life ensuring
eternal happiness be very small, still the expectation (which is
measured by product of the two) must be of sufiicient magni-
tude to make it w^orth while to be religious. The argument,
if worth anything, would apply equally to any religion which
promised eternal happiness to those who accepted its doctrines.
If any conclusion may be drawn from the statement, it is the
undesirability of applying mathematics to questions of morality
of which some of the data are necessarily outside the range
of an exact science. It is only fair to add that no one
had more contempt than Pascal for those who changed
their opinions according to the prospect of material benefit,
and this isolated passage is at variance with the spirit of his
writings.
The last mathematical work of Pascal was that on the cycloid
in 1658. The cycloid is the curve traced out by a point on the
circumference of a circular hoop which rolls along a straight
line. Galileo, in 1630, had called attention to this curve, the
shape of which is particularly graceful, and had suggested that
the arches of bridges should be built in this form.^ Four years
later, in 1634, Roberval found the area of the cycloid ; Descartes
thought little of this solution and defied him to find its tangents,
the same challenge being also sent to Fermat who at once
solved the problem. Several questions connected with the
curve, and with the surface and volume generated by its
revolution about its axis, base, or the tangent at its vertex,
were then proposed by various mathematicians. These and
some analogous questions, as well as the positions of the centres
of the mass of the solids formed, were solved by Pascal in 1658,
and the results were issued as a challenge to the world. Wallis
succeeded in solving all the questions except those connected
with the centre of mass. Pascal's own solutions were eff'ected
by the method of indivisibles, and are similar to those which
^ The bridge, by Essex, across the Cam in the grounds of Trinity College,
Cambridge, has cycloidal arches. On the history of the cycloid before Galileo,
see S. Giinther, Bibliotlieca Mathematica, 1887, vol. i, pp. 7-14.
288 HISTORY OF MATHEMATICS [ch.xv
a modern mathematician would give by the aid of the integral
calculus. He obtained by summation what are equivalent to
the integrals of sin^, sin^c^, and ^sin^, one limit being either
0 or Jtt. He also investigated the geometry of the Archi-
medean spiral. These researches, according to D'Alembert,
form a connecting link between the geometry of Archimedes and
the infinitesimal calculus of Newton.
Wallis.^ John Wallis was born at Ashford on November 22,
1616, and died at Oxford on October 28, 1703. He was educated
at Felstead school, and one day in his holidays, when fifteen
years old, he happened to see a book of arithmetic in the
hands of his brother; struck with curiosity at the odd signs
and symbols in it he borrowed the book, and in a fortnight,
with his brother's help, had mastered the subject. As it was
intended that he should be a doctor, he was sent to Emmanuel
College, Cambridge, while there he kept an " act " on the
doctrine of the circulation of the blood ; that is said to have
been the first occasion in Europe on which this theory was
publicly maintained in a disputation. His interests, however,
centred on mathematics.
He was elected to a fellowship at Queens' College, Cambridge,
and subsequently took orders, but on the whole adhered to the
Puritan party, to whom he rendered great assistance in decipher-
ing the royalist despatches. He, however, joined the moderate
Presbyterians in signing the remonstrance against the execution
of Charles I., by which he incurred the lasting hostility of the
Independents. In spite of their opposition he was appointed in
1649 to the Savilian chair of geometry at Oxford, where he
lived until his death on October 28, 1703. Besides his mathe-
matical works he wrote on theology, logic, and philosophy, and
,was the first to devise a system for teaching deaf-mutes. I
confine myself to a few notes on his more important mathematical
writings. They are notable partly for the introduction of the
^ See my History of the Study of Mathematics at Cambridge, pp. 41-46.
An edition of Wallis's mathematical works was published in three volumes at
Oxford, 1693-98.
CH. xv] WALLIS 289
use of infinite series as an ordinary part of analysis, and partly
for the fact that they revealed and explained to all students the
principles of the new methods of analysis introduced by his
contemporaries and immediate predecessors.
In 1655 Wallis published a treatise on conic sections in which
they were defined analytically. I have already mentioned that
the Geometrie of Descartes is both difficult and obscure, and to
many of his contemporaries, to whom the method was new, it
mAst have been incomprehensible. This work did something to
make the method intelligible to all mathematicians : it is the ,
earliest book in which these curves are considered and defined )
as curves of the second degree.
The most important of Wallis's works was his Arithmetica
Injinitorum, which was published in 1656. In this treatise
the" methods of analysis of Descartes and Cavalieri were
systematised and greatly extended, but their logical exposition
is open to criticism. It at once became the standard book ^
on the subject, and is constantly referred to by subsequent
writers. It is prefaced by a short tract on conic sections.
He commences by proving the law of indices: shews that
x^^ x'~'^,x~^ ... represents 1, l/.r, 1/^^ _ . that c»^^'^ represents the
square root of x, that x^'^ represents the cube root of x^, and
generally that x~^ represents the reciprocal of x^, and that
xi^l'i represents the qth. root of ocP.
Leaving the numerous algebraical applications of this dis-
covery he next proceeds to find, by the method of indivisibles,
the area enclosed between the curve y = x'^, the axis of x, and
any ordinate x = h ; and he proves that the ratio of this area
to that of the parallelogram on the same base and of the
same altitude is equal to the ratio 1 :m+l. He apparently
assumed that the same result would be true also for the
curve i/ = ax'^, where a is any constant, and m any number
positive or negative ; but he only discusses the case of the
parabola in which m = 2, and that of the hyperbola in which
m= -I : in the latter case his interpretation of the result
is incorrect. He then shews that similar results might be
u
290 HISTORY OF MATHEMATICS [ch. xv
written down for any carve of the form y = ^ax^ ; and hence
that, if the ordinate y of a curve can be expanded in powers
of the abscissa x, its quadrature can be determined : thus he
says that if the equation of a curve were y = x^ + x'^ + x^+...,
its area would be x + ^x^ + ^x^-\- .... He then applies this
to the quadrature of the curves 7/ = (x-x^y, y = (x-x^y,
y = {x- x^y, y = {x- x'^Yj etc. taken between the limits x = 0 and
x=\; and shews that the areas are respectively 1, \, -^q, -j^-^,
etc. He next considers curves of the form y = ^~*^ and estab-
lishes the theorem that the area bounded by the curve, the
axis of X, and the ordinate x=l, is to the area of the rectangle
on the same base and of the same altitude as m:m + l. This
is equivalent to finding the value of I x^'''^dx. He illustrates
J 0
this by the parabola in which m = 2. He states, but does
not prove, the corresponding result for a curve of the form
y = xPiq,
Wallis shewed considerable ingenuity in reducing the equations
of curves to the forms given above, but, as he was unacquainted
with the binomial theorem, he could not effect the quadrature of
the circle, whose equation is y = (x - x'^Y''^, since he was unable to
expand this in powers of x. He laid down, however, the principle
of interpolation. Thus, as the ordinate of the circle y = (x- ^^y/a
is the geometrical mean between the ordinates of the curves
y = (x- x^y and y = (x- x^y, it might be supposed that, as an
approximation, the area of the semicircle I (x-x'^y^dx^ which
J 0
is Jtt, might be taken as the geometrical mean between the
values of
j (x-x^fdx and l(x-x^ydx,
Jo Jo
that is, 1 and J; this is equivalent to taking 4 J^ or 3*26 ...
as the value of tt. ]But, Wallis argued, we have in fact a
series 1, J, -^jj, y|-g-, ..., and therefore the term interpolated
between 1 and J ought to be so chosen as to obey the law
CH. xv] WALLIS 291
of this series. This, by an elaborate method, which I need not
describe in detail, leads to a value for the interpolated term
which is equivalent to taking
,^2.2.4.4.6 .6.8.8...
'^""l .3.3.5.5.7.7.9...-
The mathematicians of the seventeenth century constantly used
interpolation to obtain results which we should attempt to obtain
by direct analysis.
In this work also the formation and properties of continued
fractions are discussed, the subject having been brought into
prominence by Brouncker's use of these fractions.
A few years later, in 1659, Wallis published a tract con-
taining the solution of the problems on the cycloid which had
been proposed by Pascal. In this he incidentally explained
how the principles laid down in his Arithmetica Infinitormn
could be used for the rectification of algebraic curves; and
gave a solution of the problem to rectify the semi -cubical
parabola oi^^ay^^ which had been discovered in 1657 by his^
pupil William Neil. Since all attempts to rectify the ellipse
and hyperbola had been (necessarily) ineffectual, it had been
supposed that no curves could be rectified, as indeed Descartes
had definitely asserted to be the case. The logarithmic spiral
had been rectified by Torricelli, and was the first curved line
(other than the circle) whose length was determined by mathe-
matics, but the extension by Neil and Wallace to an algebraical
curve was novel. The cycloid was the next curve rectified ; this
was done by Wren in 1658.
Early in 1658 a similar discovery, independent of that of
Neil, was made by van Heuraet,i and this was published by
van Schooten in his edition of Descartes's Geometria in 1659.
Van Heuraet's method is as follows. He supposes the curve
to be referred to rectangular axes ; if this be so, and if (r, y)
be the co-ordinates of any point on it, and n the length of the
^ On van Heuraet, see the BihliotJieca Mathematica, 1887, vol. i,
pp. 76-80.
292 HISTORY OF MATHEMATICS [ch. xv
normal, and if another point whose co-ordinates are {x, ^) be
taken such that rj -.h — n-.y, where A is a constant ; then, if ds
be the element of the length of the required curve, we have by
similar triangles ds : dx = n -.y. Therefore hds = rjdx. Hence,
if the area of the locus of the point (x, rj) can be found, the
first curve can be rectified. In this way van Heuraet effected
the rectification of the curve y^ = ax'^; but added that the
rectification of the parabola y^ = ax is impossible since it
requires the quadrature of the hyperbola. The solutions given
by Neil and Wallis are somewhat similar to that given by van
Heuraet, though no general rule is enunciated, and the analysis
is clumsy. A third method was suggested by Fermat in 1660,
but it is inelegant and laborious.
The theory of the collision of bodies was propounded by
the Royal Society in 1668 for the consideration of mathe-
maticians. Wallis, Wren, and Huygens sent correct and
similar solutions, all depending on what is now called the
conservation of momentum; but, while Wren and Huygens
confined their theory to perfectly elastic bodies, Wallis con-
sidered also imperfectly elastic bodies. This was followed in
1669 by a work on statics (centres of gravity), and in 1670 by
one on dynamics : these provide a convenient synopsis of what
was then known on the subject.
In 1685 Wallis published an Algebra, preceded by a
historical account of the development of the subject, which
contains a great deal of valuable information. The second
edition, issued in 1^93 and forming the second volume of his
Opera, was considerably enlarged. This algebra is noteworthy
as containing the first systematic use of formulae. A given
magnitude is here represented by the numerical ratio which
it bears to the unit of the same kind of magnitude : thus,
when Wallis wants to compare two lengths he regards each as
containing so many units of length. This perhaps will be
made clearer if I say that the relation between the space
described in any time by a particle moving with a uniform
velocity would be denoted by Wallis by the formula s = vtf
CH.xv] WALLIS. FERMAT 293
where s is the number representing the ratio of the space
described to the unit of length; while previous writers would
have denoted the same relation by stating what is equivalent ^
to the proposition Sj :j,^ = v^t^ ''^•2h- ^^ ^^ curious to note that
Wallis rejected as absurd the n^wuisiiaLidea of a negative
number as being less than nothing, but accepted the view that
it is something greater than infinity. The latter opinion may
be tenable and not inconsistent with the former, but it is hardly
a more simple one.
Fermat.2 While Descartes was laying the foundations of
analytical geometry, the same subject was occupying the atten-
tion of another and not less distinguished Frenchman. This
was Fermat. Pierre de Fermat, who was born near Montauban
in 1601, and died at Castres on January 12, 1665, was the son
of a leather-merchant; he was educated at home; in 1631 he
obtained the post of councillor for the local parliament at
Toulouse, and he discharged the duties of the office with scrupu-
lous accuracy and fidelity. There, devoting most of his leisure
to mathematics, he spent the remainder of his life — a life which,
but for a somewhat acrimonious dispute with Descartes on the
validity of certain analysis used by the latter, was unruffled by
any event which calls for special notice. The dispute was chiefly
due to the obscurity of Descartes, but the tact and courtesy of
Fermat brought it to a friendly conclusion. Fermat was a good
scholar, and amused himself by conjecturally restoring the work
of Apollonius on plane loci.
Except a few isolated papers, Fermat published nothing in
his lifetime, and gave no systematic exposition of his methods.
Some of the most striking of his results were found after his
death on loose sheets of paper or written in the margins of
^ See ex. gr. Newton's Frincipia, bk. i, sect, i, lemma 10 or 11.
"^ The best edition of Fermat's works is that in three volumes, edited by
S. P. Tannery and C. Henry, and published by the French government,
1891-6. Of earlier editions, I may mention one of his papers and corre-
spondence, printed at Toulouse in two volumes, 1670 and 1679 : of which a
summary, with notes, was published by E. Brassinne at Toulouse in 1853,
and a reprint was issued at Berlin in 1861.
294 HISTORY OF MATHEMATICS [oh. xv
works which he had read and annotated, and are unaccompanied
by any proof. It is thus somewhat difficult to estimate the
dates and originality of his work. He was constitutionally
modest and retiring, and does not seem to have intended his
papers to be published. It is probable that he revised his notes
as occasion required, and that his published works represent the
final form of his researches, and therefore cannot be dated much
earlier than 1660. I shall consider separately (i) his investiga-
tions in the theory of numbers ; (ii) his use in geometry of
analysis and of infinitesimals ; and (iii) his method of treating
questions of probability.
(i) The theory of numbers appears to have been the favourite
study of Fermat. He prepared an edition of Diophantus, and
the notes and comments thereon contain numerous theorems of
considerable elegance. Most of the proofs of Fermat are lost,
and it is possible that some of them were not rigorous — an
induction by analogy and the intuition of genius sufficing to
lead him to correct results. The following examples will illus-
trate these investigations.
{a) If 2^ be a prime and a be prime to jo, then aP~'^-\ is
divisible by jo, that is, a^~^ - 1 =0 (mod. p). A proof of this,
first given by Euler, is well known. A more general theorem
is that a*^(^) -1=0 (mod. n), where a is prime to n, and ^ (n) is
the number of integers less than n and prime to it.
(b) An odd prime can be expressed as the difference of two
square integers in one and only one way. Fermat's proof is as
follows. Let n be the prime, and suppose it equal to x^ - y^,
that is, to {oc + y) {x-y). Now, by hypothesis, the only integral
factors of n are n and unity, hence x + y=^n and x-y = l.
Solving these equations we get x = ^{n+ 1) and y = \{n- 1).
(c) He gave a proof of the statement made by Diophantus
that the sum of the squares of two integers cannot be of the
form 4:71— 1 ; and he added a corollary which I take to mean
that it is impossible that the product of a square and a prime
of the form 47i - 1 [even if multiplied by a number prime to the
latter], can be either a square or the sum of two squares.
CH. xv] FERMAT 295
For example, 44 is a multiple of 11 (which is of the form
4 X 3 - 1) by 4, hence it cannot be expressed as the sum of two
squares. He also stated that a number of the form a^ + b'^,
where a is prime to b, cannot be divided by a prime of the
form 4?t - 1.
(d) Every prime of the form 47^+1 is expressible, and that
in one way only, as the sum of two squares. This problem was
first solved by Euler, who shewed that a number of the form
2*"- (47?, + 1) can be always expressed as the sum of two squares.
(e) If a, b, c, be integers, such that a^ + 6^ = c^, then ab
cannot be a square. Lagrange gave a solution of this.
(/) The determination of a number x such that x^n + 1 may
be a square, where n is a given integer which is not a square.
Lagrange gave a solution of this.
(g) There is only one integral solution of the equation
^2 + 2 = y^; and there are only two integral solutions of the
equation a;^ + 4 = y^. The required solutions are evidently for
the first equation x = 6, and for the second equation x=2 and
x = ll. This question was issued as a challenge to the English
mathematicians Wallis and Digby.
(h) No integral values of x, y^ z can be found to satisfy the
equation x*^ + y'^^ = 2**, if n be an integer greater than 2. This
proposition ^ has acquired extraordinary celebrity from the fact
that no general demonstration of it has been given, but there
is no reason to doubt that it is true.
Probably Fermat discovered its truth first for the case ti = 3,
and then for the case ti = 4. His proof for the former of these
cases is lost, but that for the latter is extant, and a similar
proof for the case of ?i = 3 was given by Euler. These proofs
depend upon shewing that, if three integral values of x^ y, z can
be found which satisfy the equation, then it will be possible to
find three other and smaller integers which also satisfy it : in
this way, finally, we shew that the equation must be satisfied by
three values which obviously do not satisfy it. Thus no integral
^ On this curious proposition, see my Mathematical Recreations^ sixth
edition, 1914, pp. 40-43.
296 HISTOEY OF MATHEMATICS [ch. xv
solution is jDOSsible. It would seem that this method is in-
applicable to any cases except those oi n = 3 and n = 4:.
Fermat's discovery of the general theorem was made later.
A proof can be given on the assumption that a number can be
resolved into the product of powers of primes in one and only
one way. The assumption is true of real numbers, but it is
not true when complex factors are admitted. For instance,
10 can be expressed as the product of 3 + ^ and 3 — i, or of
3 + 2 and 3-1, or of 2, 2 + i, and 2 - i. It is possible that
Fermat made some such erroneous supposition, but, on the
whole, it seems more likely that he discovered a rigorous
demonstration.
In 1823 Legendre obtained a proof for the case of n = 5;
in 1832 Lejeune Dirichlet gave one for n=14:, and in 1840
Lame and Lebesgue gave proofs for n = 7. The proposition
appears to be true universally, and in 1849 Kummer, by means
of ideal primes, proved it to be so for all numbers except those
(if any) which satisfy three conditions. It is not certain whether
any number can be found to satisfy these conditions, but there
is no number less than 6857 which does so. The general
problem has also been discussed by Sophie Germain. I may
add that, to prove the truth of the proposition, when n is
greater than 4 obviously it is sufficient to confine ourselves to
cases where n is a, prime.
The following extracts, from a letter now in the university
library at Leyden, will give an idea of Fermat's methods ; the
letter is undated, but it would appear that, at the time Fermat
wrot6 it, he had proved the proposition (h) above only for the
case when n = 3.
Je ne m'en servis au commencement que pour demontrer les propo-
sitions negatives, comme par exemple, qu'il n'y a aucu nombre moindre
de I'unite qu'un multiple de 3 qui soit compose d'uu quarre et du tri^ile
d'un autre quarre. Qu'il n'y a aucun triahgle rectangle de nombres dont
I'aire soit uu nombre quarre. La preuve se fait par aTraycoyriv ttjp els
ddijuarop en cette maniere. S'il y auoit aucun triangle rectangle en
nombres entiers, qui eust son aire esgale a un quarre, il y auroit un
autre triangle moindre que celuy la qui auroit la mesme propriete. S'il
CH. xv] FERMAT 297
y en auoit un second moindre que le ])remier qui eust la niesnie pro-
piiete il y en auroit par un pareil raisonnement un troisieme moindre
que ce second qui auroit la mesme propriete et enfin un quatrieme, un
einquieme etc. a I'infini en descendant. Or est il qu'estant donne un
nombre il n'y en a point infinis en descendant moindres que celuy la,
j'entens parler tousjours des nonibres entiers. D'ou on conclud qu'il est
done impossible qu'il y ait aucun triangle rectangle dont I'aire soit
quarre. Vide foliii p<ost sequcns. ...
Je fus longtemps sans pouuoir appliquer ma metliode aux questions
affirmatiues, parce que le tour et le biais pour y venir est beaucoup plus
malaise que celuy dont je me sers aux negatives. De sorte que lors qu'il
me falut demonstrer que tout nombre premier qui surpasse de I'unite un
multiple de 4, est compose de deux quarrez je me treuuay en belle peine.
Mais enfin une meditation diverses fois reiteree me donna les lumieres qui
me manquoient. Et les questions affirmatiues passerent par ma metliode
a I'ayde de quelques nouueaux principes qu'il y fallust joindre par
necessite. Ce progres de mon raisonnement en ces questions affirmatives
estoit tel. Si un nombre premier pris a discretion qui surpasse de I'unite
un multiple de 4 n'est point compose de deux quarrez il y aura un nombre
premier de mesme nature moindre que le donne ; et ensuite un troisieme
encore moindre, etc. en descendant a I'infini jusques a ce que uous arriviez
an nombre 5, qui est le moindre de tons ceux de cette nature, lequel il s'en
suivroit n'estre pas compose de deux quarrez, ce qu'il est pourtant d'ou on
doit inferer par la deduction a I'impossible que tons ceux de cette nature
sont par consequent composez de 2 quarrez.
II y a infinies questions de cette espece. Mais il y en a quelques
autres qui demandent de nouveaux principes pour y appliquer la descente,
et la recherche en est quelques fois si mal aisee, qu'on n'y pent venir
qu'auec une peine extreme. Telle est la question suiuante que Bachet sur
Diophante avoiie n'avoir jamais pen demonstrer, sur le suject de laquelle
M''. Descartes fait dans une de ses lettres la mesme declaration, jusques la
qu'il confesse qu'il la juge si difficile, q^j'il ne voit point de voye pour la
resoudre. Tout nombre est quarre, on compose de deux, de trois, on de
quatre quarrez. Je I'ay enfin rangee sous ma metliode et je demonstre
que si un nombre donne n'estoit point de cette nature il y en auroit un
moindre qui ne le seroit pas non plus, puis un troisieme moindre que le
second &c. a I'infini, d'ou Ton infere que tons les nombres sont de cette
nature. . . .
J'ay ensuite considere certaines questions qui bien que negatives ne
restent pas de receuoir tres-grande difficulte, la methode pour y pratiquer
la descente estant tout a fait diuerse des precedentes comnie il sera ais6
d'esprouuer. Telles sont les suiuantes. II n'y a aucun cube diuisible en
deux cubes. II n'y a qu'un seul quarre en entiers qui augmente du binaire
298 HISTORY OF MATHEMATICS [ch. xv
fasse im cube, ledit quarre est 25. II n'y a que deux quarrez en entiers
lesquels augmentes de 4 fassent cube, lesdits quarrez sont 4 et 121....
Apres auoir couru toutes ces questions la plupart de diuerses {sic) nature
et de differente fafon de demonstrer, j'ay passe a I'inuention des regies
generales pour resoudre les equations simples et doubles de Diopliante.
On propose par exemple 2 quarr. +7957 esgaux a un quarre (hoc est
2a;i + 7967ocquadr.) J'ay une regie generale pour resoudre cette equation
si elle est possible, on decouvrir son impossibilite. Et ainsi en tons les cas et
en tons nombres taut des quarrez que des unitez. On propose cette
equation double 2a; + 3 et 3a; + 5 esgaux chaucon a un quarre. Bachet se
glorifie en ses commentaires sur Diophante d'auoir trouve une regie en deux
cas particuliers. Je la donne generale en toute sorte de cas. Et determine
par regie si elle est possible ou non....
Voila sommairement le conte de mes recherches sur le suject des
nombres. Je ne I'ay escrit que parce que j'apprehende que le loisir
d'estendre et de raettre au long toutes ces demonstrations et ces methodes
me manquera. En tout cas cette indication seruira aux S9auants pour
trouver d'eux mesmes ce que je n'estens point, principalement si M^". de
Carcaui et Frenicle leur font part de quelques demonstrations par la
descente que je leur ay enuoyees sur le suject de quelques propositions
negatiues. Et pent estre la posterite me scaura gre de luy avoir fait
connoistre que les anciens n'ont pas tout sceu, et cette relation pourra
passer dans I'esprit de ceux qui viendront apres moy pour traditio
lampadis ad filios, comme parle le grand Chancelier d'Angleterre, suiuant
le sentiment et la deuise duquel j'adjousteray, multi pertransibunt et
augebitur scientia.
(ii) I next proceed to mention Fermat's use in geometry of
analysis and of infinitesimals. It would seem from his corre-
spondence that he had thought out the principles of analytical
geometry for himself before reading Descartes's Geometrie, and
had realised that from the# equation, or, as he calls it, the
" specific property," of a curve all its properties could be
deduced. His extant papers on geometry deal, however, mainly
with the application of infinitesimals to the determination of the
tangents to curves, to the quadrature of curves, and to questions
of maxima and minima ; probably these papers are a revision of
his original manuscripts (which he destroyed), and were written
about 1663, but there is no doubt that he was in possession of
the general idea of his method for finding maxima and minima
as early as 1628 or 1629.
CH. xv] FERMAT 299
He obtained the subtangent to the ellipse, cycloid, cissoid,
conchoid, and quadratrix by making the ordinates of the curve
and a straight line the same for two points whose abscissae were
X and x-e\ but there is nothing to indicate that he was aware
that the process was general, and, though in the course of his
work he used the principle, it is probable that he never separated
it, so to speak, from the symbols of the particular problem he
was considering. The first definite statement of the method was
due to Barrow,^ and was published in 1669.
Fermat also obtained the areas of parabolas and hyperbolas
of any order, and determined the centres of mass of a few simple
laminae and of a paraboloid of revolution. As an example of
his method of solving these questions I will quote his solution of
the problem to find the area between the parabola y^ =px^, the
axis of X, and the line x = a. He says that, if the several ordin-
ates at the points for which x is equal to a, a(l - e), a{\ - ef^...
be drawn, then the area will be split into a number of little
rectangles whose areas are respectively
ae{2M'-f\ ae{l - e){pa%l - eyf\ ....
The sum of these is p^^\^'^el{l -(1 -ef'^} ; and by a subsidi-
ary proposition (for he was not acquainted with the binomial
theorem) he finds the limit of this, when e vanishes, to be
1/3 5/3
f^ a . The theorems last mentioned were published only
after his death ; and probably they were not written till after he
had read the works of Cavalieri and Wallis.
Kepler had remarked that the values of a function immedi-
ately adjacent to and on either side of a maximum (or minimum)
value must be equal. Fermat applied this principle to a few
examples. Thus, to find the maximum value of x{a-x),
his method is essentially equivalent to taking a consecutive
value of X, namely x-e where e is very small, and putting
x{a - x) = {x - e){a - X -V e). Simplifying, and ultimately putting
e = 0, we get x = Ja. This value of x makes the given expression
a maximum.
^ See below, pp. 311-12.
300 HISTORY OF MATHEMATICS [ch. xv
(iii) Fermat must share with Pascal the honour of having
founded the theory of probabilities. I have already mentioned
the problem proposed to Pascal, and which he communicated
to Fermat, and have there given Pascal's solution. Fermat's
solution depends on the theory of combinations, and will be
sufficiently illustrated by the following example, the substance
of which is taken from a letter dated August 24, 1654, which
occurs in the correspondence with Pascal. Fermat discusses the
case of two players, A and B, where A wants two points to win
and B three points. Then the game will be certainly decided
in the course of four trials. Take the letters a and b, and write
down all the combinations that can be formed of four letters.
These combinations are 16 in number, namely, aaaa, axiab, aaba^
aabb ; abaa, abab, abba, abbb ; baxia, baab, baba, babb ; bbaa,
bbabj bbba, bbbb. Now every combination in which a occurs
twice or oftener represents a case favourable to A, and every
combination in which b occurs three times or oftener represents
a case favourable to B. Thus, on counting them, it will be
found that there are 11 cases favourable to A, and 5 cases
favourable to B ; and, since these cases are all equally likely, ^'s
chance of winning the game is to J5's chance as 11 is to 5.
The only other problem on this subject which, as far as I
know, attracted the attention of Fermat was also proposed to
him by Pascal, and was as follows. A person undertakes to
throw a six with a die in eight throws ; supposing him to have
made three throws without success, what portion of the stake
should he be allowed to take on condition of giving up his
fourth throw 1 Fermat's reasoning is as follows. The chance
of success is 1/6, so that he should be allowed to take 1/6 of the
stake on condition of giving up his throw. But, if we wish to
estimate the value of the fourth throw before any throw is
made, then the first throw is worth 1/6 of the stake; the second
is worth 1/6 of what remains, that is, 5/36 of the stake ; the
third throw is worth 1/6 of what now remains, that is, 25/216
of the stake ; the fourth throw is worth 1/6 of what now remains,
that is, 125/1296 of the stake.
CH.xv] FERMAT. HUYGENS 301
Fermat does not seem to have carried the matter much
further, but his correspondence with Pascal shows that his views
on the fundamental principles of the subject were accurate :
those of Pascal were not altogether correct.
Fermat's reputation is quite unique in the history of science.
The problems on numbers which he had proposed long defied
all efforts to solve them, and many of them yielded only to the
skill of Euler. One still remains unsolved. This extraordinary
achievement has overshadowed his other work, but in fact it is
all of the highest order of excellence, and we can only regret
that he thought fit to write so little.
Huygens.^ Christian Huygens was born at the Hague on
April 14, 1629, and died in the same town on June 8, 1695.
He generally wrote his name as Hugens, but I follow the usual
custom in spelling it as above : it is also sometimes written as
Huyghens. His life was uneventful, and there is little more
to record in it than a statement of his various memoirs and
researches.
In 1651 he published an essay in which he shewed the
fallacy in a system of quadratures proposed by Gregoire de
Saint- Vincent, who was well versed in the geometry of the
Greeks, but had not grasped the essential points in the more
modern methods. This essay was followed by tracts on the
quadrature of the conies and the approximate rectification of
the circle.
In 1654 his attention was directed to the improvement of the
telescope. In conjunction with his brother he devised a new
and better way of grinding and polishing lenses. As a result
of these improvements he was able during the following two
years, 1655 and 1656, to resolve numerous astronomical ques-
tions ; as, for example, the nature of Saturn's appendage. His
astronomical observations required .some exact means of measuring
^ A new edition of all Hnygens's works and correspondence was issued at
the Hague in ten volumes, 1888-1905. An earlier edition of his works was
published in six vohimes, four at Leyden in 1724, and two at Amsterdam
in 1728 (a life by s'Gravesande is prefixed to the first volume): his scientific
correspondence was published at the Hague in 1833.
302 HISTORY OF MATHEMATICS [ch. xv
time, and he was thus led in 1656 to invent the pendulum clock,
as described in his tract Horologium, 1658. The time-pieces
previously in use had been balance-clocks.
In the year 1657 Huygens wrote a small work on the calculus
of probabilities founded on the correspondence of Pascal and
Fermat. He spent a couple of years in England about this
time. His reputation was now so great that in 1665 Louis
XIV. offered him a pension if he would live in Paris, which
accordingly then became his place of residence.
In 1668 he sent to the Eoyal Society of London, in »answer
to a problem they had proposed, a memoir in which (simul-
taneously with Wallis and Wren) he proved by experiment that
the momentum in. a certain direction before the collision of two
bodies is equal to the momentum in that direction after the
collision. This was one of the points in mechanics on which
Descartes had been mistaken.
The most important of Huygens's work was his Horologium
Oscillatorium published at Paris in 1673. The first chapter is
devoted to pendulum clocks. The second chapter contains a
complete account of the descent of heavy bodies under their own
weights in a vacuum, either vertically down or on smooth curves.
Amongst other propositions he shews that the cycloid is tauto-
chronous. In the third chapter he defines evolutes and
involutes, proves some of their more elementary properties, and
illustrates his methods by finding the evolutes of the cycloid
and the parabola. These are the earliest instances in which the
envelope of a moving line was determined. In the fourth
chapter he solves the problem of the compound pendulum, and
shews that the centres of oscillation and suspension are inter-
changeable. In the fifth and last chapter he discusses again
the theory of clocks, points out that if the bob of the pendulum
were, by means of cycloidal checks, made to oscillate in a cycloid
the oscillations would be isochronous ; and finishes by shewing
that the centrifugal force on a body which moves round a circle
of radius r with a uniform velocity v varies directly as v^
and inversely as r. This work contains the first attempt to
CH.xv] HUYGENS 303
apply dynamics to bodies of finite size and not merely to
particles.
In 1675 Huygens proposed to regulate the motion of watches .
by the use of the balance spring, in the theory of which he had
been perhaps anticipated in a somewhat ambiguous and incom-
plete statement made by Hooke in 1658. Watches or portable
clocks had been invented early in the sixteenth century, and by
the end of that century were not very uncommon, but they were
clumsy and unreliable, being driven by a main spring and
regulated by a conical pulley and verge escapement ; moreover,
until 1687 they had only one hand. The first watch whose
motion was regulated by a balance spring was made at Paris
under Huygens's directions, and presented by him to Louis XIY.
The increasing intolerance of the Catholics led to his return
to Holland in 1681, and after the revocation of the edict of
Nantes he refused to hold any further communication with
France. He now devoted himself to the construction of lenses
of enormous focal length: of these three of focal lengths 123
feet, 180 feet, and 210 feet, were subsequently given by him to
the Royal Society of London, in whose possession they still
remain. It was about this time that he discovered the achro-
matic eye-piece (for a telescope) which is known by his name. \
In 1689 he came from Holland to England in order to make
the acquaintance of Newton, whose Principia had been published
in 1687. Huygens fully recognized the intellectual merits of
the work, but seems to have deemed any theory incomplete
which did not explain gravitation by mechanical means.
On his return in 1690 Huygens published his treatise on \
light in which the undulatory theory was expounded and
explained. Most of this had been written as early as 1678.
The general idea of the theory had been suggested by Robert
Hooke in 1664, but he had not investigated its consequences
in any detail. Only three ways have been suggested in which
light can be produced mechanically. Either the eye may be
supposed to send out something which, so to speak, feels the
object (as the Greeks believed) ; or the object perceived may
304 HISTORY OF MATHEMATICS [ch. xv
send out something which hits or affects the eye (as assumed in
the emission theory) ; or there may be some medium between
the eye and the object, and the object may cause some change
in the form or condition of this intervening medium and thus
affect the eye (as Hooke and Huygens supposed in the wave or
undulatory theory. According to this last theory space is filled
with an extremely rare ether, and light is caused by a series of
waves or vibrations in this ether which are set in motion by the
pulsations of the luminous body. From this hypothesis Huygens
deduced the laws of reflexion and refraction, explained the
phenomena of double refraction, and gave a construction for
the extraordinary ray in biaxal crystals; while he found by
experiment the chief phenomena of polarization.
The immense reputation and unrivalled powers of Newton
led to disbelief in a theory which he rejected, and to the general
adoption of Newton's emission theory. Within the present
century crucial experiments have been devised which give differ-
ent results according as one or the other theory is adopted ; all
these experiments agree with the results of the undulatory theory
and differ from the results of the Newtonian theory ; the latter
is therefore untenable. Until, however, the theory of interfer-
ence, suggested by Young, was worked out by Fresnel, the
hypothesis of Huygens failed to account for all the facts, and
even now the properties which, under it, have to be attributed
to the intervening medium or ether involve difficulties of which
we still seek a solution. Hence the problem as to how the
effects of light are really produced cannot be said to be finally
solved.
Besides these works Huygens took part in most of the con-
troversies and challenges which then played so large a part in
the mathematical world, and wrote several minor tracts. In one
of these he investigated the form and properties of the catenary.
In another he stated in general terms the rule for finding maxima
and minima of which Fermat had made use, and shewed that
the subtangent of an algebraical curve f{x, y) = 0 was equal to
yfylfxi where fy is the derived function of /(a?, y) regarded as a
CH. xv] HUYGENS. BACHET 305
function of y. In some posthumous works, issued at Leyden in
1703, lie further shewed how from the focal lengths of the
component lenses the magnifying power of a telescope could be
determined; and explained some of the phenomena connected
with haloes and parhelia.
I should add that almost all his demonstrations, like those
of Newton, are rigidly geometrical, and he would seem to have
made no use of the differential or fluxional calculus, though he
admitted the validity of the methods used therein. Thus, even
when first written, his works were expressed in an archaic
language, and perhaps received less attention than their intrinsic
merits deserved.
I have now traced the development of mathematics for a
period which we may take roughly as dating from 1635 to 1675,
under the influence of Descartes, Cavalieri, Pascal, Wallis^ Fer-
mat, and Huygens. The life of Newton partly overlaps this
period; his works and influence are considered in the next
chapter.
I may dismiss the remaining mathematicians of this time ^
with comparatively slight notice. The most eminent of them
are Bachet, Barrow, Brouncker, Collins, De la Hire, de La-
loubere, Frenicle, James Gregory, Hooke, Hudde, Nicholas Mer-
cator, Mersenne, Pell, Roherval, Roemer, Rolle, Saint-Vincent,
Sluze, Torricelli, Tschirnhausen, van Schooten, Viviani, and
Wren. In the following notes I have arranged the above-
mentioned mathematicians so that as far as possible their chief
contributions shall come in chronological order.
Bachet. Claude Gaspard Bachet de Meziriac was born at
Bourg in 1581, and died in 1638. He wrote the Froblemes
plaisa7its, of which the first edition was issued in 1612, a
second and enlarged edition was brought out in 1624; this
contains an interesting collection of arithmetical tricks and
questions, many of which are quoted in my Mathematical Recrea-
tions and Essays. He also wrote Les elements arithmetiques,
^ Notes on several of these mathematicians will be found in C. Hutton's
Maihematical Dictionary and Tracts, 5 volumes, London, 1812-1815.
X
306 HISTORY OF MATHEMATICS [ch. xv
which exists in manuscript ; and a translation of the Arithmetic
of Diophantus. Bachet was the earliest writer who discussed
the solution of indeterminate equations by means of continued
fractions.
Mersenne. Marin Mersenne, born in 1588 and died at Paris
in 1648, was a Franciscan friar, who made it his business to be
acquainted and correspond with the French mathematicians of
that date and many of their foreign contemporaries. In 1634
he published a translation of Galileo's mechanics; in 1644 he
issued his Cogitata Physico-Mathematica, by which he is best
known, containing an account of some experiments in physics ;
he also wrote a synopsis of mathematics, which was printed in
1664.
The preface to the Cogitata contains a statement (possibly
due to Fermat) that, in order that 2^-1 may be prime, the
only values of p, not greater than 257, which are possible are
1, 2, 3, 5, 7, 13, 17, 19, 31, 67, 127, and 257; the number 67
is probably a misprint for 61. With this correction the state-
ment appears to be true, and it has been verified for all except
twenty-one values of p, namely, 71, 89, 101, 103, 107, 109, 127,
137, 139, 149, 157, 163, 167, 173, 181, 193, 199, 227, 229, 241,
and 257. Of these values, Mersenne asserted that p = \2l and
j[> = 257 make 2^- 1 a prime, and that the other nineteen values
make 2^—1 a composite number. It has been asserted that
the statement has been verified when j9 = 89 and 127, but these
verifications rest on long numerical calculations made by single
computators and not published ; until these demonstrations
have been confirmed we may say that twenty-one cases still
await verification or require further investigation. The factors
of 2^-1 when jo = 89 are not known, the calculation merely
shewing that the number could not be prime. It is most likely
that these results are particular cases of some general theorem
on the subject which remains to be discovered. ^
The theory of perfect numbers depends directly on that of
1 On this curious proposition, see my Mathematical Recreations, sixth
edition, 1914, chap. xv.
CH. XV] MERSENNE. ROBERVAL. VAN SCHOOTEN 307
Mersenne's numbers. It is probable that all perfect numbers
are included in the formula 2p~\2p-1), where 2P-1 is a
prime, Euclid proved that any number of this form is
perfect; Euler shewed that the formula includes all even
perfect numbers; and there is reason to believe — though a
rigid demonstration is wanting — that an odd number cannot
be perfect. If we assume that the last of these statements is
true, then every perfect number is of the above form. Thus,
if ^ = 2, 3, 5, 7, 13, 17, 19, 31, 61, then, by Mersenne's rule,
the corresponding values of 2^- 1 are prime; they are 3, 7, 31,
127,8191,131071,524287,2147483647,2305843009213693951;
and the corresponding perfect numbers are 6, 28, 496, 8128,
33550336, 8589869056,13743869132*, 230^843Q08139952128,
and 2658455991569831744654692615953842176.
Roberval.^ Gilles Personier (de) Boberval, born at Roberval
in 1602 and died at Paris in 1675, described himself from the
place of his birth as de Roberval, a seigniorial title to which he
had no right. He discussed the nature of the tangents to
curves, solved some of the easier questions connected with the
cycloid, generalized Archimedes's theorems on the spiral, wrote
on mechanics, and on the method of indivisibles, which he rendered
more precise and logical. He was a professor in the university
of Paris, and in correspondence with nearly all the leading
mathematicians of his time.
Van Schooten. Frans van Schooten, to whom we owe an
edition of Vieta's works, succeeded his father (who had taught
mathematics to Huygens, Hudde, and Sluze) as professor at
Leyden in 1646. He brought out in 1659 a Latin translation of
Descartes's Geometrie, and in 1657 a collection of mathematical
exercises in which he recommended the use of co-ordinates in
space of three dimensions. He died in 1661.
Saint- Vincent. 2 Gregoire de Saint -Vincent^ a Jesuit, born
at Bruges in 1584 and died at Ghent in 1667, discovered the
1 A complete edition of his works was included in the old Afemoires of the
Academy of Sciences published in 1693.
2 See L. A. J. Quetelet's Histoire des sciences chez les Beiges, Brussels,
1866.
308 HISTOKY OF MATHEMATICS [ch. xv
expansion of log(l -\-x) in ascending powers of x. Although a
circle-squarer he is worthy of mention for the numerous theorems
of interest which he discovered in his search after the impossible,
and Montucla ingeniously remarks that " no one ever squared
the circle with so much ability or (except for his principal object)
with so much success." He wrote two books on the subject, one
published in 1647 and the other in 1668, which cover some two
or three thousand closely printed pages ; the fallacy in the quad-
rature was pointed out by Huygens. In the former work he
used indivisibles. An earlier work entitled Theoremata Mathe-
matica, published in 1624, contains a clear account of the method
of exhaustions, which is applied to several quadratures, notably
that of the hyperbola.
Torricelli.1 Evangelista Torricelli, born at Faenza on
Oct. 15, 1608, and died at Florence in 1647, wrote on the
quadrature of the cycloid and conies ; the rectification of the
logarithmic spiral; the theory of the barometer; the value of
gravity found by observing the motion of two weights connected
by a string passing over a fixed pulley ; the theory of projectiles ;
and the motion of fluids.
Hudde. Johann Hudde, burgomaster of Amsterdam, was
born there in 1633, and died in the same town in 1704. He
wrote two tracts in 1659 : one was on the reduction of equations
which have equal roots ; in the other he stated what is equiva-
lent to the proposition that if f{x^ y) = 0 be the algebraical
equation of a curve, then the subtangent is - y^- / ^ ; but being
i' I
ignorant of the notation of the calculus his enunciation is
involved.
Fr^nicle.^ Bernard Frenicle de Bessy , born in Paris circ.
1605 and died in 1670, wrote numerous papers on combinations
and on the theory of numbers, also on magic squares. It may
^ Torricelli's mathematical writings were published at Florence in 1644,
under the title Opera Geometrica ; see also a memoir by G. Loria, Bihliotheca
mathematical series 3, vol. i, pp. 75-89, Leipzig, 1900.
2 Frenicle's miscellaneous works, edited by De la Hire, were published in
the Mhnoires de I'Academie, vol. v, 1691.
CH.xv] DE LALOUBERE. N. MERCATOR 309
be interesting to add that he challenged Huygens to solve the
following system of equations in integers, x- + i/'^ — z^, x^ = u- + v^,
X ~y^u-v. A solution was given by M. Pepin in 1880.
De Laloubdre. Antoine de Lalouhere, a Jesuit, born in
Languedoc in 1600 and died at Toulouse in 1664, is chiefly
celebrated for an incorrect solution of Pascal's problems on
the cycloid, which he gave in 1660, but he has a better claim
to distinction in having been the first mathematician to study
the properties of the helix.
N. Mercator. Nicholas Mercator (sometimes known as
Kauffmann) was born in Holstein about 1620, but resided most
of his life in England. He went to France in 1683, where he
designed and constructed the fountains at Versailles, but the
payment agreed on was refused unless he would turn Catholic ;
he died of vexation and poverty in Paris in 1687. He wrote a
treatise on logarithms entitled Logarithmo-technica, published in
1668, and discovered the series
log (1 + A') = 07 - \x'^ +lx^-\x^+ ... ',
he proved this by writing the equation of a hyperbola in the form
v=^ = \ -x + x'^ -x^-\- ....
1 -\-x
to which Wallis's method of quadrature could be applied. The
same series had been independently discovered by Saint-Vincent.
Barrow.^ Isaxic Barrow was born in London in 1630,
and died at Cambridge in 1677. He went to school first at
Charterhouse (where he was so troublesome that his father was
heard to pray that if it pleased God to take any of his children
he could best spare Isaac), and subsequently to Felstead. He
completed his education at Trinity College, Cambridge ; after
taking his degree in 1648, he was elected to a fellowship in
1649 ; he then resided for a few years in college, but in 1655 he
was driven out by the persecution of the Independents. He
spent the next four years in the East of Europe, and after many
^ Barrow's mathematical works, edited by W. Whewell, were issued at
Cambridge in 1860.
310 HISTORY OF MATHEMATICS [ch. xv
adventures returned to England in 1659. He was ordained
the next year, and appointed to the professorship of Greek at
Cambridge. In 1662 he was made professor of geometry at
Gresham College, and in 1663 was selected as the first occupier
of the Lucasian chair at Cambridge. He resigned the latter
to his pupil Newton in 1669, whose superior abilities he recog-
nized and frankly acknowledged. For the remainder of his
life he devoted himself to the study of divinity. He was
appointed master of Trinity College in 1672, and held the post
until his death.
He is described as " low in stature, lean, and of a pale com-
plexion," slovenly in his dress, and an inveterate smoker. He
was noted for his strength and courage, and once when travelling
in the East he saved the ship by his own prowess from capture
by pirates. A ready and caustic wit made him a favourite of
Charles II., and induced the courtiers to respect even if they
did not appreciate him. He wrote with a sustained and
somewhat stately eloquence, and with his blameless life and
scrupulous conscientiousness was an impressive personage of
the time.
His earliest work was a complete edition of the Elements
of Euclid, which he issued in Latin in 1655, and in English
in 1660; in 1657 he published an edition of the Data. His
lectures, delivered in 1664, 1665, and 1666, were published in
1683 under the title Lectiones Matheniaticae -, these are mostly
on the metaphysical basis for mathematical truths. His
lectures for 1667 were published in the same year, and suggest
the analysis by which Archimedes was led to his chief results.
In 1669 he issued his Lectiones Opticae et Geometricae. It is
said in the preface that Newton revised and corrected these
lectures, adding matter of his own, but it seems probable from
Newton's remarks in the fiuxional controversy that the additions
were confined to the parts which dealt with optics. This, which
is his most important work in mathematics, was republished
with a few minor alterations in 1674. In 1675 he pubHshed an
edition with numerous comments of the first four books of the
CH. XV]
BARROW
311
Conies of Apollonius, and of the extant works of Archimedes
and Theodosius.
In the optical lectures many problems connected with the
reflexion and refraction of light are treated with ingenuity.
The geometrical focus of a point seen by reflexion or refraction
is defined ; and it is explained that the image of an object is
the locus of the geometrical foci of every point on it. Barrow
also worked out a few of the easier properties of thin lenses,
and considerably simplified the Cartesian explanation of the
rainbow.
The geometrical lectures contain some new ways of deter-
mining the areas and tangents of curves. The most celebrated
of these is the method given for the determination of tangents
to curves, and this is sufiiciently important to require a detailed
notice, because it illustrates the way in which Barrow, Hudde,
and Sluze were working on the lines suggested by Fermat
towards the methods of the differential calculus. Fermat had
observed that the tangent at a point P on a curve was deter-
mined if one other point besides P on it were known ; hence,
if the length of the subtangent MT could be found (thus
O T N M
determining the point T), then the line TP would be the
required tangent. Now Barrow remarked that if the abscissa
and ordinate at a point Q adjacent to P were drawn, he got a
small triangle PQP (which he called the differential triangle,
because its sides PP and PQ were the differences of the
abscissae and ordinates of P and Q), so that
312 HISTORY OF MATHEMATICS [ch. xv
TM'.MP^QR.RP.
To find QR : RP lie supposed that x, y were the co-ordinates of
P, and x-e^ y-a those of Q (Barrow actually used p for x and
m for y, but I alter these to agree with the modern practice).
Substituting the co-ordinates of Q in the equation of the curve,
and neglecting the squares and higher powers of e and a as
compared with their first powers, he obtained e : a. The ratio
a\e was subsequently (in accordance with a suggestion made
by Sluze) termed the angular coefficient of the tangent at
the point.
Barrow applied this method to the curves (i) x'^ix^ -f y'^) = r^y^ ;
(ii) x^ + y^ = r^; (iii) x^ + y^ = rxy, called la galaiide ; (iv)
y = {r - x) tan ttx/^t, the quadratrix ; and (v) y = r tan irxj^r. It
will be sufficient here if I take as an illustration the simpler
case of the parabola y'^=px. Using the notation given
above, we have for the point P, y'^^px; and for the point
Qj (y - of- —p{x - e). Subtracting we get 2ay - a^ =p&. But,
if a be an infinitesimal quantity, a^ must be infinitely smaller
and therefore may be neglected when compared with the
quantities 2ay &,nd pe. Hence 2ay=pej that is, e :a = 2y :p.
Therefore TM : y = e -.a = 2y : p. Hence TM = 2y^/p = 2x,
This is exactly the procedure of the differential calculus,
except that there we have a rule by which we can get the ratio
a/e or dyjdx directly without the labour of going through a
calculation similar to the above for every separate case.
Brouncker. William, Viscount Brouncker, one of the
founders of the Royal Society of London, born about 1620,
and died on April 5, 1684, was among the most brilliant
mathematicians of this time, and was in intimate relations
with Wallis, Fermat, and other leading mathematicians. I
mentioned above his curious reproduction of Brahmagupta's
solution of a certain indeterminate equation. Brouncker proved
that the area enclosed between the equilateral hyperbola xy=^\,
the axis of x, and the ordinates x=\ and ^r = 2, is equal
either to
CH.xvJ BROUNCKER. JAMES GREGORY 313
111 ^111
He also worked out other similar expressions for different
areas bounded by the hyberbola and straight lines. He wrote
on the rectification of the parabola and of the cycloid. ^ It is
noticeable that he used infinite series to express quantities
whose values he could not otherwise determine. In answer to
a request of Wallis to attempt the quadrature of the circle he
shewed that the ratio of the area of a circle to the area of the
circumscribed square, that is, the ratio of tt to 4, is equal to
the ratio of
1^ 12 32 52 72
1 + 2 +2 +2 +2 +...
to 1. Continued fractions 2 had been employed by Bombelli
in 1572, and had been systematically used by Cataldi in his
treatise on finding the square roots of numbers, published at
Bologna in 1613. Their properties and theory were given by
Huygens, 1703, and Euler, 1744.
James Gregory. James Gregory, born at Drumoak near
Aberdeen in 1638, and died at Edinburgh in October 1675, was
successively professor at St. Andrews and Edinburgh. In 1660
he published his Optica Promota, in which the reflecting
telescope known by his name is described. In 1667 he issued
his Vera Circuli et Hyperbolae Quadratura, in which he shewed
how the areas of the circle and hyperbola could be obtained in
the form of infinite convergent series, and here (I believe for
the first time) we find a distinction drawn between convergent
and divergejit series. This work contains a remarkable geo-
metrical proposition to the effect that the ratio of the area of
any arbitrary sector of a circle to that of the inscribed or
circumscribed regular polygons is not expressible by a finite
^ On these investigations, see his papers in the Philosophical Trans-
actions, London, 1668, 1672, 1673, and 1678.
2 On the history of continued fractions, see papers by S. Giinther and
A. Favaro in Bonconipagni's Bulletino di hibliograjia, Rome, 1874, vol. vii, pp.
213, 451, 533 ; and Cantor, vol. ii, pp. 622, 762, 766. Bombelli used them
in 1572 ; but Cataldi introduced the usual notation for them.
314 HISTORY OF MATHEMATICS [ch. xv
number of algebraical terms. Hence lie inferred that the
quadrature of a circle was impossible; this was accepted by
Montucla, but it is not conclusive, for it is conceivable that
some particular sector might be squared, and this particular
sector might be the whole circle. This book contains also the
earliest enunciation of the expansions in series of sin x^ cos ^,
sin~^^ or arc sin x^ and cos~^^ or arc cos x. It was reprinted
in 1668 with an appendix, Geometriae Pars^ in which Gregory
explained how the volumes of solids of revolution could be
determined. In 1671, or perhaps earlier, he established the
theorem that
(9 = tan ^-ltan3 6'+i tan^ 6'-...,
the result being true only if 6 lie between - Jtt and Jtt. This
is the theorem on which many of the subsequent calculations
of approximations to the numeral value of tt have been based.
Wren. Sir Christopher Wren was born at Knoyle, Wilt-
shire, on October 20, 1632, and died in London on February
25, 1723. Wren's reputation as a mathematician has been
overshadowed by his fame as an architect, but he was Savilian
professor of astronomy at Oxford from 1661 to 1673, and for
some time president of the Royal Society. Together with
Wallis and Huygens he investigated the laws of collision of
bodies ; he also discovered the two systems of generating lines
on the hyperboloid of one sheet, though it is probable that
he confined his attention to a hyperboloid of re volution. ^
Besides these he wrote papers on the resistance of fluids, and
the motion of the pendulum. He was a friend of Newton
and (like Huygens, Hooke, Halley, and others) had made
attempts to shew that the force under which the planets move
varies inversely as the square of the distance from the sun.
Wallis, Brouncker, Wren, and Boyle (the last-named being
a chemist and physicist rather than a mathematician) w^ere the
leading philosophers who founded the Royal Society of London.
The society arose from the self-styled " indivisible college " in
^ See the Philosophical Transactions London, 1669.
CH.XV] WREN. HOOKE. COLLINS 315
London in 1645 ; most of its members moved to Oxford during
the civil war, where Hooke, who was then an assistant in Boyle's
laboratory, joined in their meetings ; the society was formally
constituted in London in 1660, and was incorporated on July
15, 1662. The French Academy was founded in 1666, and
the Berlin Academy in 1700. The Accademia dei Lincei was
founded in 1603, but was dissolved in 1630.
Hooke. Robert Iloohe, born at Freshwater on July 18,
1635, and died in London on March 3, 1703, was educated at
Westminster, and Christ Church, Oxford, and in 1665 became
professor of geometry at Gresham College, a post which he
occupied till his death. He is still known by the law which
he discovered, that the tension exerted by a stretched string
is (within certain limits) proportional to the extension, or, in
other words, that the stress is proportional to the strain. He
invented and discussed the conical pendulum, and was the
first to state explicitly that the motions of the heavenly bodies
were merely dynamical problems. He was as jealous as he was
vain and irritable, and accused both Newton and Huygens of
unfairly appropriating his results. Like Huygens, Wren, and
Halley, he made efforts to find the law of force under which the
planets move about the sun, and he believed the law to be
that of the inverse square of the distance. He, like Huygens,
discovered that the small oscillations of a coiled spiral spring
were practically isochronous, and was thus led to recommend
(possibly in 1658) the use of the balance spring in watches.
He had a watch of this kind made in London in 1675; it was
finished just three months later than a similar one made in
Paris under the directions of Huygens.
Collins. John Collins^ born near Oxford on March 5,
1625, and died in London on November 10, 1683, was a man
of great natural ability, but of slight education. Being devoted
to mathematics, he spent his spare time in correspondence with
the leading mathematicians of the time, for whom he was
always ready to do anything in his power, and he has been
described — not inaptly — as the English Mersenne. To him
316 HISTORY OF MATHEMATICS [ch. xv
we are indebted for much information on the details of the
discoveries of the period.^
Pell. Another mathematician who devoted a considerable
part of his time to making known the discoveries of others, and
to correspondence with leading mathematicians, was John Pell.
Pell was born in Sussex on March 1, 1610, and died in London
on December 10, 1685. He was educated at Trinity College,
Cambridge ; he occupied in succession the mathematical chairs
at Amsterdam and Breda ; he then entered the English diplo-
matic service; but finally settled in 1661 in London, where he
spent the last twenty years of his life. His chief works were
an edition, with considerable new matter, of the Algebra by
Branker and Rhonius, London, 1668; and a table of square
numbers, London, 1672.
Sluze. Rene Francois Walther de Sluze (Slusius), canon of
Liege, born on July 7, 1622, and died on March 19, 1685,
found for the subtangent of a curve f(x, ^) = 0 an expression
df /df
which is equivalent to - j/^ / g— ; he wrote numerous tracts, ^
and in particular discussed at some length spirals and points of
inflexion.
Viviani. Vincenzo Viviani, a pupil of Galileo and Torricelli,
born at Florence on April 5, 1622, and died there on September
22, 1703, brought out in 1659 a restoration of the lost book of
Apollonius on conic sections, and in 1701 a restoration of the
work of Aristaeus. He explained in 1677 how an angle could
be trisected by the aid of the equilateral hyperbola or the
conchoid. In 1692 he proposed the problem to construct four
windows in a hemispherical vault so that the remainder of the
surface can be accurately determined ; a celebrated problem, of
which analytical solutions were given by Wallis, Leibnitz, David
Gregory, and James Bernoulli.
^ See the Commercium Epistolicum, and S. P. Rigaud's Correspondence of
Scientific Men of the Seventeenth Century^ Oxford, 1841.
2 Some of his papers were published by Le Paige in vol. xvii of
Boncompagni's Bulletino di hihliografia, Rome, 1884.
CH.XV] TSCHIRNHAUSEN. DE LA HIRE. 317
Tschirnhausen. Ehrenfried Walther von Tschirnhaivsen was
born at Kislingswalde on April 10, 1631, and died at Dresden
on October 11, 1708. In 1682 he worked out the theory of
caustics by reflexion, or, as they were usually called, catacaustics,
and shewed that they were rectifiable. This was the second
case in which the envelope of a moving line was determined.
He constructed burning mirrors of great power. The trans-
formation by which he removed certain intermediate terms from
a given algebraical equation is well known ; it was published in
the Acta Eruditorum for 1683.
De la Hire. Philippe De la Hire (or Lahire), born in Paris
on March 18, 1640, and died there on April 21, 1719, wrote on
graphical methods, 1673 ; on the conic sections, 1685 ; a treatise
on epicycloids, 1694; one on roulettes, 1702; and, lastly,
another on conchoids, 1708. His works on conic sections and
epicycloids were founded on the teaching of Desargues, whose
favourite pupil he was. He also translated the essay of
Moschopulus on magic squares, and collected many of the
theorems on them which were previously known; this was
published in 1705.
Roemer. Olof Roemer, born at Aarhuus on September 25,
1644, and died at Copenhagen on September 19, 1710, was the
first to measure the velocity of light ; this was done in 1675 by
means of the eclipses of Jupiter's satellites. He brought the
transit and mural circle into common use, the altazimuth having
been previously generally employed, and it was on his recom-
mendation that astronomical observations of stars were subse-
quently made in general on the meridian. He was also the first
to introduce micrometers and reading microscopes into an obser-
vatory. He also deduced from the properties of epicycloids the
form of the teeth in toothed- wheels best fitted to secure a uniform
motion.
RoUe. Michel Rolle, born at Ambert on April 21, 1652,
and died in Paris on November 8, 1719, wrote an algebra in
1689, which contains the theorem on the position of the roots
of an equation which is known by his name. He published in
318 HISTORY OF MATHEMATICS [cH. xv
1696 a treatise on the solutions of equations, whether deter-
minate or indeterminate, and he produced several other minor
works. He taught that the differential calculus, which, as we
shall see later, had been introduced towards the close of the
seventeenth century, was nothing but a collection of ingenious
fallacies.
319
CHAPTER XVI.
THE LIFE AND WORKS OF NEWTON.^
The mathematicians considered in the last chapter commenced
the creation of those processes which distinguish modern mathe-
matics. The extraordinary abilities of Newton enabled him
within a few years to perfect the more elementary of those
processes, and to distinctly advance every branch of mathe-
matical science then studied, as well as to create some new
subjects. Newton was the contemporary and friend of Wallis,
Huygens, and others of those mentioned in the last chapter, but
though most of his mathematical work was done between the
years 1665 and 1686, the bulk of it was not printed — at any
rate in book-form — till some years later.
I propose to discuss the works of Newton more fully than
those of other mathematicians, partly because of the intrinsic
importance of his discoveries, and partly because this book is
mainly intended for English readers, and the development of
mathematics in Great Britain was for a century entirely in the
hands of the Newtonian school.
^ Newton's life and works are discussed in The Memoirs of Neictmi, by D.
Brewster, 2 volumes, Edinburgh, second edition, 1860. An edition of most
of Newton's works was published by S. Horsley in 5 volumes, London, 1779-
1785 ; and a bibliography of them was issued by G. J. Gray, Cambridge,
second edition, 1907 ; see also the catalogue of the Portsmouth Collection of
Newton's papers, Cambridge, 1888. My Essay on tlie Genesis, Contents, and
History of Newton's Principia, London, 1893, may be also consulted.
320 THE LIFE AND WORKS OF NEWTON [ch. xvi
Isaxic Newton was born in Lincolnshire, near Grantham, on
December 25, 1642, and died at Kensington, London, on March
20, 1727. He was educated at Trinity College, Cambridge, and
lived there from 1661 till 1696, during which time he produced
the bulk of his work in mathematics; in 1696 he was appointed
to a valuable Government office, and moved to London, where
he resided till his death.
His father, who had died shortly before Newton was born,
was a yeoman farmer, and it was intended that Newton should
carry on the paternal farm. He was sent to school at Grantham,
where his learning and mechanical proficiency excited some
attention. In 1656 he returned home to learn the business of a
farmer, but spent most of his time solving problems, making
experiments, or devising mechanical models ; his mother noticing
this, sensibly resolved to find some more congenial occupation
for him, and his uncle, having been himself educated at Trinity
College, Cambridge, recommended that he should be sent there.
In 1661 Newton accordingly entered as a student at Cam-
bridge, where for the first time he found himself among
surroundings which were likely to develop his powers. He
seems, however, to have had but little interest for general society
or for any pursuits save science and mathematics. Luckily he
kept a diary, and we can thus form a fair idea of the course of
education of the most advanced students at an English univer-
sity at that time. He had not read any mathematics before
coming into residence, but was acquainted with Sanderson's
Logic, which was then frequently read as preliminary to mathe-
matics. At the beginning of his first October term he happened
to stroll down to Stourbridge Fair, and there picked up a book
on astrology, but could not understand it on account of the
geometry and trigonometry. He therefore bought a Euclid, and
was surprised to find how obvious the propositions seemed. He
thereupon read Oughtred's Clavis and Descartes's Gconietrie, the
latter of which he managed to master by himself, though with
some difficulty. The interest he felt in the subject led him to
take up mathematics rather than chemistry as a serious study.
CH.xvi] NEWTON'S VIEWS ON GRAVITY, 1666 321
His subsequent mathematical reading as an undergraduate was
founded on Kepler's Optics, the works of Vieta, van Schooten's
Miscellanies, Descartes's Gemnetrie, and Wallis's Arithmetica
Infinitorum : he also attended Barrow's lectures. At a later
time, on reading Euclid more carefully, he formed a high
opinion of it as an instrument of education, and he used to
express his regret that he had not applied himself to geometry
before proceeding to algebraic analysis.
There is a manuscript of his, dated May 28, 1665, written in
the same year as that in which he took his B.A. degree, which
is the earliest documentary proof of his invention of fluxions.
It was about the same time that he discovered the binomial
theorem.^
On account of the plague the college was sent down during
parts of the year 1665 and 1666, and for several months at this
time Newton lived at home. This period was crowded with
brilliant discoveries. He thought out the fundamental prin-
ciples of his theory of gravitation, namely, that every particle of
matter attracts every other particle, and he suspected that the
attraction varied as the product of their masses and inversely as
the square of the distance between them. He also worked out
the fluxional calculus tolerably completely : thus in a manuscript
dated November 13, 1665, he used fluxions to find the tangent
and the radius of curvature at any point on a curve, and in
October 1666 he applied them to several problems in the theory
of equations. Newton communicated these results to his friends
and pupils from and after 1669, but they were not published in
print till many years later. It was also whilst staying at home
at this time that he devised some instruments for grinding
lenses to particular forms other than spherical, and perhaps he
decomposed solar light into different colours.
Leaving out details and taking round numbers only, his
reasoning at this time on the theory of gravitation seems to
have been as follows. He suspected that the force which
retained the moon in its orbit about the earth was the same as
1 See below, pp. 327, 341.
Y
322 THE LIFE AND WORKS OF NEWTON [ch. xvi
terrestrial gravity, and to verify this hypothesis he proceeded
thus. He knew that, if a stone were allowed to fall near the
surface of the earth, the attraction of the earth (that is, the
weight of the stone) caused it to move through 16 feet in
one second. The moon's orbit relative to the earth is nearly
a circle ; and as a rough approximation, taking it to be so, he
knew the distance of the moon, and therefore the length of its
path ; he also knew the time the moon took to go once round
it, namely, a month. Hence he could easily find its velocity at
any point such as M. He could therefore find the distance
MT through which it would move in the next second if it
were not pulled by the earth's attraction. At the end of that
second it was however at M', and therefore the earth E must
have pulled it through the distance TM' in one second (assuming
the direction of the earth's pull to be constant). Now he and
several physicists of the time had conjectured from Kepler's third
law that the attraction of the earth on a body would be found
to decrease as the body was removed farther away from the
earth inversely as the square of the distance from the centre
of the earth ; i if this were the actual law and if gravity were
the sole force which retained the moon in its orbit, then TM'
should be to 16 feet inversely as the square of the distance
' An argument leading to this result is given below on page 332,
CH. xvi] NEWTON'S VIEWS ON GRAVITY, 1666 323
of the moon from the centre of the earth to the square of
the radius of the earth. In 1679, when he repeated the
investigation, T2f' was found to have the value which was
required by the hypothesis, and the verification was complete ;
but in 1666 his estimate of the distance of the moon was
inaccurate, and when he made the calculation he found that
TM' was about one-eighth less than it ought to have been on
his hypothesis.
This discrepancy does not seem to have shaken his faith in
the belief that gravity extended as far as the moon and varied
inversely as the square of the distance ; but, from Whiston's
notes of a conversation with Newton, it would seem that
Newton inferred that some other force — probably Descartes's
vortices — acted on the moon as well as gravity. This statement
is confirmed by Pemberton's account of the investigation. It
seems, moreover, that Newton already believed firmly in the
principle of universal gravitation, that is, that every particle
of matter attracts every other particle, and suspected that the
attraction varied as the product of their masses and inversely
as the square of the distance between them; but it is certain
that he did not then know what the attraction- of a spherical
mass on any external point would be, and did not think it
likely that a particle would be attracted by the earth as if the
latter were concentrated into a single particle at its centre.
On his return to Cambridge in 1667 Newton was elected
to a fellowship at his college, and permanently took up his
residence there. In the early part of 1669, or perhaps in
1668, he revised Barrow's lectures for him. The end of the
fourteenth lecture is known to have been written byN'ewton,
but how much of the rest is due to his suggestions cannot now
be determined. As soon as this was finished he was asked by
Barrow and Collins to edit and add notes to a translation of
Kinckhuysen's Algebra ; he consented to do this, but on condition
that his name should not appear in the matter. In 1670 he also
began a systematic exposition of his analysis by infinite series,
the object of which was to express the ordinate of a curve
324 THE LIFE AND WORKS OF NEWTON [ch. xvi
in an infinite algebraical series every term of which can be
integrated by Wallis's rule ; his results on this subject had been
communicated to Barrow, Collins, and others in 1669. This
was never finished: the fragment was published in 1711, but
the substance of it had been printed as an appendix to the
Optics in 1704. These works were only the fruit of Newton's
leisure, most of his time during these two years being given up
to optical researches.
In October, 1669, Barrow resigned the Lucasian chair in
favour of Newton. During his tenure of the professorship,
it was Newton's practice to lecture "[publicly once a week, for
from half-an-hour to an hour at a time, in one term of each
year, probably dictating his lectures as rapidly as they could
be taken down; and in the week following the lecture to
devote four hours to appointments which he gave to students
who wished to come to his rooms to discuss the results of the
previous lecture. He never repeated a course, which usually
consisted of nine or ten lectures, and generally the lectures of
one course began from the point at which the preceding course
had ended. The manuscrii3ts of his lectures for seventeen out
of the first eighteen years of his tenure are extant.
When first appointed Newton chose optics for the subject
of his lectures and researches, and before the end of 1669 he
had worked out the details of his discovery of the decom-
position of a ray of white light into rays of different colours
by means of a prism. The complete explanation of the theory
of the rainbow followed from this discovery. These discoveries
formed the subject-matter of the lectures which he delivered
as Lucasian professor in the years 1669, 1670, and 1671. The
chief new results were embodied in a paper communicated
to the Royal Society in February, 1672, and subsequently
published in the Philosophical Transactions. The manuscript
of his original lectures was printed in 1729 under the title
Lectiones Opticae. This work is divided into two books, the
first of which contains four sections and the second five. The
first section of the first book deals with the decomposition
CH.XVI] NEWTON S VIEWS ON OPTICS 325
of solar light by a prism in consequence of the unequal re-
frangibility of the rays that compose it, and a description
of his experiments is added. The second section contains an
account of the method which Newton invented for the deter-
mining the coefficients of refraction of different bodies. This
is done by making a ray pass through a prism of the material
so that the deviation is a minimum ; and he proves that, if the
angle of the prism be i and the deviation of the ray be S, the
refractive index will be sin | (^.+ 8) cosec J L The third section
is on refractions at plane surfaces ; he here shews that if a
ray pass through a prism with minimum deviation, the angle
of incidence is equal to the angle of emergence; most of this
section is devoted to geometrical solutions of different problems.
The fourth section contains a discussion of refractions at curved
surfaces. The second book treats of his theory of colours and
of the rainbow.
By a curious chapter of accidents Newton failed to correct
the chromatic aberration of two colours by means of a couple
of prisms. He therefore abandoned the hope of making a
refracting telescope which should be achromatic, and instead
designed a reflecting telescope, probably on the model of a
small one which he had made in 1668. The form he used is
that still known by his name ; the idea of it was naturally
suggested by Gregory's telescope. In 1672 he invented a
reflecting microscope, and some years later he invented the
sextant which was rediscovered by J. Hadley in 1731.
His professorial lectures from 1673 to 1683 were on
algebra and the theory of equations, and are described below ;
but much of his time during these years was occupied with
other investigations, and I may remark that throughout his
life Newton must have devoted at least as much attention to
chemistry and theology as to mathematics, though his conclusions
are not of sufficient interest to require mention here. His theory
of colours and his deductions from his optical experiments were
at first attacked with considerable vehemence. The correspond-
ence which this entailed on Newton occupied nearly all his
326 THE LIFE AND WOEKS OF NEWTON [ch.xvi
leisure in tlie years 1672 to 1675, and proved extremely distaste-
ful to him. Writing on December 9, 1675, he says, "I was so
persecuted with discussions arising out of my theory of light,
that I blamed my own imprudence for parting with so substantial
a blessing as my quiet to run after a shadow." Again, on
November 18, 1676, he observes, "I see I have made myself a
slave to philosophy ; but, if I get rid of Mr. Linus's business, I
will resolutely bid adieu to it eternally, excepting what I do for
my private satisfaction, or leave to come out after me ; for I see
a man must either resolve to put out nothing new, or to become
a slave to defend it." The unreasonable dislike to have his
conclusions doubted or to be involved in any correspondence
about them was a prominent trait in Newton's character.
Newton was deeply interested in the question as to how the
effects of light were really produced, and by the end of 1675 he
had worked out the corpuscular or emission theory, and had
shewn how it would account for all the various phenomena of
geometrical optics, such as reflexion, refraction, colours, diffrac-
tion, &c. To do this, however, he was obliged to add a some-
what artificial rider, that the corpuscles had alternating fits of
easy reflexion and easy refraction communicated to them by an
ether which filled space. The theory is now known to be
untenable, but it should be noted that Newton enunciated it as
a hypothesis from which certain results would follow : it would
seem that he believed the wave theory to be intrinsically more
probable, but it was the difficulty of explaining diffraction on
that theory that led him to suggest another hypothesis.
Newton's corpuscular theory was expounded in memoirs com-
municated to the Royal Society in December 1675, which are
substantially reproduced in his Optics, published in 1704. In
the latter work he dealt in detail with his theory of fits of easy
reflexion and transmission, and the colours of thin plates, to
which he added an explanation of the colours of thick plates
[bk. II, part 4] and observations on the inflexion of light [bk.
III].
Two letters written by Newton in the year 1676 are sufficiently
CH. xvi] NEWTON ON EXPRESSIONS IN SERIES 327
interesting to justify an allusion to them.i Leibnitz, in 1674, in
a correspondence with Oldenburg, wrote saying that he possessed
" general analytical methods depending on infinite series."
Oldenburg, in reply, told him that Newton and Gregory had
used such series in their work. In answer to a request for
information, Newton wrote on June 13, 1676, giving a brief
account of his method. He here enunciated the binomial
theorem, which he stated, in ejffect, in the form that if A, B, C, D,
. . . denote the successive terms in the expansion of (P + PQ)"*''",
then
where A = P"'". He gave examples of its use. He also gave the
expansion of sin~^^, from which he deduced that of sin x : this
seems to be the earliest known instance of a reversion of series.
He also inserted an expression for the rectification of an elliptic
arc in an infinite series.
Leibnitz wrote on August 27 asking for fuller details j and
Newton, on October 24, 1676, sent, through Oldenburg, an
account of the way in which he had been led to some of his
results. The main results may be briefly summarized. He
begins by saying that altogether he had used three methods
for expansion in series. His first was arrived at from the
study of the method of interpolation. Thus, by considering
the series of expressions for (l-a?^)^''^, (l-a;^)^/^, (1 -a;^)'*/^,...,
he deduced by interpolations a rule connecting the successive
coefficients in the expansions of (1 - it;^)!/^^ (1 -a;^)^/^,... ; and
then by analogy obtained the expression for the general term
in the expansion of a binomial. He then tested his result in
various ways; for instance in the case of (1— ic^)^^, by ex-
tracting the square root of 1 - x^^ more arithmetico, and by
forming the square of the expansion of (1 - x^Y^^ which reduced
to 1 - x'^. He also used the series to determine the areas of
the circle and the hyperbola in infinite series, and found that
1 See J. Wallis, Opera, vol. iii, Oxford, 1699, p. 622 et seq.
328 THE LIFE AND WORKS OF NEWTON [ch. xvi
the results were the same as those he had arrived at by other
means.
Having established this result, he then discarded the method
of interpolation, and employed his binomial theorem to express
(when possible) the ordinate of a curve in an infinite series
in ascending powers of the abscissa, and thus by Wallis's
method he obtained expressions in infinite series for the areas
and arcs of curves in the manner described in the appendix to
his Optics and in his De Analysi per Equationes Numero Termi-
noTum Infinitas. He stales that he had employed this second
method before the plague in 1665-66, and goes on to say that
he was then obliged to leave Cambridge, and subsequently
(presumably on his return to Cambridge) he ceased to pursue
these ideas, as he found that Nicholas Mercator had employed
some of them in his Logarithmo-technica, published in 1668;
and he supposed that the remainder had been or would be found
out before he himself was likely to publish his discoveries.
Newton next explains that he had also a third method, of
which (he says) he had about 1669 sent an account to Barrow
and Collins, illustrated by applications to areas, rectification,
cubature, &c. This was the method of fluxions ; but Newton
gives no description of it here, though he adds some illustrations
of its use. The first illustration is on the quadrature of the
curve represented by the equation
which he says can be effected as a sum of {m + 1 )ln terms if
(m + 1 )jn be a positive integer, and which he thinks cannot
otherwise be effected except by an infinite series.^ He also gives
a list of other forms which are immediately integrable, of which
the chief are
rjQmn-\ /p(m+i/2)n-l
a + ox"' + cx^'^ a + bx^^ + cx^'^ ^ ■ / j
^mn - 1 (a + hx"") ^'/^ (c + dx'^) - \ x"^'^ - « - 1 (a + hx'^) (c + dx'^) " ^^^ ;
^ This is not so, the integration is possible if jw + (wi + l)/?j be an integer.
CH.xvi] CORRESPONDENCE WITH LEIBNITZ 329
where m is a positive integer and n is any number whatever.
Lastly, he points out that the area of any curve can be easily
determined approximately by the method of interpolation
described below in discussing his Methodus Differentialis.
At the end of his letter Newton alludes to the solution of the
" inverse problem of tangents," a subject on which Leibnitz had
asked for information. He gives formulae for reversing any
series, but says that besides these formulae he has two methods
for solving such questions, which for the present he will not
describe except by an anagram which, being read, is as follows,
" Una methodus consistit in extractione fluentis quantitatis ex
aequatione simul involvente fluxionem ejus : altera tantum in
assumptione seriei pro quantitate qualibet incognita ex qua
caetera commode derivari possunt, et in collatione terminorum
homologorum aequationis resultantis, ad eruendos terminos
assumptae seriei."
He implies in this letter that he is worried by the questions
he is asked and the controversies raised about every new matter
which he produces, which shew his rashness in publishing "quod
umbram captando eatenus perdideram quietem meam, rem prorsus
substantialem."
Leibnitz, in his answer, dated June 21, 1677, explains his
method of drawing tangents to curves, which he says proceeds
" not by fluxions of lines, but by the differences of numbers " ;
and he introduces his notation of dx and dy for the infini-
tesimal differences between the co-ordinates of two consecutive
points on a curve. He also gives a solution of the problem to
find a curve whose subtangent is constant, which shews that he
could integrate.
In 1679 Hooke, at the request of the Royal Society, wrote
to Newton expressing a hope that he would make further com-
munications to the Society, and informing him of various facts
then recently discovered. Newton replied saying that he had
abandoned the study of philosophy, but he added that the
earth's diurnal motion might be proved by the experiment of
observing the deviation from the perpendicular of a stone
330 THE LIFE AND WORKS OF NEWTON [ch. xvi
dropped from a height to the ground — an experiment which
was subsequently made by the Society and succeeded. Hooke
in his letter mentioned Picard's geodetical researches ; in these
Picard used a value of the radius of the earth which is substan-
tially correct. This led Newton to repeat, with Picard's data,
his calculations of 1666 on the lunar orbit, and he thus verified
his supposition that gravity extended as far as the moon and
varied inversely as the square of the distance. He then pro-
ceeded to consider the general theory of motion of a particle
under a centripetal force, that is, one directed to a fixed point,
and showed that the vector would sweep over equal areas in equal
times. He also proved that, if a particle describe an ellipse under
a centripetal force to a focus, the law must be that of the inverse
square of the distance from the focus, and conversely, that the
orbit of a particle projected under the influence of such a force
would be a conic (or, it may be, he thought only an ellipse).
Obeying his rule to publish nothing which could land him in a
scientific controversy these results -were locked up in his note-
books, and it was only a specific question addressed to him five
years later that led to their publication.
The Universal Arithmetic, which is on algebra, theory of
equations, and miscellaneous problems, contains the substance
of Newton's lectures during the years 1673 to 1683. His
manuscript of it is still extant ; Whiston ^ extracted a somewhat
reluctant permission from Newton to print it, and it was
published in 1707. Amongst several new theorems on various
points in algebra and the theory of equations Newton here
enunciates the following important results. He explains that
the equation whose roots are the solution of a given problem
will have as many roots as there are different possible cases;
^ William Whiston, born in Leicestershire on December 9, 1667, educated
at Clare College, Cambridge, of which society he was a fellow, and died in
London on August 22, 1752, wrote several works on astronomy. He acted as
Newton's deputy in the Lucasian chair from 1699, and in 1703 succeeded him
as professor, but he was expelled in 1711, mainly for theological reasons. He
was succeeded by Nicholas Saunderson, the blind mathematician, who was
born in Yorkshire in 1682, and died at Christ's College, Cambridge, on
April 19, 1739.
CH. x\ i] NEWTON'S LECTURES ON ALGEBRA 331
and he considers how it happens that the equation to which
a problem leads may contain roots which do not satisfy the
original question. He extends Descartes's rule of signs to give
limits to the number of imaginary roots. He uses the principle
of continuity to explain how two real and unequal roots may
become imaginary in passing through equality, and illustrates
this by geometrical considerations; thence he shews that
imaginary roots must occur in pairs. Newton also here gives
rules to find a superior limit to the positive roots of a numerical
equation, and to determine the approximate values of the
numerical roots. He further enunciates the theorem known by
his name for finding the sum of the nth powers of the roots of
an equation, and laid the foundation of the theory of symmetri-
cal functions of the roots of an equation.
The most interesting theorem contained in the work is his
attempt to find a rule (analogous to that of Descartes for real
roots) by which the number of imaginary roots of an equation
can be determined. He knew that the result which he obtained
was not universally true, but he gave no proof and did not
explain what were the exceptions to the rule. His theorem is as
follows. Suppose the equation to be of the nth degree arranged
in descending powers of x (the coefficient of x'^ being positive),
and suppose the n+1 fractions
-. ?i 2 ?i - 1 3 n -p + 1 p + 1 2 n ^
' tT^i' ^ii:^^ 2'"*' n-p "^' ■■'T7"r=T'
to be formed and written below the corresponding terms of the
equation, then, if the square of any term when multiplied by the
corresponding fraction be greater than the product of the terms
on each side of it, put a plus sign above it : otherwise put a
minus sign above it, and put a plus sign above the first and last
terms. Now consider any two consecutive terms in the original
equation, and the two symbols written above them. Then Ave
may have any one of the four foiloA\ang cases : (a) the terms of
the same sign and the symbols of the same sign ; (/?) the terms
of the same sign and the symbols of opposite signs; (y) the
332 THE LIFE AND WORKS OF NEWTON [ch. xvi
terms of opposite signs and the symbols of the same sign ; (S)
the terms of opposite signs and the symbols of opposite signs.
Then it has been shewn that the number of negative roots will
not exceed the number of cases (a), and the number of positive
roots will not exceed the number of cases (y) ; and therefore the
number of imaginary roots is not less than the number of cases
(fS) and (8). In other words the number of changes of signs in
the row of symbols written above the equation is an inferior
limit to the number of imaginary roots. Newton, however,
asserted that "you may almost know how many roots are
impossible" by counting the changes of sign in the series of
symbols formed as above. That is to say, he thought that in
general the actual number of positive, negative, and imaginary
roots could be got by the rule and not merely superior or
inferior limits to these numbers. But though he knew that the
rule was not universal he could not find (or at any rate did
not state) what were the exceptions to it : this problem was
subsequently discussed by Campbell, Maclaurin, Euler, and
other writers; at last in 1865 Sylvester succeeded in proving
the general result. ^
In August, 1684, Halley came to Cambridge in order to con-
sult Newton about the law of gravitation. Hooke, Huygens,
Halley, and Wren had all conjectured that the force of the
attraction of the sun or earth on an external particle varied
inversely as the square of the distance. These wTiters seem
independently to have shewn that, if Kepler's conclusions were
rigorously true, as to which they were not quite certain, the
law of attraction must be that of the inverse square. Probably
their argument was as follows. If v be the velocity of a planet,
r the radius of its orbit taken as a circle, and T its periodic
time, v = 27rr/T. But, if /be the acceleration to the centre of
the circle, we have /= v^/r. Therefore, substituting the above
value of y, /= iir^jT^. Now, by Kepler's third law, T^ varies
as r^ ; hence / varies inversely as r^. They could not, however,
^ See the Proceedings of the London Mathematical Society, 1865, vol. i.
no. 2.
CH. xvi] NEWTON'S DE MOTU, 1684 333
deduce from the law the orbits of the planets. Halley explained
that their investigations were stopped by their inability to solve
this problem, and asked Newton if he could find out what the
orbit of a planet would be if the law of attraction were that of
the inverse square. New^ton immediately replied that it was an
ellipse, and promised to send or wi'ite out afresh the demonstra-
tion of it which he had found in 1679. This was sent in
November, 1684.
Instigated by Halley, Newton now returned to the problem
of gravitation ; and before the autumn of 1684, he had worked
out the substance of propositions 1-19, 21, 30, 32-35 in the
first book of the Frincipia. These, together with notes on the
laws of motion and various lemmas, were read for his lectures
in the Michaelmas Term, 1684.
In November Halley received Newton's promised communi-
cation, which probably consisted of the substance of proposi-
tions 1, 11, and either proposition 17 or the first corollary of
proposition 13 ; thereupon Halley again went to Cambridge,
where he saw "a curious treatise, De Motu, drawn up since
August." Most likely this contained Newton's manuscript
notes of the lectures above alluded to : these notes are now
in the university library, and are headed ^^ De Motu Cor-
porum." Halley begged that the results might be published,
and finally secured a promise that they should be sent to the
Koyal Society : they were accordingly communicated to the
Society not later than February, 1685, in the paper De Motu,
which contains the substance of the following propositions in
the PmiczJ9za, book i, props. 1, 4, 6, 7, 10, 11, 15, 17, 32;
book II, props. 2, 3, 4.
It seems also to have been due to the influence and tact of
Halley at this visit in November, 1684, that Newton undertook
to attack the whole problem of gravitation, and practically
pledged himself to publish his results : these are contained in
the Principia. As yet Newton had not determined the attrac-
tion of a spherical body on an external point, nor had he
calculated the details of the planetary motions even if the
334 THE LIFE AND WORKS OF NEWTON [ch. xvi
members of the solar system could be regarded as points. The
first problem was solved in 1685, probably either in January
or February. "No sooner," to quote from Dr. Glaisher's
address on the bicentenary of the publication of the Principia^
"had Newton proved this superb theorem — and we know from
his own words that he had no expectation of so beautiful a
result till it emerged from his mathematical investigation —
than all the mechanism of the universe at once lay spread before
him. When he discovered the theorems that form the first
three sections of book i, when he gave them in his lectures of
1684, he was unaware that the suji and earth exerted their
attractions as if they were but points. How different must
these propositions have seemed to Newton's eyes when he
realized that these results, which he had believed to be only
approximately true when applied to the solar system, were
really exact ! Hitherto they had been true only in so far as he
could regard the sun as a point compared to the distance of
the planets, or the earth as a point compared to the distance
of the moon— a distance amounting to only about sixty times
the earth's radius — but now they were mathematically true, ex-
cepting only for the slight deviation from a perfectly spherical
form of the sun, earth, and planets. We can imagine the effect
of this sudden transition from approximation to exactitude in
stimulating Newton's mind to still greater efforts. It was now
in his power to apply mathematical analysis with absolute
precision to the actual problems of astronomy."
Of the three fundamental principles applied in the Principia
we may say that the idea that every particle attracts every
other particle in the universe was formed at least as early as
1666 ; the law of equable description of areas, its consequences,
and the fact that if the law of attraction were that of the
inverse square the orbit of a particle about a centre of force
would be a conic were proved in 1679 ; and, lastly, the discovery
that a sphere, whose density at any point depends only on the
distance from the centre, attracts an external point as if the
whole mass were collected at its centre was made in 1685.
CH. xvi] NEWTON'S Pi?/7\^C/P/^ 1685-1687 335
It was this last discovery tliat enabled him to apply the first
two principles to the phenomena of bodies of finite size.
The draft of the first book of the Principia was finished
before the summer of 1685, but the corrections and additions
took some time, and the book was not presented to the Royal
Society until April 28, 1686. This book is given up to the
consideration of the motion of particles or bodies in free space
either in known orbits, or under the action of known forces,
or under their mutual attraction ; and in particular to in-
dicating how the effects of disturbing forces may be calculated.
In it also Newton generalizes the law of attraction into a
statement that every particle of matter in the universe attracts
every other particle with a force which varies directly as the
product of their masses, and inversely as the square of the
distance between them ; and he thence deduces the law of
attraction for spherical shells of constant density. The book
is prefaced by an introduction on the science of dynamics,
which defines the limits of mathematical investigation. His
object, he says, is to apply mathematics to the phenomena
of nature ; among these phenomena motion is one of the
most important; now motion is the effect of force, and,
though he does not know what is the nature or origin of
force, still many of its effects can be measured ; and it is
these that form the subject-matter of the work.
The second book of the Principia was completed by the
summer of 1686. This book treats of motion in a resisting
medium, and of hydrostatics and hydrodynamics, with special
applications to waves, tides, and acoustics. He concludes it
by shewing that the Cartesian theory of vortices was in-
consistent both with the known facts and with the laws of
motion.
The next nine or ten months were devoted to the third
book. Probably for this originally he had no materials ready.
He commences by discussing when and how far it is justi-
fiable to construct hypotheses or theories to account for
known phenomena. He proceeds to apply the theorems
336 THE LIFE AND WORKS OF NEWTON [ch. xvi
obtained in the first book to the chief phenomena of the
solar system, and to determine the masses and distances of
the planets and (whenever sufficient data existed) of their
satellites. In particular the motion of the moon, the various
inequalities therein, and the theory of- the tides are worked
out in detail. He also investigates the theory of comets,
shews that they belong to the solar system, explains how
from three observations the orbit can be determined, and
illustrates his results by considering certain special comets.
The third book as we have it is but little more than a sketch
of what Newton had finally proposed to himself to accomplish ;
his original scheme is among the "Portsmouth papers," and
his notes shew that he continued to work at it for some years
after the publication of the first edition of the Principia : the
most interesting of his memoranda are those in which by
means of fluxions he has carried his results beyond the point
at which he was able to translate them into geometry.^
The demonstrations throughout the work are geometrical,
but to readers of ordinary ability are rendered unnecessarily
difficult by the absence of illustrations and explanations, and
by the fact that no clue is given to the method by which
Newton arrived at his results. The reason why it was pre-
sented in a geometrical form appears to have been that the
infinitesimal calculus was then unknown, and, had Newton
used it to demonstrate results which were in themselves
opposed to the prevalent philosophy of the time, the contro-
versy as to the truth of his results would have been hampered
by a dispute concerning the validity of the methods used
in proving them. He therefore cast the whole reasoning
into a geometrical shape which, if somewhat longer, can at
any rate be made intelligible to all mathematical students.
So closely did he follow the lines of Greek geometry that he
constantly used graphical methods, and represented forces,
velocities, and other magnitudes in the Euclidean way by
^ For a fuller account of the Principia see my Essay on the Genesis,
Contents, and History of Newton's Frincijoia, Loudon, 1893.
CH. xvi] NEWTON'S PRINCIPIA 337
straight lines {ex. gr. book i, lemma 10), and not by a certain
number of units. The latter and modern method had been
introduced by Wallis, and must have been familiar to Newton.
The effect of his confining himself rigorously to classical
geometry is that the Principia is written in a language which
is archaic, even if not unfamiliar.
The adoption of geometrical methods in the Principia for
purposes of demonstration does not indicate a preference on
Newton's part for geometry over analysis as an instrument
of research, for it is known now that Newton used the fluxional
calculus in the first instance in finding some of the theorems,
especially those towards the end of book i and in book ii ;
and in fact one of the most important uses of that calculus is
stated in book ii, lemma 2. But it is only just to remark
that, at the time of its publication and for nearly a century
afterwards, the differential and fluxional calculus were not fully
developed, and did not possess the same superiority over the
method he adopted which they do now ; and it is a matter for
astonishment that when Newton did employ the calculus he
was able to use it to so good an effect.
The printing of the work was slow, and it was not finally
published till the summer of 1687. The cost was borne by
Halley, who also corrected the proofs, and even put his own
researches on one side to press the printing forward. The
conciseness, absence of illustrations, and synthetical character
of the book restricted the numbers of those who were able to
appreciate its value; and, though nearly all competent critics
admitted the validity of the conclusions, some little time
elapsed before it affected the current beliefs of educated men.
I should be inclined to say (but on this point opinions differ
widely) that within ten years of its publication it was generally
accepted in Britain as giving a correct account of the laws of
the universe; it was similarly accepted within about twenty
years on the continent, except in France, where the Cartesian
hypothesis held its ground until Voltaire in 1733 took up the
advocacy of the Newtonian theory.
z
338 THE LIFE AND WOKKS OF NEWTON [ch. xvi
The manuscript of the Principia was finished by 1686.
Newton devoted the remainder of that year to his paper on
physical optics, the greater part of which is given up to the
subject of diffraction.
In 1687 James II. having tried to force the university to
admit as a master of arts a Roman Catholic priest who refused
to take the oaths of supremacy and allegiance, Newton took a
prominent part in resisting the illegal interference of the king,
and was one of the deputation sent to London to protect the
rights of the university. The active part taken by Newton in
this affair led to his being in 1689 elected member for the
university. This parliament only lasted thirteen months, and
on its dissolution he gave up his seat. He was subsequently
returned in 1701, but he never took any prominent part in
politics.
On his coming back to Cambridge in 1690 he resumed his
mathematical studies and correspondence, but probably did not
lecture. The two letters to Wallis, in which he explained his
method of fluxions and fluents, were written in 1692 and pub-
lished in 1693. Towards the close of 1692 and throughout the
two following years, Newton had a long illness, suffering from
insomnia and general nervous irritability. Perhaps he never
quite regained his elasticity of mind, and, though after his
recovery he shewed the same power in solving any question
propounded to him, he ceased thenceforward to do original
work on his own initiative, and it was somewhat difficult to
stir him to activity in new subjects.
In 1694 Newton began to collect data connected with the
irregularities of the moon's motion with the view of revising the
part of the Principia which dealt with that subject. To render
the observations more accurate, he forwarded to Flamsteed ^ a
^ John Flamsteed, born at Derby in 1646 and died at Greenwich in 1719,
was one of the most distinguished astronomers of this age, and the first
astronomer-royal. Besides much valuable work in astronomy, he invented
the system (published in 1680) of drawing maps by projecting the surface of
the sphere on an enveloping cone, which can then be unwrapped. His life
by R. F. Baily was published in London in 1835, but various statements in
CH. xvi] THE LIFE AND WORKS OF NEWTON 339
table of corrections for refraction which he had previously made.
This was not published till 1721, when Halley communicated it
to the Royal Society. The original calculations of Newton and
the papers connected with them are in the Portsmouth collection,
and shew that Newton obtained it by finding the path of a ray,
by means of quadratures, in a manner equivalent to the solution
of a differential equation. As an illustration of Newton's
genius, I may mention that even as late as 1754 Euler failed to
solve the same problem. In 1782 Laplace gave a rule for con-
structing such a table, and his results agree substantially with
those of Newton.
I do not suppose that Newton would in any case have pro-
duced much more original work after his illness ; but his
appointment in 1696 as warden, and his promotion in 1699
to the mastership of the Mint, at a salary of .£1500 a year,
brought his scientific investigations to an end, though it was
only after this that many of his previous investigations were
published in the form of books. In 1696 he moved to London,
in 1701 he resigned the Lucasian chair, and in 1703 he was
elected president of the Royal Society.
In 1704 Newton published his Optics, which contains the
results of the papers already mentioned. To the first edition
of this book were appended two minor works which have no
special connection with optics ; one being on cubic curves, the
other on the quadrature of curves and on fluxions. Both of
them were mani;scripts with which his friends and pupils
were familiar, but they were here published urhi et orhi for the
first time.
The first of these appendices is entitled Enumeratio Linearum
Tertii Ordinis ; ^ the object seems to be to illustrate the use of
analytical geometry, and as the application to conies was well
known, Newton selected the theory of cubics.
it should be read side by side with those in Brewster's life of Newton.
Flamsteed was succeeded as astronomer-royal by Edmund Halley (see below,
pp. 379-380).
^ On this work and its bibliography^ see my memoir in the Transactions
of the London Mathematical Society, 1891, vol. xxii, pp. 104-143.
340 THE LIFE AND WORKS OF NEWTON [ch. xvi
He begins with some general theorems, and classifies curves
according as their equations are algebraical or transcendental ;
the former being cut by a straight line in a number of points
(real or imaginary) equal to the degree of the curve, the latter
being cut by a straight line in an infinite number of points.
Newton then shews that many of the most important properties
of conies have their analogues in the theory of cubics, and he
discusses the theory of asymptotes and curvilinear diameters.
After these general theorems, he commences his detailed
examination of cubics by pointing out that a cubic must have
at least one real point at infinity. If the asymptote or tangent
at this point be at a finite distance, it may be taken for the
axis of y. This asymptote will cut the curve in three points
altogether, of which at least two are at infinity. If the third
point be at a finite distance, then (by one of his general theorems
on asymptotes) the equation can be written in the form
x'lp- + hy = ax^ + hx'^ + cx + d,
where the axes of x and y are the asymptotes of the hyperbola
which is the locus of the middle points of all chords drawn
parallel to the axis of y ; while, if the third point in which this
asymptote cuts the curve be also at infinity, the equation can be
written in the form
xy = ax^ + bx'^ + cx + d.
Next he takes the case where the tangent at the real point
at infinity is not at a finite distance. A line parallel to the
direction in which the curve goes to infinity may be taken as
the axis of y. Any such line will cut the curve in three points
altogether, of which one is by hypothesis at infinity, and one is
necessarily at a finite distance. He then shews that if the
remaining point in which this line cuts the curve be at a finite
distance, the equation can be written in the form
y2 = ^^3 ^ ^^2 ^cx + d;
while if it be at an infinite distance, the equation can be
written in the form
CH. xvi] NEWTON ON CUBIC CURVES 341
y = ax^ + bx'^ + cx + d.
Any cubic is therefore reducible to one of four characteristic
forms. Each of these forms is then discussed in detail, and the
possibility of the existence of double points, isolated ovals, &c.,
is worked out. The final result is that in all there are seventy-
eight possible forms which a cubic may take. Of these Newton
enumerated only seventy-two; four of the remainder were
mentioned by Stirling in 1717, one by Nicole in 1731, and one
by Nicholas Bernoulli about the same time.
In the course of the work Newton states the remarkable
theorem that, just as the shadow of a circle (cast by a luminous
point on a plane) gives rise to all the conies, so the shadows of
the curves represented by the equation y^ = ax^ -\- hx^ + ex -\- d
give rise to all the cubics. This remained an unsolved puzzle
until 1731, when Nicole and Clairaut gave demonstrations of
it; a better proof is that given by Murdoch in 1740, which
depends on the classification of these curves into five species
according as to whether their points of intersection with the axis
of X are real and unequal, real and two of them equal (two
cases), real and all equal, or two imaginary and one real.
In this tract Newton also discusses double points in the
plane and at infinity, the description of curves satisfying given
conditions, and the graphical solution of problems by the use of
curves.
The second appendix to the Optics is entitled De Quadratura
Curvarum. Most of it had been communicated to Barrow in
1668 or 1669, and probably was familiar to Newton's pupils
and friends from that time onwards. It consists of two parts.
The bulk of the first part is a statement of Newton's method
of effecting the quadrature and rectification of curves by means
of infinite series ; it is noticeable as containing the earliest use
in print of literal indices, and a printed statement of the
binomial theorem, but these novelties are introduced only
incidentally. The main object is to give rules for developing a
function of a? in a series in ascending powers of x, so as to
342 THE LIFE AND WORKS OF NEWTON [ch. xvi
enable mathematicians to effect the quadrature of any curve
in which the ordinate y can be expressed as an explicit
algebraical function of the abscissa x. Wallis had shewn how
this quadrature could be found when y was given as a sum of a
number of multiples of powers of x^ and Newton's rules of
expansion here established rendered possible the similar quad-
rature of any curve whose ordinate can be expressed as the sum
of an infinite number of such terms. In this way he effects the
quadrature of the curves
but naturally the results are expressed as infinite series. He
then proceeds to curves whose ordinate is given as an implicit
function of the abscissa ; and he gives a method by which y can
be expressed as an infinite series in ascending powers of x^
but the application of the rule to any curve demands in general
such complicated numerical calculations as to render it of little
value. He concludes this part by shewing that the rectification
of a curve can be effected in a somewhat similar way. His
process is equivalent to finding the integral with regard to x
of (1+^-)^ in the form of an infinite series. I should add
that Newton indicates the importance of determining whether
the series are convergent — an observation far in advance of
his time — but he knew of no general test for the purpose ;
and in fact it was not until Gauss and Cauchy took up the
question that the necessity of such limitations was commonly
recognized.
The part of the appendix which I have just described is
practically the same as Newton's manuscript De Analysi per
Equationes Numero Terminorum Infinitas, which was subse-
quently printed in 1711. It is said that this was originally
intended to form an appendix to Kinckhuysen's Algebra,
which, as I have already said, he at one time intended to edit.
The substance of it was communicated to Barrow, and by him
to Collins, in letters of July 31 and August 12, 1669; and a
CH. xvi] NEWTON'S METHOD OF FLUXIONS 343
summary of part of it was included in the letter of October 24,
1676, sent to Leibnitz.
It should be read in connection with Newton's Methodiis
Differentialis, also published in 1711. Some additional
theorems are there given, and he discusses his method of
interpolation, which had been briefly described in the letter
of October 24, 1676. The principle is this. If y = ^{x) be a
function of x^ and if, when x is successively put equal to
aj, a.2.y...j the values of y be known and be h^^ b^,..., then a
parabola whose equation is y=p + qx + rx- + ... can be drawn
through the points (a^, 5j), {a 2, ^2), . . . , and the ordinate of this
parabola may be taken as an approximation to the ordinate of
the curve. The degree of the parabola will of course be one
less than the number of given points. Newton points out
that in this way the areas of any curves can be approximately
determined.
The second part of this appendix to the Optics contains a
description of Newton's method of fluxions. This is best con-
sidered in connection with Newton's manuscript on the same
subject which was published by John Colson in 1736, and of
which it is a summary.
The invention of the infinitesimal calculus was one of the
great intellectual achievements of the seventeenth century. This
method of analysis, expressed in the notation of fluxions and
fluents, was used by Newton in or before 1666, but no account
of it was published until 1693, though its general outline was
known by his friends and pupils long anterior to that year, and
no complete exposition of his methods was given before 1736.
The idea of a fluxion or differential coefficient, as treated at
this time, is simple. When two quantities — e.(/. the radius of a
sphere and its volume — are so related that a change in one
causes a change in the other, the one is said to be a function of
the other. The ratio of the rates at which they change is
termed the differential coefficient or fluxfon of the one with
regafd~To the other, and the process by which this ratio is
determined is known as differentiation. Knowing the differential
344 THE LIFE AND WORKS OF NEWTON [ch. xvi
coefficient and one set of corresponding values of the two
quantities, it is possible by summation to determine the relation
between them, as Cavalieri and others had shewn ; but often the
process is difficult. If, however, we can reverse the process of
differentiation we can obtain this result directly. This process
of reversal is termed integration. It was at once seen that
problems connected with the quadrature of curves, and the
determination of volumes (which were soluble by summation, as
had been shewn by the employment of indivisibles), were
reducible to integration. In mechanics also, by integration,
velocities could be deduced from known accelerations, and
distances traversed from known velocities. In short, wherever
things change according to known laws, here was a possible
method of finding the relation between them. It is true that,
when we try to express observed phenomena in the language of
the calculus, we usually obtain an equation involving the
variables, and their differential coefficients — and possibly the
solution may be beyond our powers. Even so, the method is
often fruitful, and its use marked a real advance in thought and
power.
I proceed to describe somewhat fully Newton's methods as
described by Colson. Newton assumed that all geometrical
magnitudes might be conceived as generated by continuous
motion ; thus a line may be considered as generated by the
motion of a point, a surface by that of a line, a solid by that of
a surface, a plane angle by the rotation of a line, and so on.
The quantity thus generated was defined by him as the fluent
or flowing quantity. The velocity of the moving magnitude
was defined as the fluxion of the fluent. This seems to be the
earliest definite recognition of the idea of a continuous function,
though it had been foreshadowed in some of Napier's papers.
Newton's treatment of the subject is as follows. There are
two kinds of problems. The object of the first is to find the
fluxion of a given quantity, or more generally " the relation of
the fluents being given, to find the relation of their fluxions."
This is equivalent to differentiation. The object of the second
CH. xvi] NEWTON'S METHOD OF FLUXIONS 345
or inverse method of fluxions is from the fluxion or some
relations involving it to determine the fluent, or more generally
"an equation being proposed exhibiting the relation of the
fluxions of quantities, to find the relations of those quantities,
or fluents, to one another."^ This is equivalent either to
integration which Newton termed the method of quadrature,
or to the solution of a differential equation which was called
by Newton the inverse method of tangents. The methods for
solving these problems are discussed at considerable length.
Newton then went on to apply these results to questions
connected with the maxima and minima of quantities, the
method of drawing tangents to curves, and the curvature of
curves (namely, the determination of the centre of curvature,
the radius of curvature, and the rate at which the radius of
curvature increases). He next considered the quadrature of
curves, and the rectification of curves.^ In finding the maxi-
mum and minimum of functions of one variable we regard the
change of sign of the difference between two consecutive values
of the function as the true criterion ; but his argument is that
when a quantity increasing has attained its maximum it can
have no further increment, or when decreasing it has attained
its minimum it can have no further decrement; consequently
the fluxion must be equal to nothing.
It has been remarked that neither Newton nor Leibnitz
produced a calculus, that is, a classified collection of rules ; and
that the problems they discussed were treated from first prin
ciples. That, no doubt, is the usual sequence in the history of
such discoveries, though the fact is frequently forgotten by
subsequent writers. In this case I think the statement, so far
as Newton's treatment of the differential or fluxional part of
the calculus is concerned^ is incorrect, as the foregoing account
sufficiently shews.
If a flowing quantity or fluent were represented by x,
Newton denoted its fluxion by x, the fluxion of x or second
^ Colson's edition of Newton's manuscript, pp. xxi, xxii.
^ Ibid. pp. xxii, xxiii.
346 THE LIFE AND WORKS OF NEWTON [ch. xvi
fluxion of X by x, and so on. Similarly the fluent of x was
denoted by | ^ L or sometimes by x or \x\. The infinitely small
part by which a fluent such as x increased in a small interval of
time measured by o was called the moment of the fluent ; and
its value was shewn ^ to be ±o. Newton adds the important
remark that thus we may in any problem neglect the terms
multiplied by the second and higher powers of o, and we can
always find an equation between the co-ordinates x, y of a
point on a curve and their fluxions cc, ij. It is an application of
this principle which constitutes one of the chief values of the
calculus ; for if we desire to find the eff'ect produced by
several causes on a system, then, if we can find the ejQfect pro-
duced by each cause when acting alone in a very small time,
the total effect produced in that time will be equal to the sum
of the separate eff'ects. I should here note the fact that Vince
and other English writers in the eighteenth century used x to
denote the increment of x and not the velocity\with which it
increased ; that is, x in their writings stands for what Newton
would have expressed by xo and what Leibnitz would have
written as dx. *v_J
I need not discuss in detail the manner in which Newton
treated the problems above mentioned. I will only add that,
in spite of the form of his definition, the introduction into
geometry of the idea of time was evaded by supposing that
some quantity {ex. gr. the abscissa of a point on a curve)
increased equably; and the required results then depend on
the rate at which other quantities {ex. gr. the ordinate or
radius of curvature) increase relatively to the one so chosen.^
The fluent so chosen is what we now call the independent
variable ; its fluxion was termed the " principal fluxion " ; and,
of course, if it were denoted by x, then x was constant, and
consequently x = 0.
There is no question that Newton used a method of fluxions
in 1666, and it is practically certain that accounts of it were
^ Colson's edition of Newton's manuscript, p. 24.
2 Ibid. p. 20.
CH. xvi] THE LIFE AND WORKS OF NEWTON 347 /
communicated in manuscript to friends and pupils from and
after 1669. The manuscript, from which most of the above
summary has been taken, is believed to have been written
between 1671 and 1677, and to have been in circulation at
Cambridge from that time onwards, though it is probable that
parts were rewritten from time to time. It was unfortunate that
it was not published at once. Strangers at a distance naturally
judged of the method by the letter to Wallis in 1692, or by the
Tractatus de Quadratura Curvarum, and were not aware that
it had been so completely developed at an earlier date. This
was the cause of numerous misunderstandings. At the same
time it must be added that all mathematical analysis was leading
up to the ideas and methods of the infinitesimal calculus. Fore-
shadowings of the principles and even of the language of that
calculus can be found in the writings of Napier, Kepler, Cava-
lieri, Pascal, Fermat, Wallis, and Barrow. It was Newton's
good luck to come at a time when everything was ripe for the
discovery, and his ability enabled him to construct almost at
once a complete calculus.
The infinitesimal calculus can also be expressed in the notation
of the differential calculus : a notation which was invented by
Leibnitz probably in 1675, certainly by 1677, and was published
in 1684, some nine years before the earliest printed account of
Newton's method of fluxions. But the question whether the
general idea of the calculus expressed in that notation was
obtained by Leibnitz from Newton, or whether it was discovered
independently, gave rise to a long and bitter controversy. The
leading facts are given in the next chapter.
The remaining events of Newton's life require little or no
comment. In 1705 he was knighted. From this time onwards
he devoted much of his leisure to theology, and wrote at great
length on prophecies and predictions, subjects which had always
been of interest to him. His Universal Arithmetic was pub-
lished by Whiston in 1707, and his Analysis by Infinite Series
in 1711 ; but Newton had nothing to do with the preparation
of either of these for the press. His evidence before the House
348 THE LIFE AND WORKS OF NEWTON [ch. xvi
\
of Commons in 1714 on tlie determination of longitude at sea
marks an important epoch in the history of navigation.
The dispute with Leibnitz as to whether he had derived the
ideas of the differential calculus from Newton or invented it
independently originated about 1708, and occupied inuch
of Newton's time, especially between the years 1709 and
1716.
In 1709 Newton was persuaded to allow Cotes to prepare
the long-talked-of second edition of the Principia ; it was issued
in March 1713. A third edition was published in 1726 under
the direction of Henry Pemberton. In 1725 Newton's health
began to fail. He died on March 20, 1727, and eight days
later was buried in Westminster Abbey.
His chief works, taking them in their order of publication,
are the Principia, published in 1687; the Optics (with appen-
dices on cubic curves, the quadrature and rectification of curves
by the use of infinite series, and the method of fluxions), pub-
lished in 1704; the Universal Arithmetic, published ja^l707;
the Analysis per Series, Fluxiones, &c., and the Methodui Diffe-
rentialis, published in 1711; the Lectiones Opticae, published in
1729 ; the Method of Fluxions, &c. (that is, Neivton^s manuscript
on fluxions), translated by J. Colson and published in 1736 ; and
the Geometria Analytica, printed in 1779 in the first volume of
Horsley's edition of Newton's works.
In appearance Newton was short, and towards the close of
his life rather stout, but well set, with a square lower jaw,
brown eyes, a broad forehead, and rather sharp features. His
hair turned grey before he was thirty, and remained thick and
white as silver till his death.
As to his manners, he dressed slovenly, was rather languid,
and was often so absorbed in his own thoughts as to be any-
thing but a lively companion. Many anecdotes of his extreme
absence of mind when engaged in any investigation have been
preserved. Thus once when riding home from Grantham he
dismounted to lead his horse up a steep hill ; when he turned at
the top to remount, he found that he had the bridle in his hand.
CH.XVI] THE LIFE AND WORKS OF NEWTON 349
while his horse had slipped it and gone away. Again, on the
few occasions when he sacrificed his time to entertain his friends,
if he left them to get more wine or for any similar reason, he
would as often as not be found after the lapse of some time
working out a problem, oblivious alike of his expectant guests
and of his errand. He took no exercise, indulged in no amuse-
ments, and worked incessantly, often spending eighteen or nine-
teen hours out of the twenty-four in writing.
In character he was religious and conscientious, with an
exceptionally high standard of morality, having, as Bishop
Burnet said, " the whitest soul " he ever knew. Newton was
always perfectly straightforward and honest ; but in his con-
troversies with Leibnitz, Hooke, and others, though scrupulously
just, he was not generous ; and it would seem that he frequently
took offence at a chance expression when none was intended.
He modestly attributed his discoveries largely to the admirable
work done by his predecessors ; and once explained that, if he
had seen farther than other men, it was only because he had
stood on the shoulders of giants. He summed up his own
estimate of his work in the sentence, "I do not know what I
may appear to the world ; but to myself I seem to have been
only like a boy, playing on the sea-shore, and diverting myself,
in now and then finding a smoother pebble, or a prettier shell
than ordinary, whilst the great ocean of truth lay all undis-
covered before me." He was morbidly sensitive to being
involved in any discussions. I believe that, with the exception
of his papers on optics, every one of his works was published
only under pressure from his friends and against his own wishes.
There are several instances of his communicating papers and
results on condition that his name should not be published :
thus when in 1669 he had, at Collin s's request, solved some
problems on harmonic series and on annuities which had previ-
ously baffled investigation, he only gave permission that his
results should be published " so it be," as he says, " without my
name to it; for I see not what there is desirable in public
esteem, were I able to acquire and maintain it : it would per-
350 THE LIFE AND WORKS OF NEWTON [ch. xvi
haps increase my acquaintance, the thing which I chiefly study
to decline."
Perhaps the most wonderful single illustration of his powers
was the composition in seven months of the first book of the
Principia, and the expression of the numerous and complex
results in classical geometrical form. As other illustrations of
his ability I may mention his solutions of the problem of Pappus,
of John Bernoulli's challenge, and of the question of orthogonal
trajectories. The problem of Pappus, here alluded to, is to find
the locus of a point such that the rectangle under its distances
from two given straight lines shall be in a given ratio to the
rectangle under its distances from two other given straight lines.
Many geometricians from the time of ApoUonius had tried to
find a geometrical solution and had failed, but what had proved
insuperable' to his predecessors seems to have presented little
difficulty to Newton who gave an elegant demonstration that
the locus was a conic. Geometry, said (Lagrange when recom-
mending the study of analysis to his pupils, is a strong bow,
but it is one which only a Newton can fully utilize. As another
example I may mention that in 1696 John Bernoulli challenged
mathematicians (i) to determine the brachistochrone, and (ii)
to find a curve such that if any line drawn from a fixed point 0
cut it in P and Q then OP^^ + 0^'* would be constant. Leibnitz
solved the first of these questions after an interval of rather
more than six months, and then suggested they should be sent
as a challenge to Newton and others. Newton received the
problems on Jan. 29, 1697, and the next day gave the complete
solutions of both, at the same time generalising the second
question. An almost exactly similar case occurred in 1716
when Newton was asked to find the orthogonal trajectory of a
family of curves. In five hours Newton solved the problem in
the form in which it was propounded to him, and laid down the
principles for finding trajectories.
It is almost impossible to describe the effect of Newton's
writings without being suspected of exaggeration. But, if the
state of mathematical knowledge in 1669 or at the death of
CH. xvi] NEWTON'S INVESTIGATIONS 351
Pascal or Fermat be compared with what was known in 1700
it will be seen how immense was the advance. In fact we
may say that it took mathematicians half a century or more
before they were able to assimilate the work produced in those
years.
In pure geometry Newton did not establish any new methods,
but no modern writer has shewn the same power in using those
of classical geometry. In algebra and the theory of equations
he introduced the system of literal indices, established the
binomial theorem, and created no inconsiderable part of the
theory of equations : one rule which he enunciated in this
subject remained till a few years ago an unsolved riddle which
had overtaxed the resources of succeeding mathematicians. In
analytical geometry, he introduced the modern classification of
curves into algebraical and transcendental; and established
many of the fundamental properties of asymptotes, multiple
points, and isolated loops, illustrated by a discussion of cubic
curves. The fluxional or infinitesimal calculus was invented by
Newton in or before the year 1666, and circulated in manuscript
amongst his friends in and after the year 1669, though no
account of the method was printed till 1693. The fact that the
results are nowadays expressed in a different notation has led
to Newton's investigations on this subject being somewhat
overlooked.
Newton, further, was the first to place dynamics on a
satisfactory basis, and from. dynamics he deduced the theory of
statics : this was in the introduction to the Principia published
in 1687. The theory of attractions, the application of the
principles of mechanics to the solar system, the creation of
physical astronomy, and the establishment of the law of uni-
versal gravitation are due to him, and were first published in the.
same work, but of the nature of gravity he confessed his
ignorance, though he found inconceivable the idea of action at
a distance. The particular questions connected with the motion
of the earth and moon were worked out as fully as was then
possible. The theory of hydrodynamics was created in the
352 THE LIFE AND WORKS OF NEWTON [ch. xvi
second book of the Principia, and he added considerably to the
theory of hydrostatics which may be said to have been first
discussed in modern times by Pascal. The theory of the pro-
pagation of waves, and in particular the application to determine
the velocity of sound, is due to Newton and was published in
1687. In geometrical optics, he explained amongst other things
the decomposition of light and the theory of the rainbow ; he
invented the reflecting telescope known by his name, and the
sextant. In physical optics, he suggested and elaborated the
emission theory of light.
The above list does not exhaust the subjects he investigated,
but it will serve to illustrate how marked was his influence on
the history of mathematics. On his writings and on their
effects, it will be enouglrtp quote the remarks of two or three
of those whe were subsequently concerned with the subject-
matter of the Principia. Lagrange described the Principia as
the greatest production of the human mind, and said he felt
dazed at such an illustration of what man's intellect might be
capable. In describing the effect of his own writings and those
of Laplace it was a favourite remark of his that Newton was
not only the greatest genius that had ever existed, but he was
also the most fortunate, for as there is but one universe, it can
happen but to one man in the world's history to be the inter-
preter of its laws. Laplace, who is in general very sparing of
his praise, makes of Newton the one exception, and the words
in which he enumerates the causes which "will always assure
to the Principia a pre-eminence above all the other productions
of human genius " have been often quoted. Not less remarkable
is the homage rendered by Gauss ; for other great mathematicians
or philosophers he used the epithets magnus, or clarus, or
clarissimus : for Newton alone he kept the prefix summus.
Finally Biot, who had made a special study of Newton's works,
sums up his remarks by saying, "comme geometre et comme
experimentateur Newton est sans egal ; par la reunion de ces
deux genres de genies a leur plus haut degre, il est sans
exemple."
353
CHAPTER XVII.
LEIBNITZ AND THE MATHEMATICIANS OF THE FIRST HALF
OF THE EIGHTEENTH CENTURY. ^
I HAVE briefly traced in the last chapter the nature and extent
of Newton's contributions to science. Modern analysis is,
however, derived directly from the works of Leibnitz and the
elder Bernoullis ; and it is immaterial to us whether the funda-
mental ideas of it were obtained by them from Newton, or
discovered independently. The English mathematicians of
the years considered in this chapter continued to use the
language and notation of Newton ; they are thus somewhat
distinct from their continental contemporaries, and I have there-
fore grouped them together in a section by themselves.
Leibnitz and the Bernoullis.
Leibnitz.2 Gottfried Wilhelm Leibnitz (or Leibniz) was born
at Leipzig on June 21 (O.S.), 1646, and died at Hanover on
November 14, 1716. His father died before he was six, and the
teaching at the school to which he was then sent was ineffi-
* See Cantor, vol. iii ; other authorities for the matheniaticiaus of the
period are mentioned in the footnotes.
2 See the life of Leibnitz by G. E. Guhraner, two volumes and a supple-
ment, Breslau, 1842 and 1846. Leibnitz's mathematical papers have been
collected and edited by C. J. Gerhardt in seven volumes, Berlin and Halle,
1849-63.
2a
354 LEIBNITZ [ch. XviT
cient, but his industry triumphed over all difficulties ; by the
time he was twelve he had taught himself to read Latin easily,
and had begun Greek; and before he was twenty he had
mastered the ordinary text-books on mathematics, philosophy,
theology, and law. Kef used the degree, of doctor of laws at
Leipzig by those who were jealous of his youth and learning, he
moved to Nuremberg. An essay which he there wrote on the
study of law was dedicated to the Elector of Mainz, and led to
his appointment by the elector on a commission for the revision
of some statutes, from which he was subsequently promoted to
the diplomatic service. In the latter capacity he supported
(unsuccessfully) the claims of the German candidate for the
crown of Poland. The violent seizure of various small places in
Alsace in 1670 excited universal alarm in Germany as to the
designs of Louis XIV. ; and Leibnitz drew up a scheme by which
it was proposed to offer German co-operation, if France liked to
take Egypt, and use the possession of that country as a basis for
attack against Holland in Asia, provided France would agree to
leave Germany undisturbed. This bears a curious resemblance
to the similar plan by which Napoleon I. proposed to attack
England. In 1672 Leibnitz went to Paris on the invitation of
the French government to explain the details of the scheme, but
nothing came of it.
At Paris he met Huygens who was then residing there,
and their conversation led Leibnitz to study geometry, which
he described as opening a new world to him; though as a
matter of fact he had previously written some tracts on various
minor points in mathematics, the most important being a paper
on combinations written in 1668, and a description of a new
calculating machine. In January, 1673, he was sent on a
political mission to London, where he stopped some months and
made the acquaintance of Oldenburg, Collins, and others ; it
was at this time that he communicated the memoir to the Royal
Society in which he was found to have been forestalled by
Mouton.
In 1673 the Elector of Mainz died, and in the following year
CH. xvii] LEIBNITZ 355
Leibnitz entered the service of the Brunswick family; in 1676
he again visited London, and then moved to Hanover, where,
till his death, he occupied the well-paid post of librarian in the
ducal library. His pen was thenceforth employed in all the
political matters which affected the Hanoverian family, and his
services were recognized by honours and distinctions of various
kinds ; his memoranda on the various political, historical, and
theological questions which concerned the dynasty during the
forty years from 1673 to 1713 form a valuable contribution to
the history of that time.
Leibnitz's appointment in the Hanoverian service gave him
more time for his favourite pursuits. He used to assert that as
the first-fruit of his increased leisure, he invented the differential
and integral calculus in 1674, but the earliest traces of the use
of it in his extant note-books do not occur till 1675, and it was
not till 1677 that we find it developed into a consistent system ;
it was not published till 1684. Most of his mathematical
papers were produced within the ten years from 1682 to 1692,
and many of them in a journal, called the Acta Eruditcyt^um,
founded by himself and Otto Mencke in 1682, which had a
wide circulation on the continent.
Leibnitz occupies at least as large a place in the history of
philosophy as he does in the history of mathematics. Most of
his philosophical writings were composed in the last twenty or
twenty-five years of his life ; and the point as to whether his
views were original or whether they were appropriated from
Spinoza, whom he visited in 1676, is still in question among
philosophers, though the evidence seems to point to the origin-
ality of Leibnitz. As to Leibnitz's system of philosophy it will
be enough to say that he regarded the ultimate elements of the
universe as individual percipient beings whom he called monads.
According to him the monads are centres of force, and substance
is force, while s^ace, matter, and motion are merely phenomenal ;
finally, the existence "of God is inferred from the existing
harmony among the monads. His services to literature were
almost as considerable as those to philosophy ; in particular, I
356 LEIBNITZ [ch. xvii
may single out his overthrow of the then prevalent belief that
Hebrew was the primeval language of the human race.
In 1 700 the academy of Berlin was created on his advice, and
he drew up the first body of statutes for it. On the accession
in 1714 of his master, George I., to the thtone of England,
Leibnitz was thrown aside as a useless tool ; he was forbidden
to come to England ; and the last two years of his life were
spent in neglect and dishonour. He died at Hanover in 1716.
He was overfond of money and personal distinctions; w^as
unscrupulous, as perhaps might be expected of a professional
diplomatist of that time ; but possessed singularly attractive
manners, and all who once came under the charm of his personal
presence remained sincerely attached to him. His mathematical
reputation was largely augmented by the eminent position that
he occupied in diplomacy, philosophy, and literature ; and the
power thence derived was considerably increased by his influence
in the management of the Acta Ervditormn.
The last years of his life — from 1709 to 1716 — were em-
bittered by the long controversy with John Keill, Newton, and
others, as to whether he had discovered the differential calculus
independently of Newton's previous investigations, or whether
he had derived the fundamental idea from Newton, and merely
invented another notation for it. The controversy ^ occupies a
place in the scientific history of the early years of the eighteenth
century quite disproportionate to its true importance, but it so
materially affected the history of mathematics in western
Europe, that I feel obliged to give the leading facts, though I
1 The case in /avour of the independent invention by Leibnitz is stated in
Gerhardt's Ledbnizens niathematische Schriften ; and in the third volume of
M. Cantor's Geschichte der Mathematik. The arguments on the other side
are given in H. Sloman's Leibnitzens Anspruch auf die Erfindung der
Differe7izialrechnung, Leipzig, 1857, of which an English translation, with
atlditions by Br. Sloman, was published at Cambridge in 1860. A summary
of the evidence will be Ibund in G. A. Gibson's memoir, Proceedings of the
Edinburgh Matheviatical Society, vol. xiv, 1896, pp. 148-174. The history
of the invention of the calculus is given in an article on it in the ninth edition
of the Encyclopaedia Britannica, and in P. Mansion's Esquisse de Vhistoiro.
du calcid injinitisimal, Gand, 1887.
CH. xvii] LEIBNITZ 357
am reluctant to take up so much space with questions of a
personal character.
The ideas of the infinitesimal calculus can be expressed
either in the notation of fluxions or in that of differentials.
The former was used by Newton in 1666, but no distinct
account of it was printed till 1693. The earliest use of the
latter in the note-books of Leibnitz may be probably referred to
1675, it was employed in the letter sent to Newton in 1677, and
an account of it was printed in the memoir of 1684 described
below. There is no question that the differential notation is due
to Leibnitz, and the sole question is as to whether the general
idea of the calculus was taken from Newton or discovered
independently.
The case in favour of the independent invention by Leibnitz
rests on the ground that he published a description of his
method some years before Newton printed anything on fluxions,
that he always alluded to the discovery as being his own inven-
tion, and that for some years this statement was unchallenged ;
while of course there must be a strong presumption that he
acted in good faith. To rebut this case it is necessary to shew
(i) that he saw some of Newton's papers on the subject in or
before 1675, or at least 1677, and (ii) that he thence derived the
fundamental ideas of the calculus. Tlie fact that his claim was
unchallenged for some years is, in the particular circumstances
of the case, immaterial.
That Leibnitz saw some of Newton's manuscripts was always
intrinsically probable; but when, in 1849, C J. Gerhardt^
examined Leibnitz's papers he found among them a manuscript
copy, the existence of which had been previously unsuspected,
in Leibnitz's handwriting, of extracts from Newton's De Analysi
per Equationes Numero Terminorum Injinitas (which was
printed in the De Qv/ldratura Curvarum in 1704), together
with notes on their expression in the differential notation.
The question of the date at which these extracts were made is
therefore all-important. Tschirnhausen seems to have possessed
^ Gerhardt, Leibnizens tnatlievudische Schriften, vol. i, p. 7.
358 LEIBNITZ [ch. Xvii
a copy of Newton's De Analysi in 1675, and as in that year
he and Leibnitz were engaged together on a piece of work,
it is not impossible that these extracts were made then. It
is also possible that they may have been made in 1676, for
Leibnitz discussed the question of analysis by infinite series
with Collins and Oldenburg in that year, and it is a priori
probable that they would have then shewn him the manuscript
of Newton on that subject, a copy of which was possessed by
one or both of them. On the other hand it may be supposed
that Leibnitz made the extracts from the printed copy in or
after 1704. Leibnitz shortly before his death admitted in a
letter to Conti that in 1676 Collins had shewn him some
Newtonian papers, but implied that they were of little or no
value, — presumably he referred to Newton's letters of June 13
and Oct. 24, 1676, and to the letter of Dec. 10, 1672, on the
method of tangents, extracts from which accompanied i the
letter of June 13, — but it is remarkable that, on the receipt of
these letters, Leibnitz should have made no further inquiries,
unless he was already aware from other sources of the method
followed by Newton.
Whether Leibnitz made no use of the manuscript from
which he had copied extracts, or whether he had previously
invented the calculus, are questions on which at this distance
of time no direct evidence is available. It is, however, worth
noting that the unpublished Portsmouth Papers shew that
when, in 1711, Newton went carefully into the whole dispute,
he picked out this manuscript as the one w^hich had probably
somehow fallen into the hands of Leibnitz. 2 At that time
there w'as no direct evidence that Leibnitz had seen this
manuscript before it was printed in 1704, and accordingly
Newton's conjecture was not published; but Gerhardt's dis-
covery of the copy made by Leibnitz tends to confirm the
accuracy of Newton's judgment in the matter. It is said by
^ Gerhardt, vol. i, p. 91.
2 Catalogue of Portsmovih Papers, pp. xvi, xvii, 7, 8.
CH. xvii] LEIBNITZ 359
those who question Leibnitz's good faith that to a man of his
ability the manuscript, especially if supplemented by the letter
of Dec. 10, 1672, would supply sufficient hints to give him a
clue to the methods of the calculus, though as the fluxional
notation is not employed in it anyone' who used it would have
to invent a notation ; but this is denied by others.
There was at first no reason to suspect the good faith of
Leibnitz; and it was not until the appearance in 1704 of an
anonymous review of Newton's tract on quadrature, in which
it was implied that Newton had borrowed the idea of the
fluxional calculus from Leibnitz, that any responsible mathe-
matician 1 questioned the statement that Leibnitz had invented
the calculus independently of Newton. It is universally
admitted that there was no justification or authority for the
statements made in this review, which was rightly attributed
to Leibnitz. But the subsequent discussion led to a critical
examination of the whole question, and doubt was expressed as
to whether Leibnitz had not derived the fundamental idea from
Newton. The case against Leibnitz as it appeared to Newton's
friends was summed up in the Commercium Epistoliciim issued
in 1712, and detailed references are given for all the facts
mentioned.
No such summary (with facts, dates, and references) of the
case for Leibnitz was issued by his friends ; but John Bernoulli
attempted to indirectly weaken the evidence by attacking the
personal character of Newton : this was in a letter dated June 7,
1713. The charges were false, and, when pressed for an
explanation of them, Bernoulli most solemnly denied having
written the letter. In accepting the denial Newton added in a
private letter to him the following remarks, which are interesting
as giving Newton's account of why he was at last induced to
take any part in the controversy. "I have never," said he,
" grasped at fame among foreign nations, but I am very
desirous to preserve my character for honesty, which the
^ In 1699 Duillier had accused Leibnitz of plagiarism from Newton, but
Duillier was not a person of much importance.
360 LEIBNITZ [ch. xvii
author of that epistle, as if by the authority of a great
judge, had endeavoured to wrest from me. Now that I am
old, I have little pleasure in mathematical studies, and I have
never tried to propagate my opinions over the world, but have
rather taken care not to involve myself in disputes on account
of them."
Leibnitz's defence or explanation of his silence is given in
the following letter, dated April 9, 1716, from him to Conti.
*' Pour repondre de point en point a Fouvrage public contre
moi, il falloit un autre ouvrage aussi grand pour le moins que
celui-la : il falloit entrer dans un grand detail de quantite de
minuties passees il y a trente a quarante ans, dont je ne me
souvenois guere : il me falloit chercher mes vieilles lettres,
dont plusieurs se sont perdues, outre que le plus souvent je
n'ai point garde les minutes des miennes : et les autres sont
ensevelies dans un grand tas de papiers, que je ne pouvois
debrouiller qu'avec du temps et de la patience ; mais je n'en
avois guere le loisir, etant charge presentement d'occupations
d'une toute autre nature."
The death of Leibnitz in 1716 only put a temporary stop
to the controversy which was bitterly debated for many years
later. The question is one of difficulty ; the evidence is con-
flicting and circumstantial; and every one must judge for
himself which opinion seems most reasonable. Essentially it
is a case of Leibnitz's word against a number of suspicious
details pointing against him. His unacknowledged possession
of a copy of part of one of Newton's manuscripts may be
explicable ; but the fact that on more than one occasion he
deliberately altered or added to important documents {ex. gr.
the letter of June 7, 1713, in the Charta Volans, and that of
April 8, 1716, in the Acta Urtiditorum), before publishing
them, and, what is worse, that a material date in one of his
manuscripts has been falsified ^ (1675 being altered to 1673),
makes his own testimony on the subject of little value. It
^ Cantor, who advocates Leibnitz's claims, thinks that the falsification
must he taken to be Leibnitz's act : see Cantor, vol. iii, p. 176.
CH.xvii] LEIBNITZ 361
must be recollected that what he is alleged to have received was
rather a number of suggestions than an account of the calculus ;
and it is possible that as he did not publish his results of
1677 until 1684, and that as the notation and subsequent
development of it were all of his own invention, he may have
been led, thirty years later, to minimize any assistance which
he had obtained originally, and finally to consider that it was
immaterial. During the eighteenth century the prevalent
opinion was against Leibnitz, but to-day the majority of writers
incline to think it more likely that the inventions were
independent.
If we must confine ourselves to one system of notation then
there can be no doubt that that which was invented by Leibnitz
is better fitted for most of the purposes to which the infinites-
imal calculus is applied than that of fluxions, and for some
(such as the calculus of variations) it is indeed almost essential.
It should be remembered, however, that at the beginning of the
eighteenth century the methods of the infinitesimal calculus had
not been systematized, and either notation was equally good.
The development of that calculus was the main work of the
mathematicians of the first half of the eighteenth century. The
differential form was adopted by continental mathematicians.
The application of it by Euler, Lagrange, and Laplace to the^\
principles of mechanics laid down in the Principia was the great ^
achievement of the last half of that century, and finally demon-
strated the superiority of the differential to the fluxional calculus.
The translation of the Princi'pia into the language of modern
analysis, and the filling in of the details of the Newtonian theory
by the aid of that analysis, were effected by Laplace.
The controversy with Leibnitz was regarded in England as
an attempt by foreigners to defraud Newton of the credit of
his invention, and the question was complicated on both sides
by national jealousies. It was therefore natural, though it was \
unfortunate, that in England the geometrical and fluxional
methods as used by Newton were alone studied and employed.
For more than a century the English school was thus out of
362 LEIBNITZ [ch. xvii
toucli with continental mathematicians. The consequence was
that, in spite of the brilliant band of scholars formed by Newton,
the improvements in the methods of analysis gradually effected
on the continent were almost unknown in Britain. It was not
until 1820 that the value of analytical methods was fully recog-
nized in England, and that Newton's countrymen again took any
large share in the development of mathematics.
Leaving now this long controversy I come to the discussion
of the mathematical papers produced by Leibnitz, all the more
important of which w^ere published in the Acta Eruditorum.
They are mainly concerned with applications of the infinitesimal
calculus and with various questions on mechanics.
The only papers of first-rate importance which he produced
are those on the differential calculus. The earliest of these was
one published in the Acta Eruditorum for October, 1684, in
which he enunciated a general method for finding maxima and
minima, and for drawing tangents to curves. One inverse
problem, namely, to find the curve whose subtangent is constant,
was also discussed. The notation is the same as that with
which we are familiar, and the differential coefficients of x*^ and
of products and quotients are determined. In 1686 he wrote
a paper on the principles of the new calculus. In both of these
papers the principle of continuity is explicitly assumed, while
his treatment of the subject is based on the use of infinitesimals
and not on that of the limiting value of ratios. In answer to
some objections which were raised in 1694 by Bernard Nieuwentyt,
who asserted that dyjdx stood for an unmeaning quantity like
0/0, Leibnitz explained, in the same way as Barrow had
previously done, that the value of dyjdx in geometry could be
expressed as the ratio of two finite quantities. I think that
Leibnitz's statement of the objects and methods of the infinites-
imal calculus as contained in these papers, which are the three
most important memoirs on it that he produced, is somewhat
obscure, and his attempt to place the subject on a metaphysical
basis did not tend to clearness ; but the fact that all the results
of modern mathematics are expressed in the language invented
CH.xvii] LEIBNITZ 363
by Leibnitz has proved the best monument of his work. Like
Newton, he treated integration not only as a summation, but as
the inverse of differentiation.'
In 1686 and 1692 he wrote papers on osculating curves.
These, however, contain some bad blunders, as, for example, the
assertion that an osculating circle will necessarily cut a curve
in four consecutive points : this error was pointed out by John
Bernoulli, but in his article of 1692 Leibnitz defended his
original assertion, and insisted that a circle could never cross a
curve where it touched it.
In 1692 Leibnitz wrote a memoir in w^hich he laid the
foundation of the theory of envelopes. This was further de-
veloped in another paper in 1694, in which he introduced for the
first time the terms "co-ordinates" and "axes of co-ordinates."
Leibnitz also published a good many papers on mechanical
subjects ; but some of them contain mistakes which shew that
he did not understand the principles of the subject. Thus, in
1685, he wTote a memoir to find the pressure exerted by a
sphere of weight W placed between two inclined planes of com-
plementary inclinations, placed so that the lines of greatest
slope are perpendicular to the line of the intersection of the
planes. He asserted that the pressure on each plane must
consist of two components, "unum quo decliviter descendere
tendit, alteram quo planum declive premit." He further said
that for metaphysical reasons the sum of the two pressures must
be equal to W. Hence, if E and R' be the required pressures,
and a and J'^ - « the inclinations of the planes, he finds that
i? = 1 17(1 - sin a -I- cos a) and JR' = ^W {I - cos a + sin a).
The true values are E= TFcos a and E' = Tl^sin a. Nevertheless
some of his papers on mechanics are valuable. Of these the
most important were two, in 1689 and 1694, in which he solved
the problem of finding an isochronous curve; one, in 1697, on
the curve of quickest descent (this was the problem sent as a
challenge to Newton); and two, in 1691 and 1692, in which
he stated the intrinsic equation of the curve assumed by a
364 LEIBNITZ [ch. xvii
flexible rope suspended from two i^oints, that is, the catenary,
but gave no proof. This last problem had been originally
proposed by Galileo.
In 1689, that is, two years after the Principia had been
published, he wrote on the movements of the i^lanets which he
stated were produced by a motion of the ether. Not only
were the equations of motion which he obtained wrong, but his
deductions from them were not even in accordance with his own
axioms. In another memoir in 1706, that is, nearly twenty
years after the Principia had been written, he admitted that
he had made some mistakes in his former paper, but adhered
to his previous conclusions, and summed the matter up by
saying "it is certain that gravitation generates a new force at
each instant to the centre, but the centrifugal force also generates
another away from the centre. . . . The centrifugal force may
be considered in two aspects according as the movement is
treated as along the tangent to the curve or as along the arc
of the circle itself." It seems clear from this paper that he did
not really understand the principles of dynamics, and it is hardly
necessary to consider his work on the subject in further detail.
Much of it is vitiated by a constant confusion between momentum
and kinetic energy: when the force is "passive" he uses the
first, which he calls the vis mortua, as the measure of a force ;
when the force is "active" he uses the latter, the double of
which he calls the vis viva.
The series quoted by Leibnitz comprise those for e*,
log (1+^), sin ^, vers ^, and tan~i<r; all of these had been
previously published, and he rarely, if ever, added any demon-
strations. Leibnitz (like Newton) recognised the importance
of James Gregory's remarks on the necessity of examining
whether infinite series are convergent or divergent, and proposed
a test to distinguish series whose terms are alternately positive
and negative. In 1693 he explained the method of expansion
by indeterminate coefficients, though his applications were not
free from error.
To sum the matter up briefly, it seems to me that Leibnitz's
CH. xvii] LEIBNITZ 365
work exhibits great skill in analysis, but much of it is un-
finished, and when he leaves his symbols and attempts to in-
terpret his results he frequently commits blunders. No doubt
the demands of politics, philosophy, and literature on his time
may have prevented him from elaborating any problem com-
pletely or writing a systematic exposition of his views, though
they are no excuse for the mistakes of principle which occur in
his papers. Some of his memoirs contain suggestions of
methods which have now become valuable means of analysis,
such as the use of determinants and of indeterminate co-
efficients ; but when a writer of manifold interests like Leibnitz
throws out innumerable suggestions, some of them are likely to
turn out valuable; and to enumerate these (which he did not
work out) without reckoning the others (which are wrong) gives
a false impression of the value of his work. But in spite of
this, his title to fame rests on a sure basis, for by his advocacy
of the differential calculus his name is inseparably connected
with one of the chief instruments of analysis, as that of
Descartes — another philosopher — is similarly connected with
analytical geometry.
Leibnitz was but one amongst several continental writers
whose papers in the Acta E^mditorum familiarised mathe-
maticians with the use of the differential calculus. Among the
most important of these were James and John Bernoulli, both
of whom were warm friends and admirers of Leibnitz, and to
their devoted advocacy his reputation is largely due. Not only
did they take a prominent part in nearly every mathematical
question then discussed, but nearly all the leading mathe-
maticians on the continent during the first half of the eighteenth
century came directly or indirectly under the influence of one
or both of them.
The Bernoullis^ (or as they are sometimes, and perhaps
more correctly, called, the Bernouillis) were a family of Dutch
origin, who were driven from Holland by the Spanish persecu-
^ See the account in the AUgemeine deutsche Biographie, vol. ii, Leipzig,
J875, pp. 470-483.
366 JAMES BERNOULLI [ch. xvii
tions, and finally settled at Bale in Switzerland. The first
member of the family who attained distinction in mathematics
was James.
James Bernoulli.^ Jacoh or James Bernoulli was born at
Bale on December 27, 1654; in 1687 he was appointed to a
chair of mathematics in the university there ; and occupied it
until his death on August 16, 1705.
He was one of the earliest to realize how powerful as an
instrument of analysis was the infinitesimal calculus, and he
applied it to several problems, but he did not himself invent
any new processes. His great influence was uniformly and
successfully exerted in favour of the use of the differential cal-
culus, and his lessons on it, which were wTitten in the form of
two essays in 1691 and are published in the second volume of
his works, shew how completely he had even then grasped the
principles of the new analysis. These lectures, which contain
the earliest use of the term integral, were the first published
attempt to construct an integral calculus ; for Leibnitz had
treated each problem by itself, and had not laid down any
general rules on the subject.
The most important discoveries of James Bernoulli were
X his solution of the problem to find an isochronous curve; his
j2 t proof that the construction for the "catenary which had been
given by Leibnitz was correct, and his extension of this to
strings of variable density and under a central force; his
3» determination of the form taken by an elastic rod fixed at one
end and acted on by a given force at the other, the elastica ;
also of a flexible rectangular sheet with two sides fixed hori-
zontally and filled with a heavy liquid, the lintearia ; and, lastly,
of a sail filled with wind, the velaria. In 1696 he offered a
reward for the general solution of isoperimetrical figures, that
is, of figures of a given species and given perimeter which shall
^ See the eloge by B. de Fontenelle, Paris, 1766 ; also Montucla's Histoire,
vol. ii. A collected edition of the works of James Bernoulli was published
in two volumes at Geneva in 1744, and an account of his life is prefixed to
the first volume.
CH. xvii] JAMES AND JOHN BERNOULLI 367
include a maximum area : his own solution, published in 1701,
is correct as far as it goes. In 1698 he published an essay on
the differential calculus and its applications to geometry. He
here investigated the chief properties of the equiangular spiral,
and especially noticed the manner in which various curves
deduced from it reproduced the original curve : struck by this
fact he begged that, in imitation of Archimedes, an equiangular
spiral should be engraved on his tombstone with the inscription
eadetii numero mutata resurgo. He also brought out in 1695
an edition of Descartes's Geometrie. In his Ars Conjectaiidi^
published in 1713, he established the fundamental principles of
the calculus of probabilities; in the course of the work he
defined the numbers known by his name ^ and explained their
use, he also gave some theorems on finite differences. His
higher lectures were mostly on the theory of series ; these were
published by Nicholas Bernoulli in 1713.
John Bernoulli. 2 John Bernoulli^ the brother of James
Bernoulli, was born at Bale on August 7, 1667, and died there
on January 1, 1748. He occupied the chair of mathematics
at Groningen from 1695 to 1705; and at Bale, where he
succeeded his brother, from 1705 to 1748. To all who did not
acknowledge his merits in a manner commensurate with his
own view of them he behaved most unjustly : as an illustration
of his character it may be mentioned that he attempted to sub-
stitute for an incorrect solution of his own on the problem of
isoperimetrical curves another stolen from his brother James,
while he expelled his son Daniel from his house for obtaining
a prize from the French Academy which he had expected to
receive himself. He was, however, the most successful teacher
of his age, and had the faculty of inspiring his pupils with
^ A bibliography of Bernoulli's Numbers was given by G. S. Ely, in the
American Journal of Mathematics^ 1882, vol. v, pp. 228-235.
2 D'Alembert wrote a eulogistic eloge on the work and influence of John
Bernoulli, but he explicitly refused to deal with his private life or quarrels ;
see also Montucla's Histoire, vol. ii. A collected edition of the works of
John Bernoulli was published at Geneva in four volumes in 1742, and his
correspondence with Leibnitz was published in two volumes at the same
place in 1745.
368 JOHN BERNOULLI [ch. xvii
almost as passionate a zeal for mathematics as he felt himself.
The general adoption on the continent of the differential rather
than the fluxional notation was largely due to his influence.
Leaving out of account his innumerable controversies, the
chief discoveries of John Bernoulli were the exponential cal-
culus, the treatment of trigonometry as a branch of analysis,
the conditions for a geodesic, the determination of orthogonal
trajectories, the solution of the brachistochrone, the statement
that a ray of light pursues such a path that ^jxds is a minimum,
and the enunciation of the principle of virtual work. I believe
that he was the first to denote the accelerating effect of gravity
by an algebraical sign g, and he thus arrived at the formula
-y^ = 2gh : the same result would have been previously expressed
by the proportion v^^ : v^^ = h^ : h^. The notation (^ix to indicate
a function 1 of x was introduced by him in 1718, and displaced
the notation X or g proposed by him in 1698 ; but the general
adoption of symbols like /, F, ^, ^, . . . to represent functions,
seems to be mainly due to Euler and Lagrange.
Several members of the same family, but of a younger
generation, enriched mathematics by their teaching and
writings. The most important of these were the three sons of
John ; namely, Nicholas, Daniel, and John the younger ; and
the two sons of John the younger, who bore the names of
John and James. To make the account complete I add here
their respective dates. Nicholas Bernoulli, the eldest of the
three sons of John, was born on Jan. 27, 1695, and was
drowned at Petrograd, where he was professor, on July 26,
1726. Daniel Bernoulli, the second son of John, was born on
Feb. 9, 1700, and died on March 17, 1782; he was professor
first at Petrograd and afterwards at Bale, and shares with
Euler the unique distinction of having gained the prize proposed
annually by the French Academy no less than ten times : I refer
to him again a few pages later. John Bernoulli, the younger,
1 On the meaning assigned at first to the word function see a note by
M. Cantor, L" Intermediaire des mathematiciens, January 1896, vol. iii, pp.
22-23.
CH. xvii] JOHN BERNOULLI 369
a brother of Nicholas and Daniel, was born on May 18, 1710,
and died in 1790; he also was a professor at Bale. He left
two sons, John and James : of these, the former, who was born
on Dec. 4, 1744, and died on July 10, 1807, was astronomer-
royal and director of mathematical studies at Berlin ; while
the latter, who was born on Oct. 17, 1759, and died in
July 1789, was successively professor at Bale, Verona, and
Petrograd.
The development of analysis on the continent.
Leaving for a moment the English mathematicians of the
first half of the eighteenth century we come next to a number
of continental writers who barely escape mediocrity, and to
whom it will be necessary to devote but few words. Their
writings mark the steps by which analytical geometry and the
differential and integral calculus were perfected and made
familiar to mathematicians. Nearly all of them were pupils
of one or other of the two elder Bernoullis, and they were so
nearly contemporaries that it is difficult to arrange them
chronologically. The most eminent of them are Cramer^ de
Gua, de Montmort^ Fagnaiio, Vllospital, Nicole, Parent^
Riccati, Saurin, and Varignon.
L'Hospital. Guillaume Francois Antoine V Hospital, Mar-
quis de St.-Mesme, born at Paris in 1661, and died there on
Feb. 2, 1704, was among the earliest pupils of John Bernoulli,
who, in 1691, spent some months at I'Hospital's house in
Paris for the purpose of teaching him the new calculus. It
seems strange, but it is substantially true, that a knowledge of
the infinitesimal calculus and the power of using it was then
confined to Newton, Leibnitz, and the two elder Bernoullis —
and it will be noticed that they were the only mathematicians
who solved the more difficult problems then proposed as chal-
lenges. There was at that time no text-book on the subject,
and the credit of putting together the first treatise which
explained the principles and use of the method is due to
2b
370 L'HOSPITAL. VARIGNON [oh. xvii
I'Hospital; it was published in 1696 under the title Analyse des
infiniment x>etiU. This contains a partial investigation of
the limiting value of the ratio of functions which for a certain
value of the variable take the indeterminate form 0 : 0, a
problem solved by John Bernoulli in 1704. This work had
a wide circulation; it brought the differential notation into
general use in France, and helped to make it known in
Europe. A supplement, containing a similar treatment of
the integral calculus, together with additions to the differential
calculus which had been made in the following half century,
was published at Paris, 1754-56, by L. A. de Bougainville.
L'Hospital took part in most of the challenges issued
by Leibnitz, the BernouUis, and other continental mathe-
maticians of the time; in particular he gave a solution of
the brachistochrone, and investigated the form of the solid
of least resistance of which Newton in the Principia had
stated the result. He also wrote a treatise on analytical
conies, which -was published in 1707, and for nearly a century
was deemed a standard work on the subject.
Varignon.^ Pierre Varignon, born at Caen in 1654, and
died in Paris on Dec. 22, 1722, was an intimate friend of
Newton, Leibnitz, and the BernouUis, and, after I'Hospital, was
the earliest and most powerful advocate in France of the use of
the differential calculus. He realized the necessity of obtaining
a test for examining the convergency of series, but the
analytical difficulties were beyond his powers. He simplified
the proofs of many of the leading propositions in mechanics,
and in 1687 recast the treatment of the subject, basing it on
the composition of forces. His works were published at Paris
in 1725.
De Montmort. Nicole. Pierre Raymond de Moiitmort,
born at Paris on Oct. 27, 1678, and died there on Oct. 7,
1719, was interested in the subject of finite differences. He
determined in 1713 the sum of n terms of a finite series of
the form
^ See the 4loge by B. de Fontenelle, Paris, 1766.
CH. xvii] DE MONTMORT. NICOLE. PAEENT 371
a theorem which seems to have been independently re-
discovered by Chr. Goldbach in 1718. Francois Nicole, who
was born at Paris on Dec. 23, 1683, and died there on
Jan. 18^ 1758, published his Traite du calcul des differences
finies in 1717; it contains rules both for forming differences
and for effecting the summation of given series. Besides this,
in 1706 he wrote a work on roulettes, especially spherical
epicycloids; and in 1729 and 1731 he published memoirs on
Newton's essay on curves of the third degree.
Parent. Saurin. De Gua. Antoine Parent, born at
Paris on Sept. 16, 1666, and died there on Sept. 26, 1716,
wTote in 1700 on analytical geometry of three dimensions.
His works were collected and published in three volumes at
Paris in 1713. Joseph Saurin, born at Courtaison in 1659,
and died at Paris on Dec. 29, 1737, was the first to show how
the tangents at the multiple points of curves could be deter-
mined by analysis. Jean Paul de Gua de Halves was born at
Carcassonne in 1713, and died at Paris on June 2, 1785. He
published in 1740 a work on analytical geometry in which he
applied it, without the aid of the differential calculus, to find
the tangents, asymptotes, and various singular points of an
algebraical curve; and he further shewed how singular points
and isolated loops were affected by conical projection. He
gave the proof of Descartes's rule of signs which is to be
found in most modern works. It is not clear whether Descartes
ever proved it strictly, and Newton seems to have regarded it
as obvious.
Cramer. Gabriel Granier, born at Geneva in 1704, and
died at Bagnols in 1752, was professor at Geneva. The work
by which he is best known is his treatise on algebraic
curves 1 published in 1750, which, as far as it goes, is fairly
complete; it contains the earliest demonstration that a curve
•^ See Cantor, chapter cxvi.
372 CRAMER. RICCATI. FAGNANO [ch. xvii
of the nth. degree is in general determined if ^n{n + 3) points
on it be given. This work is still sometimes read. Besides
this, he edited the works of the two elder Bernoullis ; and
wrote on the physical cause of the spheroidal shape of the
planets and the motion of their apses, 1730, and on Newton's
treatment of cubic curves, 1746.
Riccati. Jacopo Francesco, Count Riccati, born at Venice
on May 28, 1676, and died at Treves on April 15, 1754, did
a great deal to disseminate a knowledge of the Newtonian
philosophy in Italy. Besides the equation known by his
name, certain cases of which he succeeded in integrating, he
discussed the question of the possibility of lowering the order
of a given differential equation. His works were published at
Treves in four volumes in 1758. He had two sons who wrote
on several minor points connected with the integral calculus
and differential equations, and applied the calculus to several
mechanical questions : these were Viricenzo, who was born in
1707 and died in 1775, and Giordano, who Avas born in 1709
and died in 1790.
Fagnano. Giulio Carlo, Count Fagnano, and Marquis de
Toschi, born at Sinigaglia on Dec. 6, 1682, and died on Sept. 26,
1766, may be said to have been the first writer who directed
attention to the theory of elliptic functions. Failing to rectify
the ellipse or hyperbola, Fagnano attempted to determine arcs
whose difference should be rectifiable. He also pointed out
the remarkable analogy existing between the integrals which
represent the arc of a circle and the arc of a lemniscate.
Finally he proved the formula
^ = 2^1og{(l-^)/(l+^)},
where i stands for J -l. His works were collected and
published in two volumes at Pesaro in 1750.
It was inevitable that some mathematicians should object to
methods of analysis founded on the infinitesimal calculus. The
most prominent of these were Viviani, De la Hire, and Rolle,
whose names were mentioned at the close of chapter xv.
CH.xvii] FAGNANO. CLAIRAUT 373
So far no one of the school of Leibnitz and the two elder
Bernoullis had shewn any exceptional ability, but by the action
of a number of second-rate writers the methods and language
of analytical geometry and the diflferential calculus had become
well known by about 1740. The close of this school is marked
by the appearance of Clairaut, D^Alemhert, and Daniel Bernoidli.
Their lives overlap the period considered in the next chapter,
but, though it is difficult to draw a sharp dividing line which
shall separate by a definite date the mathematicians there con-
sidered from those whose writings are discussed in this chapter,
I think that on the whole the works of these three writers are
best treated here.
Clairaut. Alexis Claude Clairaut was born at Paris on
May 13, 1713, and died there on May 17, 1765. He belongs
to the small group of children who, though of exceptional
precocity, survive and maintain their powers when grown up.
As early as the age of twelve he wrote a memoir on four
geometrical curves ; but his first important work was a treatise
on tortuous curves, published when he was eighteen — a work
which procured for him admission to the French Academy. In
1731 he gave a demonstration of the fact noted by Newton
that all curves of the third order were projections of one of five
parabolas.
In 1741 Clairaut went on a scientific expedition to measure
the length of a meridian degree on the earth's surface, and on
his return in 1743 he published his Theorie de la figure de la
terre. This is founded on a paper by Maclaurin, wherein it had
been shewn that a mass of homogeneous fluid set in rotation
about a line through its centre of mass would, under tlie mutual
attraction of its particles, take the form of a spheroid. This
work of Clairaut treated of heterogeneous spheroids and contains
the proof of his formula for the accelerating eff'ect of gravity in
a place of latitude I, namely,
^=6^{l-f (f m-€)sin2^},
where G is the value of equatorial gravity, m the ratio of the
374 CLAIRAUT. D'ALEMBERT [ch. xvii
centrifugal force to gravity at the equator, and e the ellipticity
of a meridian section of the earth. In 1849 Stokes ^ shewed
that the same result was true whatever was the interior con-
stitution or density of the earth, provided the surface was a
spheroid of equilibrium of small ellipticity.
Impressed by the power of geometry as shewn in the writings
of Newton and Maclaurin, Clairaut abandoned analysis, and his
next work, the Theorie de la lune, published in 1752, is strictly
Newtonian in character. This contains the explanation of the
motion of the apse which had previously puzzled astronomers,
and which Clairaut had at first deemed so inexplicable that he
was on the point of publishing a new hypothesis as to the law
of attraction when it occurred to him to carry the approximation
to the third order, and he thereupon found that the result was
in accordance with the observations. This was followed in 1754
by some lunar tables. Clairaut subsequently wrote various
papers on the orbit of the moon, and on the motion of comets
as affected by the perturbation of the planets, particularly on
the path of Halley's comet.
His growing popularity in society hindered his scientific
work : " engage," says Bossut, " a des soupers, a des veilles,
entraine par un goUt vif paur les femmes, voulant allier le
plaisir a ses travaux ordinaires, il perdit le repos, la sante,
enfin la vie a Page de cinquante-deux ans."
D'Alembert.2 Jean-le-Eooid D'Alembert was born at Paris
on November 16, 1717, and died there on October 29, 1783.
He was the illegitimate child of the chevalier Destouches.
Being abandoned by his mother on the steps of the little church
of St. Jean-le-Rond, which then nestled under the great porch
of Notre-Dame, he was taken to the parish commissary, who,
following the usual practice in such cases, gave him the Christian
name of Jean-le-Rond; I do not know by what authority he
^ See Ca7nbridge Philosophical Transactions, vol. viii, pp. 672-695.
^ Bertrand, Condorcet, and J. Bastien have left sketches of D'Alembert's
life. His literary works have been published, but there is no complete edition
of his scientific writings. Some papers and letters, discovered comparatively
recently, were published by C. Henry at Paris in 1887.
CH. xviij D'ALEMBERT 375
subsequently assumed the right to prefix de to his name. He
was boarded out by the parish with the wife of a glazier in a
small way of business who lived near the cathedral, and here he
found a real home, though a humble one. His father appears
to have looked after him, and paid for his going to a school
where he obtained a fair mathematical education.
An essay written by him in 1738 on the integral calculus,
and another in 1740 on "ducks and drakes" or ricochets,
attracted some attention, and in the same year he was elected
a member of the French Academy ; this was probably due to
the influence of his father. It is to his credit that he absolutely
refused to leave his adopted mother, with whom he continued
to live until her death in 1757. It cannot be said that she
sympathised with his success, for at the height of his fame she
remonstrated with him for wasting his talents on such work :
" Vous ne serez jamais qu'un philosophe," said she, " et qu'est-ce
qu'un philosophe 1 c'est un f ou qui se tourmente pendant sa vie,
pour qu'on parle de lui lorsqu'il n'y sera plus."
Nearly all his mathematical works were produced during
the years 1743 to 1754. The first of these was his Traite de
dynamique, published in 1743, in which he enunciates the prin-
ciple known by his name, namely, that the "internal forces of
inertia" (that is, the forces which resist acceleration) must be
equal and opposite to the forces which produce the acceleration.
This may be inferred from Newton's second reading of his third
law of motion, but the full consequences had not been realized
previously. The application of this principle enables us to obtain
the difi'erential equations of motion of any rigid system.
In 1744 D'Alembert published his Traite de Veqidlibre et
du mouvement des fluides, in which he applies his principle to
fluids; this led to partial differential equations which he was
then unable to solve. In 1745 he developed that part of the
subject which dealt with the motion of air in his Theorie generale
des vents, and this again led him to partial differential equations.
A second edition of this in 1746 was dedicated to Frederick the
Great of Prussia, and procured an invitation to Berlin and the
376 D'ALEMBERT [ch. xvii
offer of a pension ; he declined tlie former, but subsequently,
after some pressing, pocketed his pride and the latter. In 1747
he applied the differential calculus to the problem of a vibrating
string, and again arrived at a partial differential equation.
His analysis had three times brought him to an equation
of the form
and he now succeeded in shewing that it was satisfied by
^^ = cfi(^x + f) + ^p{x- t),
where 4* and \j^ are arbitrary functions. It may be interesting
to give his solution which was published in the transactions
of the Berlin Academy for 1747. He begins by saying that, if
— be denoted by p and ^ by q^ then
du =pdx + qdU
But, by the given equation, — = — , and therefore pdt + qdx is
also an exact differential : denote it by dv.
Therefore dv =pdt + qdx.
Hence dii + dv = (pdx + qdt) + {j^dt + qdx) — (p + q) (dx + dt),
and du -dv = (pdx + qdt) - (pdt + qdx) = (p - q) (dx - dt).
Thus u + v must be a function of x + t, and u-v must be a
function of ^ - ^. We may therefore put
qt + 'V='2cf)(x + t),
and u-v = 2\p(x- 1).
Hence u^(f>(x-\-t) + \p (x - t).
D'Alembert added that the conditions of the physical problem
of a vibrating string demand that, when x = 0, u should vanish
for all values of t. Hence identically
<i>(t) + ^(-t)=^0.
Assuming that both functions can be expanded in integral
CH. xvii] D'ALEMBERT. DANIEL BERNOULLI 377
powers of t, this requires that they should contain only odd
powers. Hence
Therefore u = <fi{x + t) + (fi (x - t).
Euler now took the matter up and shewed that the equation
of the form of the string was -^ = ^2^, and that the general
integral was u = 4> {^x + at) + ip {x - at), where <^ and i// are
arbitrary functions.
The chief remaining contributions of D'Alembert to mathe-
matics w^ere on physical astronomy, especially on the precession
of the equinoxes, and on variations in the obliquity of the
ecliptic. These were collected in his Systeme du monde, pub-
lished in three volumes in 1754.
During the latter part of his life he was mainly occupied
with the great French encyclopaedia. For this he wrote the
introduction, and numerous philosophical and mathematical
articles; the best are those on geometry and on probabilities.
His style is brilliant, but not polished, and faithfully reflects
his character, which was bold, honest, and frank. He defended
a severe criticism which he had offered on some mediocre work
by the remark, "j'aime mieux etre incivil qu'ennuye " ; and
vdth his dislike of sycophants and bores it is not surprising that
during his life he had more enemies than friends.
Daniel Bernoulli. ^ Daniel Bernoidli, whose name I mentioned
above, and who was by far the ablest of the younger Bernoullis,
was a contemporary and intimate friend of Euler, whose works
^ The only accmmt of Daniel Bernoulli's life with which I am acquainted is
the elflffe by h is friend Condorcet. Marie Jean A ntoine Nicolas Caritat, Marquis
de Condorcet, Avas born in Picardy on Sept. 17, 1743, and fell a victim to the
republican terrorists on March 28, 1794. He was secretary to the Academy,
and is the author of numerous elor/es. He is perhaps more celebrated for his
studies in philosophy, literature, and politics than in mathematics, but his
mathematical treatment of probabilities, and his discussion of differential
equations and finite differences, shew an ability which might have put him in
the first rank had he concentrated his attention on mathematics. He sacri-
ficed himself in a vain effort to guide the revolutionary torrent into a consti-
tutional channel.
378 ENGLISH SCHOOL [ch. xvii
are mentioned in the next chapter. Daniel Bernoulli was born
on Feb. 9, 1700, and died at Bale, where he was professor of
natural philosophy, on March 17, 1782. He went to Petrograd
in 1724 as professor of mathematics, but the roughness of the
social life was distasteful to him, and he was not sorry when
a temporary illness in 1733 allowed him to plead his health as
an excuse for leaving. He then returned to Bale, and held
successively chairs of medicine, metaphysics, and natural philo-
sophy there.
His earliest mathematical work was the Ejcercitationes, pub-
lished in 1724, which contains a solution of the differential
equation proposed by Riccati. Two years later he pointed out
for the first time the frequent desirability of resolving a com-
pound motion into motions of translation and motions of rota-
tion. His chief work is his Hydrodynamica, published in 1738 ;
it resembles Lagrange's Mecanique analytique in being arranged
so that all the results are consequences of a single principle,
namely, in this case, the conservation of energy. This was
followed by a memoir on the theory of the tides, to which, con-
jointly with memoirs by Euler and Maclaurin, a prize was
awarded by the French Academy : these three memoirs contain
all that was done on this subject between the publication of
Newton's Principia and the investigations of Laplace. Ber-
noulli also wrote a large number of papers on various mechanical
questions, especially on problems connected with vibrating strings,
and the solutions given by Taylor and by D'Alembert. He is
the earliest writer who attempted to formulate a kinetic theory
of gases, and he applied the idea to explain the law associated
with the names of Boyle and Mariotte.
The English mathematicians of the eighteenth century,
I have reserved a notice of the English mathematicians who
succeeded Newton, in order that the members of the English
school may be all treated together. It was almost a matter of
course that the English should at first have adopted the notation
CH. xvii] DAVID GREGORY. HALLEY 379
of Newton in the infinitesimal calculus in preference to that of
Leibnitz, and consequently the English school would in any case
have developed on somewhat different lines to that on the conti-
nent, where a knowledge of the infinitesimal calculus was derived
solely from Leibnitz and the Bernoullis. But this separation
into two distinct schools became very marked omng to the
action of Leibnitz and John Bernoulli, which was naturally
resented by Newton's friends ; and so for forty or fifty years, to
the disadvantage of both sides, the quarrel raged. The leading
members of the English school were Cotes, Demoivre, Ditton,
David Gregory^ Halley, Madaurin^ Simjjson, and Taylor. I
may, however, again remind my readers that as we approach
modern times the number of capable mathematicians in Britain,
France, Germany, and Italy becomes very considerable, but that
in a popular sketch like this book it is only the leading men
whom I propose to mention.
To David Gregory, Halley, and Ditton I need devote but few
words.
David Gregory. David Gregm-y, the nephew of the James
Gregory mentioned above, born at Aberdeen on June 24, 1661,
and died at Maidenhead on Oct. 10, 1708, was appointed
professor at Edinburgh in 1684, and in 1691 was on Newton's
recommendation elected Savilian professor at Oxford. His
chief works are one on geometry, issued in 1684 ; one on optics,
published in 1695, which contains [p. 98] the earliest suggestion
of the possibility of making an achromatic combination of lenses ;
and one on the Newtonian geometry, physics, and astronomy,
issued in 1702.
Halley. Edmund Halley, born in London in 1656, and
died at Greenwich in 1742, was educated at St. Paul's School,
London, and Queen's College, Oxford, in 1703 succeeded Wallis
as Savilian professor, and subsequently in 1720 was appointed
astronomer-royal in succession to Flamsteed, whose Historia
Coelestis Britannica he edited ; the first and imperfect edition
was issued in 1712. Halley's name will be recollected for the
generous manner in which he secured the immediate publication
380 DITTON. TAYLOR [ch. xvii
of Newton's Principia in 1687. Most of his original Avork was
on astronomy and allied subjects, and lies outside the limits of
this book ; it may be, however, said that the work is of excellent
quality, and both Lalande and Mairan speak of it in the highest
terms. Halley conjecturally restored the eighth and lost book
of the conies of Apollonius, and in 1710 brought out a magnifi-
cent edition of the whole work ; he also edited the works of
Serenus, those of Menelaus, and some of the minor works of
Apollonius. He was in his turn succeeded at Greenwich as
astronomer-royal by Bradley.^
Ditton. llumphry Ditton was born at Salisbury on May 29,
1675, and died in London in 1715 at Christ's Hospital, where
he was mathematical master. He does not seem to have paid
much attention to mathematics until he came to London about
1705, and his early death was a distinct loss to English science.
He published in 1706 a textbook on fluxions; this and another
similar work by William Jones, which was issued in 1711,
occupied in England much the same place that I'Hospital's
treatise did in France. In 1709 Ditton issued an algebra, and
in 1712 a treatise on perspective. He also wrote numerous
papers in the Philosophical Transactions. He was the earliest
writer to attempt to explain the phenomenon of capillarity
on mathematical principles; and he invented a method for
finding the longitude, which has been since used on various
occasions.
Taylor.- Broolc Taylor, born at Edmonton on August 18,
1685, and died in London on J3ecember 29, 1731, was educated
at St. John's College, Cambridge, and was among the most
^ James Bradley, born in Gloucestershire in 1692, and died in 1762, was
the most distinguished astronomer of the first half of the eighteenth century.
Among his more important discoveries were the explanation of astronomical
aberration (1729), the cause of nutation (1748), and his empirical formula
for corrections for refraction. It is perhaps not too much to say that he was
the first astronomer who made the art of observing part of a methodical
science.
'^ An account of his life by Sir William Young is prefixed to the Contem-
platio Philosophica. This was printed at London in 1793 for private
circulation and is now extremely rare.
CH.XVII] TAYLOR 381
enthusiastic of Newton's admirers. From the year 1712 onwards
he wrote numerous papers in the Philosophical Transactions^
in which, among other things, he discussed the motion of
projectiles, the centre of oscillation, and the forms take|^ by
liquids when raised by capillarity. In 1719 he resigned the
secretaryship of the Royal Society and abandoned the study
of mathematics. His earliest Work, and that by which he is
generally known, is his Methodus Incrementorum Directa et
Inversa^ published in London in 1715. This contains [prop. 7]
a proof of the well-known theorem
/(^ + h) =f{x) + hf (x) + 1|/" (^) + . . .,
by which a function of a single variable can be expanded in
powers of it. He does not consider the convergency of the
series, and the proof which involves numerous assumptions is
not worth reproducing. The work also includes several
theorems on interpolation. Taylor was the earliest writer to
deal with theorems on the change of the independent variable ;
he was perhaps the first to realize the possibility of a calculus
of operation, and just as he denotes the nth differential coeffi-
cient of ^ by ^n, so he uses y_j to represent the integral of ^ ;
lastly, he is usually recognized as the creator of the theory of
finite differences.
The applications of the calculus to various questions given in
the Methodus have hardly received that attention they deserve.
The most important of them is the theory of the transverse
vibrations of strings, a problem which had baffled previous
investigators. In this investigation Taylor shews that the
number of half -vibrations executed in a second is
I where L is the length of the string, N its weight, P the weight
' which stretches it, and D the length of a seconds pendulum.
This is correct, but in arriving at it he assumes that every
I point of the string will pass through its position of equi-
382 TAYLOR. COTES [ch. xvii
librium at the same instant, a restriction which D'Alembert
subsequently shewed to be unnecessary. Taylor also found the
form which the string assumes at any instant.
The Methodus also contains the earliest determination of
the differential equation of the path of a ray of light when
traversing a heterogeneous medium ; and, assuming that the
density of the air depends only on its distance from the
earth's surface, Taylor obtained by means of quadratures the
approximate form of the curve. The form of the catenary and
the determination of the centres of oscillation and percussion
are also discussed.
A treatise on perspective by Taylor, published in 1719,
contains the earliest general enunciation of the principle of
vanishing points ; though the idea of vanishing points for
horizontal and parallel lines in a picture hung in a vertical
plane had been enunciated by Guido Ubaldi in his Perspectivae
Libri, Pisa, 1600, and by Stevinus in his Sciagraphia, Ley den,
1608.
Cotes. Roger Cotes was born near Leicester on July 10,
1682, and died at Cambridge on June 5, 1716. He was
educated at Trinity College, Cambridge, of which society he
was a fellow, and in 1706 was elected to the newly- created
Plumian chair of astronomy in the university, of Cambridge.
From 1709 to 1713 his time was mainly occupied in editing
the second edition of the Principia. The remark of Newton
that if only Cotes had lived " we might have known some-
thing" indicates the opinion of his abilities held by most of
his contemporaries.
Cotes's writings were collected and published in 1722
under the titles Harmonia Mensurarum and Opera Miscel-
lanea. His lectures on hydrostatics were published in 1738.
A large part of the Harmonia Mensurarum is given up
to the decomposition and integration of rational algebraical
expressions. That part which deals with the theory of partial
fractions was left unfinished, but was completed by Demoivre.
Cotes's theorem in trigonometry, which depends on forming the
CH.XVII] COTES. DEMOIVRE 383
quadratic factors of a:^ - 1, is well known. The proposition that
" if from a fixed point 0 a line be drawn cutting a curve in
^1) Q^i •••> Qni ^iid a point P be taken on the line so that the
reciprocal of OP is the arithmetic mean of the reciprocals of
OQi, OQo, ..., OQm then the locus of P will be a straight line"
is also due to Cotes. The title of the book was derived from
the latter theorem. The Opera Miscellanea contains a paper
on the method for determining the most probable result from
a number of observations. This was the earliest attempt to
frame a theory of errors. It also contains essays on Newton's
Methodus Differentialis, on the construction of tables by the
method of differences, on the descent of a body under gravity,
on the cycloidal pendulum, and on projectiles.
Demoivre. Abraham Demoivre (more correctly written
as de Moivre) was born at Yitry on May 26, 1667, and died in
London on November 27, 1754. His parents came to England
when he was a boy, and his education and friends were alike
English. His interest in the higher mathematics is said to
have originated in his coming by chance across a copy of
Newton's Principia. From the eloge on him delivered in 1754
before the French Academy it would seem that his work
as a teacher of mathematics had led him to the house of the
Earl of Devonshire at the instant when New^ton, who had
asked permission to present a copy of his work to the earl,
was coming out. Taking up the book, and charmed by the far-
reaching conclusions and the apparent simplicity of the reasoning,
Demoivre thought nothing would be easier than to master the
subject, but to his surprise found that to follow the argument
overtaxed his powers. He, however, bought a copy, and as he
had but little leisure he tore out the pages in order to carry one
or two of them loose in his pocket so that he could study them
in the intervals of his work as a teacher. Subsequently he
joined the Royal Society, and became intimately connected with
Newton, Halley, and other mathematicians of the English
school. The manner of his death has a certain interest for
psychologists. Shortly before it he declared that it was neces-
384 DEMOIVRE. MACLAUEIN [ch. xvii
sary for him to sleep some ten minutes or a quarter of an hour
longer each day than the preceding one. The day after he had thus
reached a total of something over twenty-three hours he slept
up to the limit of twenty-four hours, and then died in his sleep.
He is best known for having, together with Lambert,
created that part of trigonometry which deals with imaginary
quantities. Two theorems on this part of the subject are still
connected with his name, namely, that which asserts that
cos nx + i sin nx is one of the values of (cos x + i sin x^, and
that which gives the various quadratic factors of x'^^^ - 2px'''^ + 1 .
His chief works, other than numerous papers in the Philo-
sophical Transactions, were The Doctrine of Chances, published
in 1718, and the Miscellanea Anxilytica, published in 1730. In
the former the theory of recurring series w^as first given, and
the theory of partial fractions which Cotes's premature death
had left unfinished was completed, while the rule for finding
the probability of a compound event w^as enunciated. The
latter book, besides the trigonometrical propositions mentioned
above, contains some theorems in astronomy, but they are treated
as problems in analysis.
Maclaurin.1 Colin Maclaurin, who was born at Kilmodan
in Argyllshire in February 1698, and died at York on June 14,
1746, was educated at the university of Glasgow; in 1717
he was elected, at the early age of nineteen, professor of
mathematics at Aberdeen; and in 1725 he was appointed the
deputy of the mathematical professor at Edinburgh, and ulti-
mately succeeded him. There was some difficulty in securing a
stipend for a deputy, and Newton privately wrote offering to
bear the cost so as to enable the university to secure the services
of Maclaurin. Maclaurin took an active part in opposing the
advance of the Young Pretender in 1745; on the approach of
the Highlanders he fled to York, but the exposure in the
trenches at Edinburgh and the privations he endured in his
escape proved fatal to him.
^ A sketch of Maclaurin's life is prefixed to his posthumous account of
Newton's discoveries, London, 1748.
CH. xvii] MACLAURIN 385
His chief works are his Geometria Organica, London, 1720;
his De Linearum Geometricarum Proprietatibus^ London, 1720;
his Treatise on Fluxions, Edinburgh, 1742; his Algebra,
London, 1748 ; and his Account of Newton^ s Discoveries, London,
1748.
The first section of the first part of the Geometria Organica
is on conies ; the second on nodal cubics ; the third on other
cubics and on quartics ; and the fourth section is on general
properties of curves. Newton had shewn that, if two angles
bounded by straight lines turn round their respective summits
so that the point of intersection of two of these lines moves
along a straight line, the other point of intersection will
describe a conic ; and, if the first point move along a conic, the
second will describe a quartic. Maclaurin gave an analytical
discussion of the general theorem, and shewed how by this
method various curves could be practically traced. This work
contains an elaborate discussion on curves and their pedals,
a branch of geometry which he had created in two papers
published in the Fhilosophical Transactions for 1718 and
1719.
The second part of the work is divided into three sections
and an appendix. The first section contains a proof of Cotes's
theorem above alluded to ; and also the analogous theorem
(discovered by himself ) that, if a straight line OP^P^--- drawn
through a fixed point 0 cut a curve of the nth. degree in n
points Pp Pg'-'-j ^^^ if *^® tangents at Pj, Pg,--- cut a fixed
line Ox in points A^, -^-z^-'-i ^^®^ *^® ^^^^^ ^^ ^^® reciprocals
of the distances OA^, OA^,... is constant for all positions of
the line OP^P^-- These two theorem.s are generalizations of
those given by Newton on diameters and asymptotes. Either
is deducible from- the other. In the second and third sections
these theorems are applied to conies and cubics ; most of the
harmonic properties connected with a quadrilateral inscribed
in a conic are determined ; and in particular the theorem on
an inscribed hexagon which is known by the name of Pascal is
deduced. Pascal's essay was not published till 1779, and
2c
386 MACLAURIN [ch. xvii
the earliest printed enunciation of his theorem was that given
by Maclaurin. Amongst other propositions he shews that,
if a quadrilateral be inscribed in a cubic, and if the points
of intersection of the opposite sides also lie on the curve, then
the tangents to the cubic at any two opposite angles of the
quadrilateral will meet on the curve. In the fourth section
he considers some theorems on central force. The fifth section
contains some theorems on the description of curves through
given points. One of these (which includes Pascal's as a par-
ticular case) is that if a polygon be deformed so that while
each of its sides passes through a fixed point its angles (save
one) describe respectively curves of the mth, 7ith, pih,...
degrees, then shall a remaining angle describe a curve of the
degree 2mnp...; but if the given points be collinear, the
resulting curve will be only of the degree mnp.... This essay
was reprinted with additions in the Philosophical Transactions
for 1735.
The Treatise of Fluxions, published in 1742, was the first
logical and systematic exposition of the method of fluxions.
The cause of its publication was an attack by Berkeley on the
principles of the infinitesimal calculus. In it [art. 751, p. 610]
Maclaurin gave a proof of the theorem that
f{x)=f{0) + xf{0) + p"{0) + ....
This was obtained in the manner given in many modern text-
books by assuming that f(x) can be expanded in a form like
f{x) = ^0 + ^1^ + ^2^^ + '■-,
then, on differentiating and putting x = 0 in the successive
results, the values of Aq, A^,... are obtained; but he did not
investigate the convergency of the series. The result had been
previously given in 1730 by James Stirling in his Methodus
Differentialis [p. 102], and of course is at once deducible from
Taylor's theorem. Maclaurin also here enunciated [art. 350,
p. 289] the important theorem that, if 4i{x) be positive and
decrease as x increases from x = a io x = ao , then the series
CH. xvii] MACLAURIN 387
I cfi(a) + cf>{a+l) + (f>{a-\-2)+...
is convergent or divergent as the integral from oc = a to x = 00 of
(f)(x) is finite or infinite. The theorem had been given by
Euler ^ in 1732, but in so awkward a form that its value escaped
general attention. Maclaurin here also gave the correct theory of
maxima and minima, and rules for finding and discriminating
multiple points.
This treatise is, however, especially valuable for the solutions
it contains of numerous problems in geometry, statics, the theory
of attractions, and astronomy. To solve these Maclaurin re-
verted to classical methodic and so powerful did these processes
seem, when used by him, that Clairaut, after reading the work,
abandoned analysis, and attacked the problem of the figure of
the earth again by pure geometry. At a later tim* this part of
the book was described by Lagrange as the "chef-d'oeuvre de
geometrie qu'on pent comparer a tout ce qu'Archimede nous a
laisse de plus beau et de plus ingenieux." Maclaurin also
determined the attraction of a homogeneous ellipsoid at* an
internal point, and gave some theorems on its attraction at an
external point ; in attacking these questions he introduced
the conception of level surfaces, that is, surfaces at every point
of which the resultant attraction is perpendicular to the surface.
No further advance in the theory of attractions was made until
Lagrange in 1773 introduced the idea of the potential. Mac-
laurin also shewed that a spheroid was a possible form of
equilibrium of a mass of homogeneous liquid rotating about an
axis passing through its centre of mass. Finally he discussed
the tides ; this part had been previously published (in 1740) and
had received a prize from the French Academy.
Among Maclaurin's minor works is his Algebra, published
in 1748, and founded on Newton's Universal Arithmetic. It
contains the results of some early papers of Maclaurin ; notably
of two, written in 1726 and 1729, on the number of imaginary
roots of an equation, suggested by Newton's theorem ; and of
^ See Cantor, vol. iii, p. 663.
388 MACLAUEIN. STEWART. SIMPSON [ch. xvii
one, written in 1729, containing the well-known rule for finding
equal roots by means of the derived equation. In this book
negative quantities are treated as being not less real than
positive quantities. To this work a treatise, entitled De
Linearum Geometricarum Proprietatibus Generalibus, was added
as an appendix; besides the paper of 1720 above alluded to,
it contains some additional and elegant theorems. Maclaurin
also produced in 1728 an exposition of the Newtonian philosophy,
which is incorporated in the posthumous work printed in 1748.
Almost the last paper he wrote was one printed in the Philo-
sophical Transactions for 1743 in which he discussed from a
mathematical point of view the form of a bee's cell.
Maclaurin was one of the most able mathematicians of the
eighteenth century, but his influence on the progress of British
mathematics * was on the whole unfortunate. By himself
abandoning the use both of analysis and of the infinitesimal
calculus, he induced Newton's countrymen to confine themselves
to Newton's methods, and it was not until about 1820, when
the differential calculus was introduced into the Cambridge
curriculum, that English mathematicians made any general use
of the more powerful methods of modern analysis.
Stewart. Maclaurin was succeeded in his chair at Edinburgh
by his pupil Matthew Stewart^ born at Rothesay in 1717 and
died at Edinburgh on January 23, 1785, a mathematician of
considerable power, to whom I allude in passing, for his theorems
on the problem of three bodies, and for his discussion, treated by
transversals and involution, of the properties of the circle and
straight line.
Simpson.^ The last member of the English school whom
I need mention here is Thonias Simpson^ who was born in
Leicestershire on August 20, 1710, and died on May 14, 1761.
His father was a weaver, and he owed his education to his own
efforts. His mathematical interests were first aroused by the
1 A sketch of Simpson's life, with a bibliography of his writings, by J.
Bevis and C. Hutton, Avas published in London in 1764. A short memoir is
also prefixed to the later editions of his work on fluxions.
CH. xvii] SIMPSON 389
solar eclipse which took place in 1724, and with the aid of a
fortune-telling pedlar he mastered Cocker's Arithmetic and the
elements of algebra. He then gave up his weaving and became
an usher at a school, and by constant and laborious efforts
improved his mathematical education, so that by 1735 he w^as
able to solve several questions which had been recently proposed
and which involved the infinitesimal calculus. He next moved
to London, and in 1743 was appointed professor of mathematics
at Woolwich, a post which he continued to occupy till his death.
The works published by Simpson prove him to have been
a man of extraordinary natural genius and extreme industry.
The most important of them are his Fluxions, 1737 and 1750,
with numerous applications to physics and astronomy ; his Laws
of Chobnce and his Essays, 1740; his theory of Annuities and
Reversions (a branch of mathematics that is due to James
Dodson, died in 1757, who was a master at Christ's Hospital,
London), with tables of the value of lives, 1742; his Disserta-
tions, 1743, in which the figure of the earth, the force of
attraction at the surface of a nearly spherical body, the theory
of the tides, and the law of astronomical refraction are discussed ;
his Algebra, 1745; his Geometry, 1747; his Trigonometry,
1748, in which he introduced the current abbreviations for
the trigonometrical functions; his Select Exercises, 1752, con-
taining the solutions of numerous problems and a theory of
gunnery; and lastly, his Miscellaneous Tracts, 1754.
The work last mentioned consists of eight memoirs, and these
contain his best known investigations. The first three papers
are on various problems in astronomy; the fourth is on the
theory of mean observations ; the fifth and sixth on problems in
fluxions and algebra ; the seventh contains a general solution of
the isoperimetrical problem; the eighth contains a discussion
of the third and ninth sections of the Principia, and their
application to the lunar orbit. In this last memoir Simpson
obtained a differential equation for the motion of the apse of the
lunar orbit similar to that arrived at by Clairaut, but instead of
solving it by successive approximations, he deduced a general
390 SIMPSON [oh. XVII
solution by indeterminate coefficients. The result agrees with
that given by Clairaut. Simj^son solved this problem in 1747,
two years later than the publication of Clairaut's memoir,
but the solution was discovered independently of Clairaut's
researches, of which Simpson first heard in 1748.
391
CHAPTER XVIII.
LAGRANGE, LAPLACE, AND THEIR CONTEMPORARIES.^
CIRC. 1740-1830.
The last chapter contains the history of two separate schools
— the continental and the British. In the early years of the
eighteenth century the English school appeared vigorous and
fruitful, but decadence rapidly set in, and after the deaths of
Maclaurin and Simpson no British mathematician appeared
who is at all comparable to the continental mathematicians of
the latter half of the eighteenth century. This fact is partly
explicable by the isolation of the school, partly by its tendency
to rely too exclusively on geometrical and fiuxional methods.
Some attention was, however, given to practical science, but,
except for a few remarks at the end of this chapter, I do not
think it necessary to discuss English mathematics in detail,
until about 1820, when analytical methods again came into
vogue.
On the continent, under the influence of John Bernoulli, the
calculus had become an instrument of great analytical power
^ A fourth volume of M. Cantor's History, covering the period from 1759
to 1799, was brought out in 1907. It contains memoirs by S. Giinther
on the mathematics of the period ; by F. Cajori on arithmetic, algebra, and
.numbers ; by E. Netto on series, imaginaries, &c. ; by V. von Braunmiihl on
trigonometry ; by V. Bobynin and G. Loria on pure geometry ; by V.
Kommerell on analytical geometry ; by G. Vivanti on the infinitesimal
calculus ; and by C. R. Wallner on diflerential equations.
392 LAGRANGE, LAPLACE, ETC. [ch. xvni
expressed in an admirable notation — and for practical applica-
tions it is impossible to over-estimate the value of a good
notation. The subject of mechanics remained, however, in much
the condition in which Newton had left it, until D'Alembert, by-
making use of the differential calculus, did something to extend
it. Universal gravitation as enunciated in the Principia was
accepted as an established fact, but the geometrical methods
adopted in proving it were difficult to follow or to use in
analogous problems ; Maclaurin, Simpson, and Clairaut may
be regarded as the last mathematicians of distinction who
employed them. Lastly, the Newtonian theory of light was
generally received as correct.
The leading mathematicians of the era on which we are now
entering are Euler, Lagrange, Laplace, and Legendre. Briefly
we may say that Euler extended, summed up, and completed
the work of his predecessors; while Lagrange with almost un-
rivalled skill developed the infinitesimal calculus and theoretical
mechanics, and presented them in forms similar to those in
which we now know^ them. At the same time Laplace made
some additions to the infinitesimal calculus, and applied that
calculus to the theory of universal gravitation ; he also created
a calculus of probabilities. Legendre invented spherical har-
monic analysis and elliptic integrals, and added to the theory
, of numbers. The works of these writers are still standard
[authorities. I shall content myself with a mere sketch of the
chief discoveries embodied in them, referring any one who mshes
to know more to the works themselves, Lagrange, Laplace,
and Legendre created a French school of mathematics of which
the younger members are divided into two groups ; one (includ-
ing Poisson and Fourier) began to apply mathematical analysis to
physics, and the other (including Monge, Carnot, and Poncelet)
created modern geometry. Strictly speaking, some of the great
mathematicians of recent times, such as Gauss and Abel, were
contemporaries of the mathematicians last named ; but, except
for this remark, I think it convenient to defer any consideration
of them to the next chapter.
CH. xviii] EULER 393
The development of analysis and mechanics.
Euler.^ Leonhard Euler was born at Bale on April 15, I7t)7,
and died at Petrograd on September 7, 1783. He was
the son of a Lutheran minister who had settled at Bale, and
was educated in his native town under the direction of John
Bernoulli, with whose sons Daniel and Nicholas he formed a
lifelong friendship. When, in 1725, the younger Bernoullis
went to Russia, on the invitation of the empress, they procured
a place there for Euler, which in 1733 he exchanged for the
chair of mathematics, then vacated by Daniel Bernoulli. The
severity of the climate affected his eyesight, and in 1735 he lost
the use of one eye completely. In 1741 he moved to Berlin at
the request, or rather command, of Frederick the Great; here
he stayed till 1766, when he returned to Russia, and was
succeeded at Berlin by Lagrange. Within two or three years of
his going back to Petrograd he became blind ; but in spite
of this, and although his house, together with many of his
papers, were burnt in 1771, he recast and improved most of his
earlier works. He died of apoplexy in 1783. He was married
twice.
I think we may sum up Euler's work by saying that he
created a good deal of analysis, and revised almost all the
branches of pure mathematics which were then known, filling
up the details, adding proofs, and arranging the whole in a
consistent form. Such work is very important, and it is
fortunate for science when it falls into hands as competent as
those of Euler.
Euler wrote an immense number of memoirs on all kinds of
mathematical subjects. His chief works, in which many of the
results of earlier memoirs are embodied, are as follows.
^ The chief facts in Euler's life are given by N. Fuss, and a list of Euler's
writings is prefixed to his Correspondence, 2 vols., Petrograd, 1843 ; see also
Index Operum Euleri by J. G. Hagen, Berlin, 1896. Euler's earlier
works are discussed by Cantor, chapters cxi, cxiii, cxv, and cxvii. No
complete edition of Euler's writings has been published, though the work
has been begun twice.
M
-/;
't^crvAj^ >iJ y "(^ I
394 LAGRANGE, LAPLACE, ETC. [ch. xviii
In the first place, he wrote in 1748 his Introductio in
Analysin Injinitorum, which was intended to serve as an intro-
duction to pure analytical mathematics. This is divided into
two parts.
The first part of the Ainalysis Injinitorum contains the bulk
of the matter which is to be found in modern text -books on
algebra, theory of equations, and trigonometry. In the algebra
he paid particular attention to the exjiansion of various func-
tions in series, and to the summation of given series ; and
^pointed out explicitly that an infinite series cannot be safely
Y employed unless it is convergent. In the trigonometry, much
of which is f ounded^ on F. C. Mayer's Arithmetic of Sines, which
had been published in 1727, Euler developed the idea of John
Bernoulli, that the subject was a branch of analysis and not a
mere appendage of astronomy or geometry. He also introduced
(contemporaneously with Simpson) the current abbreviations for
the trigonometrical functions, and shewed that the trigono-
metrical and exponential functions were connected by the
relation cos ^ + ^ sin ^ = e^^.
Here, too [pp. 85, 90, 93], we meet the symbol e used to
denote the base of the Napierian logarithms, namely, the incom-
mensurable number 2*71828..., and the symbol tt used to denote
the incommensurable number 3* 141 59.... The use of a single
symbol to denote the number 2-71828... seems to be due to
Cotes, who denoted it by if ; Euler in 1731 denoted it by e.
To the best of my knowledge, Newton had been the first to
employ the literal exponential notation, and Euler, using the
form a^, had taken a as the base of any system of logarithms. It
is probable that the choice of e for a particular base was deter-
mined by its being the vowel consecutive to a. The use of a
single symbol to denote the number 3*14159... appears to have
been introduced about the beginning of the eighteenth century.
W. Jones in 1706 represented it by tt, a symbol which had been
used by Oughtred in 1647, and by Barrow a few years later, to
denote the periphery of a circle. John Bernoulli represented
the number by c; Euler in 1734 denoted it by jt>, and in
CH. XVIII
EULER 395
a letter of 1736 (in which he enunciated the theorem that the
sum of the squares of the reciprocals of the natural numbers
is T^/g) he used the letter c; Chr. Goldbach in 1742 used tt;
and after the publication of Euler's Analysis the symbol tt was
generally employed.
The numbers e and tt would enter into mathematical analysis
from whatever side the subject was approached. The latter
represents among other things the ratio of the circumference of
a circle to its diameter, but it is a mere accident that that is
taken for its definition. De Morgan in the Budget of Paradoxes
tells an anecdote which illustrates how little the usual definition
suggests its real origin. He was explaining to an actuary what
was the chance that at the end of a given time a certain propor-
tion of some group of people would be alive ; and quoted the
actuarial formula involving tt, which, in answer to a question, he
explained stood for the ratio of the circumference of a circle to
its diameter. His acquaintance, who had so far listened to the
explanation with interest, interrupted him and explained, " My
dear friend, that must be a delusion ; what can a circle have to
do with the number of people alive at the end of a given
time?"
The second part of the ATialysis Infinitorum is on analytical
geometry. Euler commenced this part by dividing curves into
algebraical and transcendental, and established a variety of pro-
positions which are true for all algebraical curves. He then
applied these to the general equation of the second degree in
two dimensions, shewed that it represents the various conic
sections, and deduced most of their properties from the general
equation. He also considered the classification of cubic, quartic,
and other algebraical curves. He next discussed the question as
to what surfaces are represented by the general equation of the
second degree in three dimensions, and how they may be dis-
criminated one from the other : some of these surfaces had not
been previously investigated. In the course of this analysis he
laid down the rules for the transformation of co-ordinates in
space. Here also we find the earliest attempt to bring the
I
396 LAGRANGE, LAPLACE, ETC. [ch. xviii
curvature of surfaces within the ___domain of mathematics, and the
first complete discussion of tortuous curves.
The Aimlyds Infinitorum was followed in 1755 by the
Institutiones Calculi Differentialis, to which it was intended as
an introduction. This is the first text-book on the differential
calculus which has any claim to be regarded as complete, and it
may be said that until recently many modern treatises on the
subject are based on it ; at the same time it should be added
that the exposition of the principles of the subject is often prolix
and obscure, and sometimes not altogether accurate.
This series of works was completed by the publication in
three volumes in 1768 to 1770 of the Institutiones Calculi
Integralis, in which the results of several of Euler's earlier
memoirs on the same subject and on differential equations are
included. This, like the similar treatise on the differential
calculus, summed up what was then known on the subject, but
many of the theorems were recast and the proofs improved.
The Beta and Gamma ^ functions were invented by Euler and
are discussed here, but only as illustrations of methods of
reduction and integration. His treatment of elliptic integrals
is superficial; it was suggested by a theorem, given by John
Landen in the Philosophical Transactions for 1775, connecting
the arcs of a hyperbola and an ellipse. Euler's works that
form this trilogy have gone through numerous subsequent
editions.
The classic problems on isoperimetrical curves, the brachisto-
chrone in a resisting medium, and the theory of geodesies (all of
which had been suggested by his master, John Bernoulli) had
engaged Euler's attention at an early date ; and in solving them
he was led to the calculus of variations. The idea of this was
given in his Curvarum Maximi Minimive Proprietate Gaudentium
Inventio, published in 1741 and extended in 1744, but the
complete development of the new calculus was first effected by
Lagrange in 1759. The method used by Lagrange is described
^ The history of the Gamma function is given in a monograph by Brunei in
the AJenioires de la societe des sciences, Bordeaux, 1886.
CH. xviii] EULER 397
in Euler's integral calculus, and is the same as that given in
most modern text-books on the subject.
In 1770 Euler published his Volhtdndige A^ileitung znr
Algebra. A French translation, with numerous and valuable
additions by Lagrange, was brought out in 1774 ; and a
treatise on arithmetic by Euler was appended to it. The first
volume treats of determinate algebra. This contains one of
the earliest attempts to place the fundamental processes on a
scientific basis : the same subject had attracted D'Alembert's
attention. \'his work also includes the proof of the binomial
theorem for an unrestricted real index which is still known by
Euler's name ;- the proof is founded on the principle ot tJie
permanence ot equivalent forms, but Euler made no attempt to
investl^a^"f^ thp^ r^.nnvprgpnfy nf tjiPi rpHps • tha.t ViP should have
omitted this essential step is the more curious as he had himself
recognized the necessity of considering the convergency of
infinite series: Vandermonde's proof given in 1764 suffers from
tne same aeiect.
The second volume of the algebra treats of indeterminate
or Diophantine algebra. This contains the solutions of some
of the problems proposed by Fermat, and which had hitherto
remained unsolved.
As illustrating the simplicity and directness of Euler's
methods I give the substance of his demonstration,^ alluded to
above, that all even perfect numbers are included in Euclid's
formula, 2^~^p, where p stands for 2"-l and is a prime.^
Let iV^ be an even perfect number. JV is even, hence it can be
written in the form 2^~\ where a is not divisible by 2. iV
is perfect, that is, is equal to the sum of all its integral sub-
divisors ; therefore (if the number itself be reckoned as one of
its divisors) it is equal to half the sum of all its integral divisors,
which we may denote by '^K Since 2A^= 2i\", we have
1 Covimentationes Arithmeticae Collectae, Petrograd, 1849, vol. ii, p. 514,
art. 107. Sylvester published an analysis of the argument in Nature,
December 15, 1887, vol. xxxvii, p. 152.
2 Euc. ix, 36 ; see above, page 307.
398 LAGRANGE, LAPLACE, ETC. [ch. xviii
2 X 2^-ia = 22«-ia = 22"-i X 2a.
.♦. 2«a = (l + 2+...+2«-l)2a-=(2«-l)2a,
therefore a : 2a = 2*^ - 1 : 2'^=^ : j9+ 1. Hence a = Xp, and
^a = X.(j) + 1) 'j and since the ratio p:p + l is in its lowest
terms, A must be a positive integer. Now, unless A = 1, we
have 1, \ p, and Xp as factors of \p ; moreover, if p be not
prime, there will be other factors also. Hence, unless A = 1 and
^ be a prime, we have
2Ap = l+A+j9 + Ap+... = (A + l)(p+l) + ....
But this is inconsistent with the result 2A/? = 2a = A(jo + 1).
Hence A must be equal to 1 and p must be a prime. There-
fore a =jo, therefore A^= 2^"! a = 2^^-i (2^* - 1). I may add the
corollary that since ^ is a prime, it follows that n is a prime ;
and the determination of what values of 7i (less than 257)
make p prime falls under Mersenne's rule.
The four works mentioned above comprise most of what
Euler produced in pure mathematics. He also wrote numerous
memoirs on nearly all the subjects of applied mathematics and
mathematical physics then studied : the chief novelties in them
are as follows.
In the mechanics of a rigid system he determined the
general equations of motion of a body about a fixed point,
which are ordinarily written in the form
and he gave the general equations of motion of a free body,
which are usually presented in the form
i.{mu) - mv9^ + mwO<^ = A', and -^ - K^O^ + h.^0^ = L.
He also defended and elaborated the theory of " least action "
which had been propounded by Maupertuis in 1751 in his
Essai de cosmologie [p. 70].
In hydrodynamics Euler established the general equations of
motion, which are commonly expressed in the form
CH.xviii] EULER 399
1 dp du du du du
p dx^ dt~ ^dx dy dz '
At the time of his death he was engaged in writing a treatise
on hydromechanics in which the treatment of the subject would
have been completely recast.
His most important works on astronomy are his Theoria
Motuwni Planetarvmi et Cometarum, published in 1744;^ his
Theoria Motus Lurmris^ published in 1753; and his Theoria
Motuum Lunae^ published in 1772. In these he attacked the
problem of three bodies : he supposed the body considered
(ex. gr. the moon) to carry three rectangular axes with it in
its motion, the axes moving parallel to themselves, and to
these axes all the motions were referred. This method is not
convenient, but it was from Euler's results that Mayer ^ con-
structed the lunar tables for which his widow in 1770 received
£5000 from the English parliament, and in irecognition of
Euler's services a sum of £300 was also voted as an honorarium
to him.
Euler was much interested in optics. In 1746 he discussed
the relative merits of the emission and undulatory theories of
light; he on the whole preferred the latter. In 1770-71 he
published his optical researches in three volumes under the
title Dioptrica.
He also wrote an elementary work on physics and the
fundamental principles of mathematical philosophy. This
originated from an invitation he received when he first went
to Berlin to give lessons on physics to the princess of Anhalt-
Dessau. These lectures were published in 1768-1772 in
three volumes under the title Lettres . . .sur quelques sujets de
physique...^ and for half a century remained a standard treatise
on the subject.
Of course Euler's magnificent works were not the only
^ Johann Tobias Mayer, born in Wlirtemberg in 1723, and died in 1762,
was director of the English observatory at Gottingen. Most of his memoirs,
other than his lunar tables, were published in 1775 under the title Opera
Inedita.
400 LAGRANGE, LAPLACE, ETC. [cii. xviii
text -books containing original matter produced at this time.
Amongst numerous writers I would specially single out Daniel
Bernoulli, Simpson, Lambert, Bezout, Trembley, and Arhogast,
as having influenced the development of mathematics. To the
two first-mentioned I have already alluded in the last chapter.
Lambert.^ Johann Heinrich Lambert was born at Miil-
hausen on August 28, 1728, and died at Berlin on September
25, 1777. He was the son of a small tailor, and had to rely
on his own efforts for his education ; from a clerk in some iron-
works he got a place in a newspaper office, and subsequently,
on the recommendation of the editor, he was appointed tutor in
a private family, which secured him the use of a good library
and sufficient leisure to use it. In 1759 he settled at Augsburg,
and in 1763 removed to Berlin where he was given a small
pension, and finally made editor of the Prussian astronomical
almanack.
Lambert's most important works were one on optics, issued
in 1759, which suggested to Arago the lines of investigation he
subsequently pursued; a treatise on perspective, published in
1759 (to which in 1768 an appendix giving practical applica-
tions were added); and a treatise on comets, printed in 1761,
containing the well-known expression for the area of a focal
sector of a conic in terms of the chord and the bounding radii.
Besides these he communicated numerous papers to the Berlin
Academy. Of these the most important are his memoir in 1768
on transcendental magnitudes, in which he proved that it is
( incommensurable (the proof is given in Legendre's Geometrie,
and is there extended to tt^) : his paper on trigonometry, read
in 1768, in which he developed Demoivre's theorems on the
trigonometry of complex variables, and introduced the hyper-
bolic sine and cosine ^ denoted by the symbols sinh x, cosh x :
^ See Lambert nach seineni Leben und Wirken, by D. Hiiber, Bale, 1829.
Most of Lambert's memoirs are collected in his Beitrage zum Gebrauche der
Mathematik, published in four volumes, Berlin, 1765-1772.
2 These functions are said to have been previously suggested by
F. C. Mayer, see Die Lehre von den Hyperhelfunktionen by S. Giinther, Halle,
1881, and BMrdge zur Geschichte der neueren Mathematik, Ansbach, 1881.
CH. xviii] LAMBERT. BEZOUT. TREMBLEY 401
his essay entitled analytical observations, published in 1771,
which is the earliest attempt to form functional equations by
expressing the given properties in the language of the differential
calculus, and then integrating his researches on non-Euclidean
geometry: lastly, his paper on vis viva, published in 1783, in
which for the first time he expressed Newton's second law of
motion in the notation of the differential calculus.
B^zout. Trembley. Arbogast. Of the other mathema-
ticians above mentioned I here add a few words. £tienne
Bezout, born at Nemours on March 31, 1730, and died on
September 27, 1783, besides numerous minor works, wrote a
The'orie generale des equations algebriques^ published at Paris in
1779, which in particular contained much new and valuable
matter on the theory of elimination and symmetrical functions
of the roots of an equation : he used determinants in a paper '
in the Histoire de Vcucademie royale^ 1764, but did not treat
of the general theory. Jean Trembley^ born at Geneva in 1749,
and died on September 18, 1811, contributed to the develop-
ment of differential equations, finite differences, and the calculus
of probabilities. Louis Francois Antoine Arbogast, born in
Alsace on October 4, 1759, and died at Strassburg, where he
was professor, on April 8, 1803, wTote on series and the deriva-
tives known by his name : he was the first writer to separate
the symbols of operation from those of quantity.
I do not wash to crowd my pages with an account of those
who have not distinctly advanced the subject, but I have
mentioned the above writers because their names are still well
known. We may, however, say that the discoveries of Euler
and' Lagrange in the subjects which they treated were so com-
plete and far-reaching that what their less gifted contemporaries
added is not of sufficient importance to require mention in a
book of this nature.
Lagrange.^ Joseph Louis Lagrange, the greatest mathe-
^ Summaries of the life aud works of Lagrange are given in the English
Cyclopaedia and the Encyclopaedia Britannica (ninth edition), of which I
have made considerable use : the former contains a bibliography of his
2d
402 LAGRANGE, LAPLACE, ETC. [ch. xviti
matician of the eighteenth century, was born at Turin on
January 25, 1736, and died at Paris on April 10, 1813. His
father, who had charge of the Sardinian military chest, was
of good social position and wealthy, but before his son grew up
he had lost most of his property in speculations, and young
Lagrange had to rely for his position on his own abilities. He
was educated at the college of Turin, but it was not until he
was seventeen that he shewed any taste for mathematics — his
interest in the subject being first excited by a memoir by Halley,i
across which he came by accident. Alone and unaided he threw
himself into mathematical studies ; at the end of a year's
incessant toil he was already an accomplished mathematician,
and was made a lecturer in the artillery school.
The first fruit of Lagrange's labours here was his letter,
written when he was still only nineteen, to Euler, in which he
solved the isoperimetrical problem which for more than half a
century had been a subject of discussion. To effect the solution
(in which he sought to determine the form of a function so
that a formula in which it entered should satisfy a certain con-
dition) he enunciated the principles of the calculus of variations.
Euler recognized the generality of the method adopted, and its
superiority to that used by himself ; and with rare courtesy he
withheld a paper he had previously written, which covered some
of the same ground, in order that the young Italian might
have time to complete his work, and claim the undisputed
invention of the new calculus. The name of this branch of
analysis was suggested by Euler. This memoir at once placed
Lagrange in the front rank of mathematicians then living.
In 1758 Lagrange established with the aid of his pupils a
society, which was subsequently incorporated as the Turin
Academy, and in the five volumes of its transactions, usually
known as the Miscellanea Taurinensia, most of his early
writings. Lagrange's works, edited by MM. J. A. Serret and G. Darboux,
were published in 14 volumes, Paris, 1867-1892. Delambre's account of his
life is printed in the first volume.
1 On the excellence of the modern algebra in certain optical problems,
Philosophical Transactions, 1693, vol. xviii, p. 960.
CH. xviii] LAGRANGE 403
writings are to be found. Many of these are elaborate memoirs.
The first volume contains a memoir on the theory of the
propagation of sound ; in this he indicates a mistake made by
Newton, obtains the general differential equation for the motion,
and integrates it for motion in a straight line. This volume
also contains the complete solution of the problem of a string
vibrating transversely; in this paper he points out a lack
of generality in the solutions previously given by Taylor,
D'Alembert, and Euler, and arrives at the conclusion that
the form of the curve at any time t is given by the equation
y=^a mi mx sin nt. The article concludes with a masterly
discussion of echoes, beats, and compound sounds. Other
articles in this volume are on recurring series, probabilities, and
the calculus of variations.
The second volume contains a long paper embodying the
results of several memoirs in the first volume on the theory and
notation of the calculus of variations ; and he illustrates its
use by deducing the principlejof least action, and by solutions
of various problems in dynamics.
The third volume includes the solution of several dynamical
problems by means of the calculus of variations ; some papers
on the integral calculus; a solution of Fermat's problem
mentioned above, to find a number x which will make {x'^n + 1 )
a square where ?i is a given integer which is not a square ; and
the general differential equations of motion for three bodies
moving under their mutual attractions.
In 1761 Lagrange stood without a rival as the foremost
mathematician living; but the unceasing labour of the pre-
ceding nine years had seriously affected his health, and the
doctors refused to be responsible for his reason or life unless
he would take rest and exercise. Although his health was
temporarily restored his nervous system never quite recovered
its tone, and henceforth he constantly suffered from attacks of
profound melancholy.
The next work he produced was in 1764 on the libration of
the moon, and an explanation as to why the same face was
404 LAGRANGE, LAPLACE, ETC. [ch. xviii
always turned to the earth, a problem which he treated by the
aid of virtual work. His solution is especially interesting as
containing the germ of the idea of generalized equations
of motion, equations which he first formally proved in
1780.
He now started to go on a visit to London, but on the way
fell ill at Paris. There he was received with marked honour,
and it was with regret he left the brilliant society of that city
to return to his provincial life at Turin. His further stay in
Piedmont was, however, short. In 1766 Euler left Berlin, and
Frederick the Great immediately wrote expressing the wish of
"the greatest king in Europe" to have "the greatest mathe-
matician in Europe " resident at his court. Lagrange accepted
the offer and spent the next twenty years in Prussia, where he
produced not only the long series of memoirs published in the
Berlin and Turin transactions, but his monumental work, the
Mecanique analytique. His residence at Berlin commenced
with an unfortunate mistake. Finding most of his colleagues
married, and assured by their wives that it was the only way
to be happy, he married ; his wife soon died, but the union was
not a happy one.
Lagrange was a favourite of the king, who used frequently
to discourse to him on the advantages of perfect regularity of
life. The lesson went home, and thenceforth Lagrange studied
his mind and body as though they w^ere haachines, and found
by experiment the exact amount of work which he was able to
do without breaking down. Every night he set himself a
definite task for the next day, and on completing any branch
of a subject he wrote a short analysis to see what points in the
demonstrations or in the subject-matter were capable of im-
provement. He always thought out the subject of his papers
before he began to compose them, and usually wrote them
straight off without a single erasure or correction.
His mental activity during these twenty years was amazing.
Not only did he produce his splendid Mecanique analytique^
but he contributed between one and two hundred papers to
CH. xviii] LAGRANGE 405
the Academies of Berlin, Turin, and Paris. Some of these are
really treatises, and all without exception are of a high order
of excellence. Except for a short time when he was ill he
produced on an average about one memoir a month. Of these
I note the following as among the most important.
First, his contributions to the fourth and fifth volumes,
1766-1773, of the Miscellanea Taurinensia ; of which the most
important was the one in 1771, in which he discussed how
numerous astronomical observations should be combined so as to
give the most probable result. And later, his contributions to
the first two volumes, 1784-1785, of the transactions of the
Turin Academy ; to the first of which he contributed a paper
on the pressure exerted by fluids in motion, and to the second
an article on integration by infinite series, and the kind of
problems for which it is suitable.
Most of the memoirs sent to Paris were on astronomical
questions, and among these I ought particularly to mention
his memoir on the Jovian system in 1766, his essay on the
problem of three bodies in 1772, his work on the secular
equation of the moon in 1773, and his treatise on cometary
perturbations in 1778. These were all written on subjects
proposed by the French Academy, and in each case the prize
was awarded to him.
The greater number of his papers during this time were,
however, contributed to the Berlin Academy. Several of them
deal with questions on algebra. In particular I may mention
the following, (i) His discussion of the solution in integers of
indeterminate quadratics, 1769, and generally of indeterminate
equations, 1770. (ii) His tract on the theory of elimination,
1770. (iii) His memoirs on a general process for solving an
algebraical equation of any degree, 1770 and 1771 ; this method
fails for equations of an order above the fourth, because it then
involves the solution of an equation of higher dimensions than
the one proposed, but it gives all the solutions of his predecessors
as modifications of a single principle, (iv) The complete solution
of a binomial equation of any degree ; this is contained in the
406 LAGRANGE, LAPLACE, ETC. [ch. xviii
memoirs last mentioned, (v) Lastly, in 1773, his treatment of
/^determinants of the second and third order, and of invariants.
Several of his early papers also deal with questions con-
nected with the neglected but singularly fascinating subject
of the theory of numbers. Among these are the following,
(i) His proof of the theorem that every integer which is not
a square can be expressed as the sum of two, three, or four
integral squares, 1770. (ii) His proof of Wilson's theorem that
if w be a prime, then |^-1 + 1 is always a multiple of n,
1771. (iii) His memoi^T^ 1773, 1775, and 1777, which
give the demonstrations of several results enunciated by Fermat,
and not previously proved, (iv) And, lastly, his method for
determining the factors of numbers of the form x^ + ay^.
There are also numerous articles on various points of analytical
geometry. In two of them, written rather later, in 1792 and
1793, he reduced the equations of the quadrics (or conicoids) to
their canonical forms.
During the years from 1772 to 1785 he contributed a long
series of memoirs which created the science of differential
equations, at any rate as far as partial differential equations
are concerned. I do not think that any previous writer had
done anything beyond considering equations of some particular
form. A large part of these results were collected in the
second edition of Euler's integral calculus which was published
in 1794.
Lagrange's papers on mechanics require no separate mention
here as the results arrived at are embodied in the Mecanique
analytique which is described below.
Lastly, there are numerous memoirs on problems in astronomy.
Of these the most important are the following, (i) On the
attraction of ellipsoids, 1773 : this is founded on Maclaurin's
work, (ii) On the secular equation of the moon, 1773 ; also
noticeable for the earliest introduction of the idea of the
potential. The potential of a body at any point is the sum
of the mass of every element of the body when divided by its
distance from the point. Lagrange shewed that if the potential
, CH.XVIIT] LAGRANGE 407
of a body at an external point were known, the attraction in
any direction could be at once found. The theory of the
potential was elaborated in a paper sent to Berlin in 1777.
(iii) On the motion of the nodes of a planet's orbit, 1774.
(iv) On the stability of the planetary orbits, 1776. (v) Two
memoirs in which the method of determining the orbit of a
comet from three observations is completely worked out, 1778
and 1783 : this has not indeed proved practically available,
but his system of calculating the perturbations by means of,
mechanical quadratures has formed the basis of most subsequent
researches on the subject, (vi) His determination of the secular
and periodic variations of the elements of the planets, 1781-1784:
the upper limits assigned for these agree closely with those
obtained later by Leverrier, and Lagrange proceeded as far as
the knowledge then possessed of the masses of the planets
permitted, (vii) Three memoirs on the method of interpolation,
1783, 1792, and 1793: the part of finite differences dealing
therewith is now in the same stage as that in which Lagrange
left it.
Over and above these various papers he composed his great
treatise, the Mecanique analytique. In this he lays down the
law of virtual work, and from that one fundamental principle,
by the aid of the calculus of variations, deduces the whole
of mechanics, both of solids and fluids. The object of the
book is to shew that the subject is implicitly included in a
single principle, and to give general formulae from which any
particular result can be obtained. The method of generalized
co-ordinates by which he obtained this result is perhaps the
most brilliant result of his analysis. Instead of following the
motion of each individual part of a material system, as
D'Alembert and Euler had done, he shewed that, if we deter-
mine its configuration by a sufficient number of variables
whose number is the same as that of the degrees of freedom
possessed by the system, then the kinetic and potential energies
of the system can be expressed in terms of these variables, and
the differential equations of motion thence deduced by simple
408 LAGRANGE, LAPLACE, ETC. [ch. xviii
diiferentiation. For example, in dynamics of a rigid system
he replaces the consideration of the particular problem by
the general equation which is now usually written in the form
Amongst other theorems here given are the proposition that the
kinetic energy imparted by given impulses to a material system
under given constraints is a maximum, and a more general state-
ment of the principle of least action than had been given by
Maupertuis or Euler. All the analysis is so elegant that
Sir William Rowan Hamilton said the work could be only
described as a scientific poem. Lagrange held that mechanics
was really a branch of pure mathematics analogous to a geometry
of four dimensions, namely, the time and the three co-ordinates
of the point in space ; ^ and it is said that he prided himself
that from the beginning to the end of the work there was not a
single diagram. At first no printer could be found who would
publish the book ; but Legendre at last persuaded a Paris firm
to undertake it, and it was issued in 1788.
In 1787 Frederick died, and Lagrange, who had found
the climate of Berlin trying, gladly accepted the offer of
Louis XVI. to migrate to Paris. He received similar invita-
tions from Spain and Naples. In France he was received with
every mark of distinction, and special apartments in the Louvre
were prepared for his reception. At the beginning of his
residence here he w^as seized with an attack of melancholy,
and even the printed copy of his Mecanique on which he had
worked for a quarter of a century lay for more than two years
unopened on his desk. Curiosity as to the results of the
French revolution first stirred him out of his lethargy, a
curiosity which soon turned to alarm as the revolution
developed. It was about the same time, 1792, that the un-
accountable sadness of his life and his timidity moved the
compassion of a young girl who insisted on marrying him, and
^ On the development of this idea, see H. Minkowski, Raum und Zeit,
Leipzig, 1909.
CH. xviii] LAGRANGE 409
proved a devoted wife to whom he became warmly attached.
Although the decree of October 1793, which ordered all
foreigners to leave France, specially exempted him by name,
he was preparing to escape when he was offered the presidency
of the commission for the reform of weights and measures
The choice of the units finally selected was largely due to him,
and it was mainly owing to his influence that the decimal
subdivision was accepted by the commission of 1799.
Though Lagrange had determined to escape from France
while there was yet time, he was never in any danger; and
the different revolutionary governments (and, at a later time.
Napoleon) loaded him with honours and distinctions. A
striking testimony to the respect in which he was held was
shown in 1796 when the French commissary in Italy was
ordered to attend in full state on Lagrange's father, and tender
the congratulations of the republic on the achievements of his
son, who " had done honour to all mankind by his genius, and
whom it was the special glory of Piedmont to have produced."
It may be added that Napoleon, when he attained power,
warmly encouraged scientific studies in France, and was a
liberal benefactor of them.
In 1795 Lagrange was appointed to a mathematical chair at
the newly-established Ecole normale, which enjoyed only a
brief existence of four months. His lectures here were quite
elementary, and contain nothing of any special importance, but
they were published because the professors had to "pledge
themselves to the representatives of the people and to each
other neither to read nor to repeat from memory," and the
discourses were ordered to be taken down in shorthand in order
to enable the deputies to see how the professors acquitted
themselves.
On the establishment of the Ecole polytechnique in 1797
Lagrange was made a professor; and his lectures there are
described by mathematicians who had the good fortune to be
able to attend them, as almost perfect both in form and matter.
Beginning with the merest elements, he led his hearers on until.
410 LAGRANGE, LAPLACE, ETC. [ch. xvni
almost unknown to themselves, they were themselves extending
the bounds of the subject : above all he impressed on his pupils
the advantage of always using general methods expressed in a
symmetrical notation.
His lectures on the differential calculus form the basis of his
Theorie cles fonctions analytiqtues which was published in 1797.
This work is the extension of an idea contained in a jDaper he
had sent to the Berlin Memoirs in 1772, and its object is to
substitute for the differential calculus a group of theorems based
on the development of algebraic functions in series. A some-
what similar method had been previously used by John Landen
in his Residual Analysis, published in London in 1758.
Lagrange believed that he could thus get rid of those diffi-
culties, connected with the use of infinitely large and infinitely
small quantities, to which some philosophers objected in the
usual treatment of the differential calculus. The book is divided
into three parts : of these, the first treats of the general theory
of functions, and gives an algebraic proof of Taylor's theorem,
the validity of which is, however, open to question ; the second
deals with applications to geometry ; and the third with appli-
cations to mechanics. Another treatise on the same lines was
his Legons sur le calcul des fonctions, issued in 1804. These
works may be considered as the starting-point for the researches
of Cauchy, Jacobi, and Weierstrass, and are interesting from the
historical point of view.
Lagrange, however, did not himself object to the use of
infinitesimals in the differential calculus; and in the preface
to the second edition of the Mecanique, which was issued in
1811, he justifies their employment, and concludes by saying
that "when we have grasped the spirit of the infinitesimal
method, and have verified the exactness of its results either by
the geometrical method of prime and ultimate ratios, or by the
analytical method of derived functions, we may employ infinitely
small quantities as a sure and valuable means of shortening and
simplifying our proofs."
His Resolution des equations numeriqtoes, published in 1798,
CH. xviii] LAGRANGE 411
was also the fruit of his lectures at the Polytechnic. In this he
gives the method of approximating to the real roots of an
equation by means of continued fractions, and enunciates several
other theorems. In a note at the end he shows how Fermat's
theorem that (X^~^ - 1 = 0 (mod p)y where j9 is a prime and a is
prime to />, may be applied to give the complete algebraical
solution of any binomial equation. He also here explains how
the equation whose roots are the squares of the diflferences of
the roots of the original equation may be used so as to give
considerable information as to the position and nature of those
roots.
The theory of the planetary motions had formed the subject
of some of the most remarkable of Lagrange's Berlin papers.
In 1806 the subject was reopened by Poisson, who, in a paper
read before the French Academy, showed that Lagrange's
formulae led to certain limits for the stability of the orbits.
Lagrange, who was present, now discussed the whole subject
afresh, and in a memoir communicated to the Academy in
1808 explained how, by the variation of arbitrary constants, the
periodical and secular inequalities of any system of mutually
interacting bodies could be determined.
In 1810 Lagrange commenced a thorough revision of the
Mecanique analytique^ but he was able to complete only about
two-thirds of it before his death.
In appearance he was of medium height, and slightly formed,
with pale blue eyes and a colourless complexion. In character
he was nervous and timid, he detested controversy, and to avoid
it willingly allowed others to take the credit for what he had
himself done.
Lagrange's interests were essentially those of a student of
pure mathematics : he sought and obtained far-reaching abstract
results, and was content to leave the applications to others.
Indeed, no inconsiderable part of the discoveries of his great
contemporary, Laplace, consists of the application of the
Lagrangian formulae to the facts of nature ; for example,
Laplace's conclusions on the velocity of sound and the secular
412 LAGRANGE, LAPLACE, ETC. [cn. xviii
acceleration of the moon are implicitly involved in Lagrange's
results. The only difficulty in understanding Lagrange is that
of the subject-matter and the extreme generality of his pro-
cesses; but his analysis is "as lucid and luminous as it is
symmetrical and ingenious."
A recent writer speaking of Lagrange says truly that he
took a prominent part in the advancement of almost every
branch of pure mathematics. Like Diophantus and Fermat, he
possessed a special genius for the theory of numbers, and in this
subject he gave solutions of many of the problems which had
been proposed by Fermat, and added some theorems of his own.
He developed the calculus of variations. To him, too, the theory
of differential equations is indebted for its position as a science
rather than a collection of ingenious artifices for the solution of
particular problems. To the calculus of finite differences he
contributed the formula of interpolation which bears his name.
But above all he impressed on mechanics (which it will be
remembered he considered a branch of pure mathematics) that
generality and completeness towards which his labours invari-
ably tended.
Laplace.^ Pierre Simoii Laplace was born at Beaumont-en-
Auge in Normandy on March 23, 1749, and died at Paris on
March 5, 1827. He was the son of a small cottager or perhaps
a farm-labourer, and owed his education to the interest excited
in some wealthy neighbours by his abilities and engaging
presence. Very little is known of his early years, for when he
became distinguished he had the pettiness to hold himself aloof
both from his relatives and from those who had assisted him.
It would seem that from a pupil he became an usher in the
school at Beaumont ; but, having procured a letter of introduc-
tion to DAlembert, he went to Paris to push his fortune. A
paper on the principles of mechanics excited D'Alembert's
^ The following account of Laplace's life and writings is mainly
founded on the articles in the English Cydo2oaedia and the Encyclopaedia
Britannica. Laplace's works were published in seven volumes by the
French government in 1843-7 ; and a new edition with considerable
additional matter was issued at Paris in six volumes, 1878-84.
CH. xviii] LAPLACE 4 1 3
interest, and on his recommendation a place in the military
school was offered to Laplace.
Secure of a competency, Laplace now threw himself into
original research, and in the next seventeen years, 1771-1787,
he produced much of his original work in astronomy. This
commenced with a memoir, read before the French Academy
in 1773, in which he shewed that the planetary motions were
stable, and carried the proof as far as the cubes of the eccen-
tricities and inclinations. This was followed by several papers
on points in the integral calculus, finite differences, differential
equations, and astronomy.
During the years 1784-1787 he produced some memoirs of
exceptional power. Prominent among these is one read in 1784,
and reprinted in the third volume of the Mecanique celeste, in
which he completely determined the attraction of a spheroid on
a particle outside it. This is memorable for the introduction
into analysis of spherical harmonics or Laplace's coefficients, as
also for the development of the use of the potential — a name
first given by Green in 1828.
If the co-ordinates of two points be (r, /z, w) and (r, //', w'),
and if r <^r, then the reciprocal of the distance between them
can be expanded in powers of r/r', and the respective coefficients
are Laplace's coefficients. Their utility arises from the fact that
every function of the co-ordinates of a point on a sphere can be
expanded in a series of them. It should be stated that the
similar coefficients for space of two dimensions, together with
some of their properties, had been previously given by Legendre
in a paper sent to the French Academy in 1783. Legendre had
good reason to complain of the way in which he was treated in
this matter.
This paper is also remarkable for the development of the
idea of the potential, which was appropriated from Lagrange, ^
who had used it in his memoirs of 1773, 1777, and 1780. Laplace
shewed that the potential always satisfies the differential equation
^ See tlie Bulletin of the New York Mathematical Society, 1892, vol. i.
pp. 66-74.
414 LAGRANGE, LAPLACE, ETC. [ch.xviii
'bx^ 'by^ dz^ '
and on this result his subsequent work on attractions was based.
The quantity V^ V has been termed the concentration of F, and
its value at any point indicates the excess of the value of V
there over its mean value in the neighbourhood of the point.
Laplace's equation, or the more general form V^F=-47r/3,
appears in all branches of mathematical physics. According to
some writers this follows at once from the fact that V^ is a
scalar operator ; or the equation may represent analytically
some general law of nature which has not been yet reduced to
words ; or possibly it might be regarded by a Kantian as the
outward sign of one of the necessary forms through which all
phenomena are perceived.
This memoir was followed by another on planetary inequali-
ties, which was presented in three sections in 1784, 1785, and
1786. This deals mainly with the explanation of the "great
inequality " of Jupiter and Saturn. Laplace shewed by general
considerations that the mutual action of two planets could never
largely affect the eccentricities and inclinations of their orbits ;
and that the peculiarities of the Jovian system were due to the
near approach to commensurability of the mean motions of
Jupiter and Saturn : further developments of these theorems
on planetary motion were given in his two memoirs of 1788
and 1789. It was on these data that Delambre computed his
astronomical tables.
The year 1787 was rendered memorable by Laplace's explana-
tion and analysis of the relation between the lunar acceleration
and the secular changes in the eccentricity of the earth's orbit :
this investigation completed the proof of the stability of the
whole solar system on the assumption that it consists of a
collection of rigid bodies moving in a vacuum. All the me-
moirs above alluded to were presented to the French Academy,
and they are printed in the Memoires presentes par divers
savans.
Laplace now set himself the task to write a work which
CH.xviii] LAPLACE 415
should "offer a complete solution of the great mechanical
problem presented by the solar system, and bring theory to
coincide so closely with observation that empirical equations
should no longer find a place in astronomical tables." The
result is embodied in the Exposition du systeme du monde and
the Mecanique celeste.
The former was published in 1796, and gives a general
explanation of the phenomena, but omits, all details. It con-
tains a summary of the history of astronomy : this summary
procured for its author the honour of admission to the forty
of the French Academy ; it is commonly esteemed one of the
masterpieces of French literature, though it is not altogether
reliable for the later periods of which it treats.
The nebular hypothesis was here enunciated.^ According to
this hypothesis the solar system has been evolved from a quantity
of incandescent gas rotating round an axis through its centre
of mass. As it cooled the gas contracted and successive
rings broke off from its outer edge. These rings in their turn
cooled, and finally condensed into the planets, while the sun
represents the central core which is still left. On this view we
should expect that the more distant planets would be older than
those nearer the sun. The subject is one of great difficulty, and
though it seems certain that the solar system has a common
origin, there are various features which appear almost inexplicable
on the nebular hypothesis as enunciated by Laplace.
Another theory which avoids many of the difficulties raised
by Laplace's hypothesis has recently found favour. According
to this, the origin of the solar system is to be found in the
gradual aggregation of meteorites which swarm through our
system, and perhaps through space. These meteorites which
are normally cold may, by repeated collisions, be heated, melted,
or even vaporized, and the resulting mass would, by the effect
of gravity, be condensed into planet -like bodies — the larger
aggregations so formed becoming the chief bodies of the solar
^ On hypotheses as to the origin of the solar system, see H. Poincare,
Hypotheses cosmogoniques, Paris, 1911.
416 LAGRANGE, LAPLACE, ETC. [ch. xviii
system. To account for these collisions and condensations it
is supposed that a vast number of meteorites were at some
distant epoch situated in a spiral nebula, and that condensations
and collisions took place at certain knots or intersections of
orbits. As the resulting planetary masses cooled, moons or rings
would be formed either by collisions of outlying parts or in the
manner suggested in Laplace's hypothesis. This theory seems
to be primarily due to Sir Norman Lockyer. It does not
conflict with any of the known facts of cosmical science, but
as yet our knowledge of the facts is so limited that it would be
madness to dogmatize on the subject. Recent investigations
have shown that our moon broke off from the earth while the
latter was in a plastic condition owing to tidal friction. Hence
its origin is neither nebular nor meteoric.
Probably the best modern opinion inclines to the view that
nebular condensation, meteoric condensation, tidal friction, and
possibly other causes as yet unsuggested, have all played their
part in the evolution of the system.
The idea of the nebular hypothesis had been outlined by
Kant^ in 1755, and he had also suggested meteoric aggrega-
tions and tidal friction as causes affecting the formation of the
solar system : it is probable that Laplace was not aware of
this.
According to the rule published by Titius of Wittemberg
in 1766 — but generally known as Bode' s law, from the fact
that attention was called to it by Johann Elert Bode in
1778 — the distances of the planets from the sun are nearly in
the ratio of "the numbers 0 + 4, 3 + 4, 6 + 4, 12 + 4, (fee, the
(n + 2)th term being {2''^ x 3) + 4. It would be an interesting
fact if this could be deduced from the nebular, meteoric, or any
other hypotheses, but so far as I am aware only one writer has
made any serious attempt to do so, and his conclusion seems
to be that the law is not sufficiently exact to be more than a
convenient means of remembering the general result.
Laplace's analytical discussion of the solar system is given
^ See Kant's Cosmogony, edited by W. Hastie, Glasgow, 1900.
CH. xviii] LAPLACE 417
in his Mecanique celeste published in five volumes. An analysis
of the contents is given in the English Cyclopaedia. The first
two volumes, published in 1799, contain methods for calculating
the motions of the planets, determining their figures, and re-
solving tidal problems. The third and fourth volumes, published
in 1802 and 1805, contain applications of these methods, and
several astronomical tables. The fifth volume, published in
1825, is mainly historical, but it gives as appendices the results
of Laplace's latest researches. Laplace's own investigations
embodied in it are so numerous and valuable that it is regret-
table to have to add that many results are appropriated from
writers with scanty or no acknowledgment, and the conclusions
— which have been described as the organized result of a century
of patient toil — are frequently mentioned as if they were due to
Laplace.
The matter of the Mecanique celeste is excellent, but it is
by no means easy reading. Biot, who assisted Laplace in
revising it for the press, says that Laplace himself was fre-
quently unable to recover the details in the chain of reasoning,
and, if satisfied that the conclusions were correct, he w^as
content to insert the constantly recurring formula, " II est aise
a voir." The Mecanique celeste is not only the translation of
the Principia into the language of the differential calculus,
but it completes parts of which Newton had been unable to
fill in the details. F. F. Tisserand's recent work may be taken
as the modern presentation of dynamical astronomy on classical
lines, but Laplace's treatise will always remain a standard
authority.
Laplace went in state to beg Napoleon to accept a copy of
his work, and the following account of the interview is well
authenticated, and so characteristic of all the parties concerned
that I quote it in full. Someone had told Napoleon that the
book contained no mention of the name of God ; Napoleon,
who was fond of putting embarrassing questions, received it
with the remark, " M. Laplace, they tell me you have written
this large book on the system of the universe, and have never
2e
418 LAGRANGE, LAPLACE, ETC. [ch. xviii
even mentioned its Creator." Laplace, who, though the most
supple of politicians, was as stiff as a martyr on every point of
his philosophy, drew himself up and answered bluntly, "Je
n'avais pas besoin de cette hypothese-lk." Napoleon, greatly
amused, told this reply to Lagrange, who exclaimed, " Ah !
c'est une belle hypothese ; ga explique beaucoup de choses."
In 1812 Laplace issued his Theorie analytique des proha-
hilites.^ The theory is stated to be only common sense ex-
pressed in mathematical language. The method of estimating
the ratio of the number of favourable cases to the whole
number of possible cases had been indicated by Laplace
in a paper written in 1779. It consists in treating the suc-
cessive values of any function as the coefficients in the expan-
sion of another function with reference to a different variable.
The latter is therefore called the generating function of the
former. Laplace then shews how, by means of interpolation,
these coefficients may be determined from, the generating func-
tion. Next he attacks the converse problem, and from the
coefficients he finds the generating function ; this is effected by
the solution of an equation in finite differences. The method
is cumbersome, and in consequence of the increased power of
analysis is now rarely used.
This treatise includes an exposition of the method of least
squares, a remarkable testimony to Laplace's command over the
processes of analysis. The method of least squares for the com-
bination of numerous observations had been given empirically
by Gauss and Legendre, but the fourth chapter of this work
contains a formal proof of it, on which the whole of the theory
of errors has been since based. This was effected only by a
most intricate analysis specially invented for the purpose, but
the form in which it is presented is so meagre and unsatis-
factory that in spite of the uniform accuracy of the results it was
at one time questioned whether Laplace had actually gone through
the difficult work he so briefly and often incorrectly indicates.
^ A summary of Laplace's reasoning is given in the article on Probability
in the Encyclopaedia Metropolitana.
CH. xviii] LAPLACE 419
In 1819 Laplace published a popular account of his work
on probability. This book bears the same relation to the
Theorie des prohabilites that the Systeme du monde does to
the Mecanique celeste.
Amongst the minor discoveries of Laplace in pure mathe-
matics I may mention his discussion (simultaneously with Van-
dermonde) of the general theory of determinants in 1772; his
proof that every equation of an even degree must have at least
one real quadratic factor ; his reduction of the solution of linear
differential equations to definite integrals ; and his solution of
the linear partial differential equation of the second order. He
was also the first to consider the difficult problems involved in
equations of mixed differences, and to prove that the solution of
an equation in finite differences of the first degree and the
second order might be always obtained in the form of a
continued fraction. Besides these original discoveries he
determined, in his theory of probabilities, the values of a
number of the more common definite integrals; and in the
same book gave the general proof of the theorem enunciated
by Lagrange for the development of any implicit function in
a series by means of differential coefficients.
In theoretical physics the theory of capillary attraction
is due to Laplace, who accepted the idea propounded by
Hauksbee in the Philosophical Transactions for 1709, that
the phenomenon was due to a force of attraction which was
insensible at sensible distances. The part which deals with
the action of a solid on a liquid and the mutual action of two
liquids was not worked out thoroughly, but ultimately was
completed by Gauss : Neumann later filled in a few details.
In 1862 Lord Kelvin (Sir William Thomson) shewed that, if
we assume the molecular constitution of matter, the laws of
capillary attraction can be deduced from the Newtonian law of
gravitation.
Laplace in 1816 was the first to point out explicitly why
Newton's theory of vibratory motion gave an incorrect value for
the velocity of sound. The actual velocity is greater than that
420. LAGRANGE, LAPLACE, ETC. [ch. xviii
calculated by Newton in consequence of the heat developed by
the sudden compression of the air which increases the elasticity
and therefore the velocity of the sound transmitted. Laplace's
investigations in practical physics were confined to those carried
on by him jointly with Lavoisier in the years 1782 to 1784 on
the specific heat of various bodies.
Laplace seems to have regarded analysis merely as a means
of attacking physical problems, though the ability with which
he invented the necessary analysis is almost phenomenal. As
long as his results were true he took but little trouble to ex-
plain the steps by which he arrived at them ; he never studied
elegance or symmetry in his processes, and it was sufiicient
for him if he could by any means solve the particular question
he was discussing.
It would have been well for Laplace's reputation if he had
been content with his scientific work, but above all things he
coveted social fame. The skill and rapidity with which he
managed to change his politics as occasion required would be
amusing had they not been so servile. As Napoleon's power
increased Laplace abandoned his republican principles (which,
since they had faithfully reflected the opinions of the party in
power, had themselves gone through numerous changes) and
begged the first consul to give him the post of minister of the
interior. Napoleon, who desired the supj^ort of men of science,
agreed to the proposal ; but a little less than six weeks saw
the close of Laplace's political career. Napoleon's memorandum
on his dismissal is as follows : " Geometre de premier rang,
Laplace ne tarda pas k se montrer administrateur plus que
mediocre; des son premier travail nous reconnumes que nous
nous etions trompe. Laplace ne saisissait aucune question sous
son veritable point de vue : il cherchait des subtilites partout,
n'avait que des idees problematiques, et portait enfin I'esprit des
' infiniment petits ' jusque dans I'administration."
Although Laplace was removed from office it was desirable
to retain his allegiance. He was accordingly raised to the
senate, and to the third volume of the Mccanique celeste he
CH.xviii] LAPLACE. LEGENDRE 421
prefixed a note that of all the truths therein contained the most
precious to the author was the declaration he thus made of his
devotion towards the peacemaker of Europe. In copies sold
after the restoration this was struck out. In 1814 it was
evident that the empire was falling ; Laplace hastened to
tender his services to the Bourbons, and on the restoration
was rewarded with the title of marquis : the contempt that his
more honest colleagues felt for his conduct in the matter may-
behead in the pages of Paul Louis Courier. His knowledge
was useful on the numerous scientific commissions on which
he served, and probably accounts for the manner in which his
political insincerity was overlooked; but the pettiness of his
character must not make us forget how great were his services
to science.
That Laplace was vain and selfish is not denied by his
warmest admirers ; his conduct to the benefactors of his youth
and his political friends was ungrateful and contemptible ; while
his appropriation of the results of those who were comparatively-
unknown seems to be well established and is absolutely in-
defensible— of those whom he thus treated three subsequently
rose to distinction (Legendre and Fourier in France and Young
in England) and never forgot the injustice of which they had
been the victims. On the other side it may be said that on
some questions he shewed independence of character, and he
never concealed his views on religion, philosophy, or science,
however distasteful they might be to the authorities in power ;
it should be also added that towards the close of his life, and
especially to the work of his pupils, Laplace was both generous and
appreciative, and in one case suppressed a paper of his own in
order that a pupil might have the sole credit of the investigation.
Legendre. Adrian Marie Legendre was born at Toulouse
on September 18, 1752, and died at Paris on January 10, 1833.
The leading events of his life are very simple and may be
summed up briefly. He was educated at the Mazarin College
in Paris, appointed professor at the military school in Paris
in 1777, was a member of the Anglo-French commission of
422 LAGRANGE, LAPLACE, ETC. [ch.xviii
1787 to connect Greenwich and Paris geodetically ; served on
several of the public commissions from 1792 to 1810; was made
a professor at the Normal school in 1795; and subsequently
held a few minor government appointments. The influence
of Laplace was steadily exerted against his obtaining office or
public recognition, and Legendre, who was a timid student,
accepted the obscurity to which the hostility of his colleague
condemned him.
Legendre's analysis is of a high order of excellence, and is
second only to that produced by Lagrange and Laplace, though
it is not so original. His chief works are his Geometrie, his
Theorie des nombres, his Exercices de calcul integral, and his
Fonctions elliptiques. These include the results of his various
papers on these subjects. Besides these he wrote a treatise
which gave the rule for the method of least squares, and two
groups of memoirs, one on the theory of attractions, and the
other on geodetical operations.
The memoirs on attractions are analyzed and discussed in
Todhunter's History of the Theories of Attraction. The earliest
of these memoirs, presented in 1783, was on the attraction
of spheroids. This contains the introduction of Legendre's
coefficients, which are sometimes called circular (or zonal)
harmonics, and which are particular cases of Laplace's co-
efficients ; it also includes the solution of a problem in which
the potential is used. The second memoir was communicated
in 1784, and is on the form of equilibrium of a mass of
rotating liquid which is approximately spherical. The third,
written in 1786, is on the attraction of confocal ellipsoids.
The fourth is on the figure which a fluid planet would assume,
and its law of density.
His papers on geodesy are three in number, and were
presented to the Academy in 1787 and 1788. The most
important result is that by which a spherical triangle may
be treated as plane, provided certain corrections are applied
to the angles. In connection with this subject he paid con-
siderable attention to geodesies.
CH.xviii] LEGENDRE 423
The method of least squares was enunciated in his Nouvelles
meihodes published in 1806, to which supplements were added
in 1810 and 1820. Gauss independently had arrived at the
same result, had used it in 1795, and published it and the
law of facility in 1809. Laplace was the earliest writer to
give a proof of it; this was in 1812.
Of the other books produced by Legendre, the one most
widely known is his Elements de geometrie which was published
in 1794, and was at one time widely adopted on the continent
as a substitute for Euclid. The later editions contain the
elements of trigonometry, and proofs of the irrationality of
TV and TT^. An appendix on the diflficult question of the theory
of parallel lines was issued in 1803, and is bound up with most
of the subsequent editions.
His Theorie des nombres was published in 1798, and ap-
pendices were added in 1816 and 1825 ; the third edition,
issued in two volumes in 1830, includes the results of his
various later papers, and still remains a standard work on the
subject. It may be said that he here carried the subject as
far as was possible by the application of ordinary algebra ; but
he did not realize that it might be regarded as a higher
arithmetic, and so form a distinct subject in mathematics.
The law of quadratic reciprocity, which connects any two
odd primes, was first proved in this book, but the result had
been enunciated in a memoir of 1785. Gauss called the pro-
position " the gem of arithmetic," and no less than six separate
proofs are to be found in his works. The theorem is as follows.
If ^ be a prime and n be prime to p, then we know that the
remainder when n^P~'^^'^ is divided by p is either -1-1 or - 1.
Legendre denoted this remainder by (n/p). When the re-
mainder is + 1 it is possible to find a square number which
when divided by p leaves a remainder ??-, that is, n is a quadratic
residue of ^j> ; when the remainder is - 1 there exists no such
square number, and n is a non -residue of p. The law of
quadratic reciprocity is expressed by the theorem that, if a
and b be any odd primes, then
424 LAGKANGE, LAPLACE, ETC. [ch. xviii
(a/^>)(Va) = (-l) '^-1X^-1'/*;
thus, if 6 be a residue of a, then a is also a residue of b, unless
both of the primes a and b are of the form 4m + 3. In other
words, if a and b be odd primes, we know that
a^b-m^ ^ I (mod b), and ^.(^-i)/2= ± 1 (mod a) ;
and, by Legendre's law, the two ambiguities will be either both
positive or both negative, unless a and b are both of the form
4m + 3. Thus, if one odd prime be a non-residue of another,
then the latter will be a non- residue of the former. Gauss
and Kummer have subsequently proved similar laws of cubic
and biquadratic reciprocity ; and an important branch of the
theory of numbers has been based on these researches.
This work also contains the useful theorem by which, when
it is possible, an indeterminate equation of the second degree
can be reduced to the form ax'^ + b?/^ + cz^ — 0. Legendre here
discussed the forms of numbers which can be expressed as the
sum of three squares ; and he proved [art. 404] that the number
of primes less than n is approximately w/(loge n - 1 '08366).
The Exercices de calcul integral was published in three
volumes, 1811, 1817, 1826. Of these the third and most of
the first are devoted to elliptic functions ; the bulk of this
being ultimately included in the Fonctions elliptiques. The
contents of the remainder of the treatise are of a miscellaneous
character ; they include integration by series, definite integrals,
and in particular an elaborate discussion of the Beta and the
Gamma functions.
The Traite des fonctions elliptiques was issued in two volumes
in 1825 and 1826, and is the most important of Legendre's
works. A third volume was added a few weeks before his
death, and contains three memoirs on the researches of Abel and
Jacobi. Legendre's investigations had commenced with a paper
written in 1786 on elliptic arcs, but here and in his other papers
he treated the subject merely as a problem in the integral
calculus, and did not see that it might be considered as a
CH.xviii] LEGENDRE. PFAFF . 425
higher trigonometry, and so constitute a distinct branch of
analysis. Tables of the elliptic integrals were constructed by
him. The modern treatment of the subject is founded on that
of Abel and Jacobi. The superiority of their methods was at
once recognized by Legendre, and almost the last act of his
life was to recommend those discoveries which he knew would
consign his own labours to comparative oblivion.
This may serve to remind us of a fact which I wish to
specially emphasize, namely, that Gauss, Abel, Jacobi, and some
others of the mathematicians alluded to in the next chapter, were
contemporaries of the members of the French school.
Pfaff. I may here mention another writer who also made
a special study of the integral calculus. This was Johann
Friederich Pfaff, born at Stuttgart on Dec. 22, 1765, and died
at Halle on April 21, 1825, who was described by Laplace as
the most eminent mathematician in Germany at the beginning
of this century, a description which, had it not been for Gauss's
existence, would have been true enough.
Pfaff was the precursor of the German school, which under
Gauss and his followers largely determined the lines on which
mathematics developed during the nineteenth century. He was
an intimate friend of. Gauss, and in fact the two mathematicians
lived together at Helmstadt during the year 1798, after Gauss
had finished his university course. Pfaff's chief work was his
(unfinished) Disqiiisitiones Analyticae on the integral calculus,
published in 1797 ; and his most important memoirs were either
on the calculus or on differential equations : on the latter subject
his paper read before the Berlin Academy in 1814 is noticeable.
The creation of Diodern geometry.
While Euler, Lagrange, Laplace, and Legendre were per-
fecting analysis, the members of another group of French
mathematicians were extending the range of geometry by
methods similar to those previously used by Desargues and
Pascal. The revival of the study of synthetic geometry is
426 CREATION OF MODERN GEOMETRY [ch. xviii
largely due to Poncelet, but the subject is also associated with
the names of Monge and L. Carnot ; its great development in
more recent times is mainly due to Steiner, von Staudt, and
Cremona.
Monge. ^ Gaspard Monge was born at Beaune on May 10,
1746, and died at Paris on July 28, 1818. He was the son of
a small pedlar, and was educated in the schools of the Oratorians,
in one of which he subsequently became an usher. A plan of
Beaune which he had made fell into the hands of an officer who
recommended the military authorities to admit him to their
training-school at Mezieres. His birth, however, precluded his
receiving a commission in the army, but his attendance at an
annexe of the school where surveying and drawing were taught
was tolerated, though he was told that he was not sufficiently
well born to be allowed to attempt problems which required
calculation. At last his opportunity came. A plan of a fortress
having to be drawn from the data supplied by certain observa-
tions, he did it by a geometrical construction. At first the
officer in charge refused to receive it, because etiquette required
that not less than a certain time should be used in making such
/drawings, but the superiority of the method over that then
/ taught was so obvious that it was accepted; and in 1768
Monge was made professor, on the understanding that the
\ results of his descriptive geometry were to be a military secret
\ confined to officers above a certain rank.
In 1780 he was appointed to a chair of mathematics in Paris,
and this with some provincial appointments which he held gave
him a comfortable income. The earliest paper of any special
importance which he communicated to the French Academy was
one in 1781, in which he discussed the lines of curvature drawn
on a surface. These had been first considered by Euler in 1760,
and defined as those normal sections whose curvature was a
maximum or a minimum. Monge treated them as the locus of
those points on the surface at which successive normals intersect,
^ On the authorities for Monge's life and works, see the note by H. Brocard
in V lntcrm6diaire des mathematiciens, 1906, vol. xiii, pp. 118, 119.
CH. xviii] MONGE 427
and thus obtained the general differential equation. He applied
his results to the central quadrics in 1795. In 1786 he pub-
lished his well-known work on statics.
Monge eagerly embraced the doctrines of the revolution.
In 1792 he became minister of the marine, and assisted the *
committee of public safety in utilizing science for the defence i
of the republic. When the Terrorists obtained power he was
denounced, and escaped the guillotine only by a hasty flight.
On his return in 1794 he was made a professor at the short-
lived Normal school, where he gave lectures on descriptive
geometry ; the notes of these were published under the regula-
tion above alluded to. In 1796 he went to Italy on the roving
commission which was sent with orders to compel the various
Italian towns to offer pictures, sculpture, or other works of art
that they might possess, as a present or in lieu of contributions
to the French republic for removal to Paris. In 1798 he
accepted a mission to Rome, and after executing it joined
Napoleon in Egypt. Thence after the naval and military
victories of England he escaped to France.
Monge then settled down at Paris, and was made professor
at the Polytechnic school, where he. gave lectures on descriptive
geometry ; these were published in 1800 in the form of a text-
Ibook entitled Geometrie descriptive. This work contains pro-
positions on the form and relative position of geometrical figures
deduced by the use of transversals. The theory of perspective \
is considered ; this includes the art of representing in two
dimensions geometrical objects which are of three dimensions,
a problem which Monge usually solved by the aid of two
diagrams, one being the plan and the other the elevation.
Monge also discussed the question as to whether, if in solving
a problem certain subsidiary quantities introduced to facilitate
the solution become imaginary, the validity of the solution is
thereby impaired, and he shewed that the result would not be
affected. On the restoration he was deprived of his offices and
honours, a degradation which preyed on his mind and which he
did not long survive.
428 MONGE. CARNOT. PONCELET [ch. xviii
Most of his miscellaneous papers are embodied in his works,
Application de Valgebre a la geometrie, published in 1805, and
Application de Vanalyse a la geometric^ the fourth edition of
which, published in 1819, was revised by him just before his
death. It contains among other results his solution of a partial
differential equation of the second order.
Carnot.^ Lazare Nicholas Marguerite Carnot, born at
Nolay on May 13, 1753, and died at Magdeburg on Aug. 22,
1823, was educated at Burgundy, and obtained a commission
in the engineer corps of Conde. Although in the army, he
continued his mathematical studies in which he felt great
interest. His first work, published in 1784, was on machines;
it contains a statement which foreshadows the principle of
energy as applied to a falling weight, and the earliest proof of
the fact that kinetic energy is lost in the collision of imperfectly
elastic bodies. On the outbreak of the revolution in 1789 he
threw himself into politics. In 1793 he was elected on the
committee of public safety, and the victories of the French army
were largely due to his powers of organization and enforcing
discipline. He continued to occupy a prominent place in every
successive form of government till 1796 when, having opposed
Napoleon's coup d'etat, he had to fly from France. He took
refuge in Geneva, and there in 1797 issued his Reflexions sur la
metaphysique du calcul infinitesimal : in this he amplifies views
previously expounded by Berkeley and Lagrange. In 1802 he
assisted Napoleon, but his sincere republican convictions were
inconsistent with the retention of ofl&ce. In 1803 he produced
his Geometric de position. This work deals with projective rather
than descriptive geometry, it also contains an elaborate discussion
of the geometrical meaning of negative roots of an algebraical
equation. In 1814 he offered his services to fight for France,
though not for the empire ; and on the restoration he was exiled.
Poncelet.^ Jean Victor Poncelet, born at Metz on July 1,
^ See the iloge by Arago, which, like most obituary notices, is a panegyric
rather than an impartial biography.
2 See La Vie et les ouvtages de Poncelet, by I. Didion and C. Dupin, Paris,
1869.
CH. XYiii] THE DEVELOPMENT OF PHYSICS 429
1788, and died at Paris on Dec. 22, 1867, held a commission
in the French engineers. Having been made a prisoner in the
French retreat from Moscow in 1812 he occupied his enforced
leisure by writing the Traite des proprietes projectives des
figures, published in 1822, which was long one of the best
known text-books on modern geometry. By means of pro-
jection, reciprocation, and homologous figures, he established
all the chief properties of conies and quadrics. He also treated
the theory of polygons. His treatise on practical mechanics in
1826, his memoir on water-mills in 1826, and his report on
the English machinery and tools exhibited at the International
Exhibition held in London in 1851 deserve mention. He
contributed numerous articles to Crelle's journal ; the most
valuable of these deal with the explanation, by the aid of the
doctrine of continuity, of imaginary solutions in geometrical
problems.
The development of mathematical physics.
It will be noticed that Lagrange, Laplace, and Legendre
mostly occupied themselves with analysis, geometry, and astro-
nomy. I am inclined to regard Cauchy and the French mathe-
maticians of the present day as belonging to a difi"erent school
of thought to that considered in this chapter, and I place them
amongst modern mathematicians, but I think that Fourier,
Poisson, and the majority of their contemporaries, are the lineal
successors of Lagrange and Laplace. If this view be correct, we
may say that the successors of Lagrange and Laplace devoted
much of their attention to the application of mathematical
analysis to physics. Before considering these mathematicians
I may mention the distinguished English experimental physicists
who were their contem})oraries, and whose merits have only
recently received an adequate recognition. Chief among these
are Cavendish and Young.
Cavendish. 1 The Honourable Henry Cavendish was born at
^ An account of his life by G. Wilson will be found in the first volume
of the publications of the Cavendish Society, London, 1851. His Electrical
430 CAVENDISH. RUMFORD [ch. xviii
Nice on October 10, 1731, and died in London on February 4,
1810. His tastes for scientific research and mathematics were
formed at Cambridge, where he resided from 1749 to 1753. He
created experimental electricity, and was one of the earliest
writers to treat chemistry as an exact science. I mention him
/ here on account of his experiment in 1798 to determine the
/^density of the earth, by estimating its attraction as compared
with that of two given lead balls : the result is that the mean
density of the earth is about five and a half times that of water.
This experiment was carried out in accordance with a suggestion
which had been first made by John Mitchell (1724-1793), a
fellow of Queens' College, Cambridge, who had died before he
was able to carry it into effect.
Rumford.i Sir Benjamin Thomson, Count Rumford, born
at Concord on March 26, 1753, and died at Auteuil on August
21, 1815, was of English descent, and fought on the side of the
loyalists in the American War of Secession : on the conclusion
of peace he settled in England, but subsequently entered the
service of Bavaria, where his powers of organization proved of
great value in civil as well as military affairs. At a later period
he again resided in England, and when there founded the Royal
Institution. The majority of his papers were communicated to
the Royal Society of London ; of these the most important is
r his memoir in which he showed that heat and work are mutually
V. convertible.
Young.^ Among the most eminent physicists of his time
was Thomas Yotrng, who was born at Milverton on June 13,
1773, and died in London on May 10, 1829. He seems as a
■boy to have been somewhat of a prodigy, being well read in
modern languages and literature, as well as in science ; he always
Researches were edited by J. C. Maxwell, and published at Cambridge in
1879.
^ An edition of Rumford's works, edited by George Ellis, accompanied by
a biography, was published by the American Academy of Sciences at Boston
in 1872.
^ Young's collected works and a memoir on his life were published by G.
Peacock, four volumes, London, 1855.
CH.XVIII] YOUNG. DALTON 431
kept up his literary tastes, and it was he who in 1819 first
suggested the key to decipher the Egyptian hieroglyphics, which
J. F. Champollion used so successfully. Young was destined
to be a doctor, and after attending lectures at Edinburgh and
Gottingen entered at Emmanuel College, Cambridge, from which
he took his degree in 1799 ; and to his stay at the University
he attributed much of his future distinction. His medical
career was not particularly successful, and his favourite maxim
that a medical diagnosis is only a balance of probabilities was
not appreciated by his patients, who looked for certainty in
return for their fee. Fortunately his private means were ample.
Several papers contributed to various learned societies from
1798 onwards prove him to have been a mathematician of
considerable power ; but the researches which have immortalised
his name are those by which he laid down the laws of inter-
ference of waves and of light, and was thus able to suggest the
means by which the chief difficulties then felt in the w^ay of the
acceptance of the undulatory theory of light could be overcome.
Dalton.i Another distinguished writer of the same period
was John Dalton^ who was born in Cumberland on September 5,
1766, and died at Manchester on July 27, 1844. Dalton
investigated the tension of vapours, and the law of the expansion
of a gas under changes of temperature. He also founded the
atomic theory in chemistry.
It will be gathered from these notes that the English school
of physicists at the beginning of this century were mostly
concerned with the experimental side of the subject. But in
fact no satisfactory theory could be formed without some similar
careful determination of the facts. The most eminent French
physicists of the same time were Fourier, Poisson, Ampere,
and Fresnel. Their method of treating the subject is more
mathematical than that of their English contemporaries, and
the two first named were distinguished for general mathematical
ability.
^ See "the Memoir of Bolton, by R. A. Smith, London, 1856 ; and W. C,
Henry's memoir in the Cavendish Society Transactions, houdon, 1854.
432 FOURIER [ch. xviii
Fourier.i The first of these French physicists was Jean
Baptiste Joseph Fourier, who was born at Auxerre on March 21,
1768, and died at Paris on May 16, 1830. He was the son of
a tailor, and was educated by the Benedictines. The commis-
sions in the scientific corps of the army were, as is still the case
in Russia, reserved for those of good birth, and being thus
L ineligible he accepted a military lectureship on mathematics.
He took a prominent part in his own district in promoting the
revolution, and was rewarded by an appointment in 1795 in the
Normal school, and subsequently by a chair in the Polytechnic
school.
Fourier went with Napoleon on his Eastern expedition in
1798, and was made governor of Lower Egypt. Cut off from
France by the English fleet, he organised the workshops on
which the French army had to rely for their munitions of war.
He also contributed several mathematical papers to the Egyptian
Institute which Napoleon founded at Cairo, with a view of
weakening English influence in the East. After the British
victories and the capitulation of the French under General
Menou in 1801, Fourier returned to France, and was made
prefect of Grenoble, and it was while there that he made his
experiments on the propagation of heat. He moved to Paris
in 1816. In 1822 he published his Theorie analytique de la
ckaleur, in which he bases his reasoning on Newton's law of
cooling, namely, that the flow of heat between two adjacent
molecules is proportional to the infinitely small difference of
their temperatures. In this work he shows that any function
I of a variable, whether continuous or discontinuous, can be
^expanded in a series of sines of multiples of the variable — a
result which is constantly used in modern analysis. Lagrange
had given particular cases of the theorem, and had implied that
the method was general, but he had not pursued the subject.
Dirichlet was the first to give a satisfactory demonstration
of it.
^ An edition of his works, edited by G. Darboux, was published in two
volumes, Paris, 1888, 1890.
CH.xviii] SADI CARNOT. POISSON 433
Fourier left an unfinished work on determinate equations
w'liicli was edited by Navier, and published in 1831 ; this
contains much original matter, in particular there is a demon-
stration of Fourier's theorem on the position of the roots of an ^
algebraical equation. Lagrange had shewn how the roots of an
algebraical equation might be separated by means of another
equation whose roots were the squares of the differences of the
roots of the original equation. Budan, in 1807 and 1811, had
enunciated the theorem generally known by the name of
Fourier, but the demonstration was not altogether satisfactory.
Fourier's proof is the same as that usually given in text-
books on the theory of equations. The final solution of the
problem was given in 1829 by Jacques Charles Frangois Sturm
(1803-1855).
Sadi Camot.^ Among Fourier's contemporaries who were
interested in the theory of heat the most eminent was Sadi
Carnot, a son of the eminent geometrician mentioned above.
Sadi Carnot was born at Paris in 1796, and died there of
cholera in August 1832 ; he was an officer in the French
army. In 1824 he issued a short work entitled Reflexions
sur la puissance motrice du feu, in which he attempted to
determine in what way heat produced its mechanical effect.
He made the mistake of assuming that heat was material, but
his essay may be taken as initiating the modern theory of\/
thermodynamics.
Poisson.2 Simeon Denis Foisson, born at Pithiviers on
June 21, 1781, and died at Paris on April 25, 1840, is almost
equally distinguished for his applications of mathematics to
mechanics and to physics. His father had been a private
soldier, and on his retirement was given some small adminis-
trative post in his native village ; when the revolution broke
out he appears to have assumed the government of the place,
^ A sketch of S. Carnot's life and an English translation of his Reflexions
was published by E. H. Thurston, London and New York, 1890.
^ Memoirs of Poisson will be found in the Encyclopaedia Britannica, the
Transactions of th^ Royal Astronomical Society, vol. v, and Arago's Eloges,
vol. ii ; the latter contains a bibliography of Poisson's papers and works.
2f
434 POISSON [cH. XVIII
and, being left undisturbed, became a person of some local
importance. The boy was put out to nurse, and he used to tell
how one day his father, coming to see him, found that the nurse
had gone out, on pleasure bent, having left him suspended by a
small cord attached to a nail fixed in the wall. This, she
explained, was a necessary precaution to prevent him from
perishing under the teeth of the various animals and animalculae
4hat roamed on the floor. Poisson used to add that his gymnastic
efforts carried him incessantly from one side to the other, and it
w^as thus in his tenderest infancy that he commenced those
studies on the pendulum that were to. occupy so large a part of
his mature age.
He was educated by his father, and destined much against
his will to be a doctor. His uncle offered to teach him the art,
and began by making him prick the veins of cabbage -leaves
with a lancet. When perfect in this, he was allowed to put on
blisters ; but in almost the first case he did this by himself, the
patient died in a few hours, and though all the medical practi-
tioners of the place assured him that " the event was a very
common one," he vowed he would have nothing more to do with
the profession.
Poisson, on his return home after this adventure, discovered
amongst the official papers sent to his father a copy of the
questions set at the Polytechnic school, and at once found his
career. At the age of seventeen he entered the Polytechnic, and
his abilities excited the interest of Lagrange and Laplace, whose
friendship he retained to the end of their lives. A memoir on
finite differences which he wrote when only eighteen was
reported on so favourably by Legendre that it was ordered to be
published in the Recueil des savants etrangers. As soon as he
had finished his course he was made a lecturer at the school,
and he continued through his life to hold various government
scientific posts and professorships. He was somewhat of a
socialist, and remained a rigid republican till 1815, when, with
a view to making another empire impossible, he joined the
legitimists. He took, however, no active part in politics, and
CH. xviii] POISSON 435
made the study of mathematics his amusement as well as his
business.
His works and memoirs are between three and four hundred
in number. The chief treatises which he wrote were his Traite
de mecaniqiie^ published in tw^o volumes, 1811 and 1833, which
was long a standard work; his Theorie nouvelle de Vaction
capillaire, 1831 ; his Theorie mathematique de la chaleur, 1835,
to which a supplement was added in 1837; and his Recherches
sur la probahilite des jugements, 1837. He had intended, if he
had lived, to write a work which should cover all mathematical
physics and in which the results of the three books last named
would have been incorporated.
Of his memoirs in pure mathematics the most important are
those on definite integrals, and Fourier's series, their application
to physical problems constituting one of his chief claims to dis-
tinction ; his essays on the calculus of variations ; and his
papers on the probability of the mean results of observations. ^ j
Perhaps the most remarkable of his memoirs in applied
mathematics are those on the theory of electrostatics and
magnetism, which originated a new branch of mathematical
physics ; he supposed that the results were due to the attrac-
tions and repulsions of imponderable particles. The most
important of those on physical astronomy are the two read in
1806 (printed in 1809) on the secular inequalities of the meaii
motions of the planets, and on the variation of arbitrary
constants introduced into the solutions of questions on
mechanics; in these Poisson discusses the question of the
stability of the planetary orbits (which Lagrange had already
^ Among Poisson's contemporaries who studied mechanics and of whose
works he made use I may mention Louis Poinsot, who was born in Paris on
Jan. 3, 1777, and died there on Dec. 5, 1859. In his Staticiue, published in
1803, he treated the subject without any explicit reference to dynamics. The
theory of couples is largely due to him (1806), as also the motion of a body
in space under the action of no forces.
2 See the Journal de Vecole polytechnique from 1813 to 1823, and the
Mimoires de Vacademie for 1823 ; the M^moires de Vacademie, 1833 ; and
the Connaissance des temps, 1827 and following years. Most of his memoirs
were published in the three periodicals here mentioned.
t
436 POISSON. AMPERE. FRESNEL. BIOT [ch. xviii
•proved to the first degree of approximation for the disturbing
forces), and shews that the result can be extended to the third
order of small quantities : these were the memoirs which led
to Lagrange's famous memoir of 1808. Poisson also published
a paper in 1821 on the libration of the moon; and another in
1827 on the motion of the earth about its centre of gravity.
His most important memoirs on the theory of attraction are
one in 1829 on the attraction of spheroids, and another in 1835
on the attraction of a homogeneous ellipsoid : the substitu-
tion of the correct equation involving the potential, namely,
S/^V= -4:7rp, for Laplace's form of it, V^V=0, was first pub-
lished^ in 1813. Lastly, I may mention his memoir in 1825
on the theory of waves.
Ampere. ^ Andre Marie Ampere was born at Lyons on
January 22, 1775, and died at Marseilles on June 10, 1836.
He was widely read in all branches of learning, and lectured
and wrote on many of them, but after the year 1809, when he
was made professor of analysis at the Polytechnic school in
Paris, he confined himself almost entirely to mathematics and
science. His papers on the connection between electricity and
magnetism were written in 1820. According to his theory,
propounded in 1826, a molecule of matter which can be
magnetized is traversed by a closed electric current, and
magnetization is produced by any cause which makes the
direction of these currents in the different molecules of the
body approach parallelism.
Fresnel. Biot. Augustin Jean Fresnel', born at Broglie on
May 10, 1788, and died at Ville-d'Avray on July 14, 1827, was
a civil engineer by profession, but he devoted his leisure to the
study of physical optics. The undulatory theory of light, which
Hooke, Huygens, and Euler had supported on a priori grounds,
had been based on experiment by the researches of Young.
Fresnel deduced the mathematical consequences of these experi-
ments, and explained the phenomena of interference both of
^ In the Bulletin des sciences of the Societe philomatique.
"^ See C. A. Valson's ^tude sur la vie et les ouvrages d' Ampere, Lyons, 1885.
CH. xviii] ARAGO 437
ordinary and polarized light. Fresnel's friend and contemporary,
Jean Baptiste Biot, who was born at Paris on April 21, 1774,
and died there in 1862, requires a word or two in passing.
Most of his mathematical work was in connection with the
subject of optics, and especially the polarization of light. His
systematic works were produced within the years 1805 and
1817; a selection of his more valuable memoirs was published
in Paris in 1858.
Arago.^ Frangois Jean Dominique Arago was born' at
Estagel in the Pyrenees on February 26, 1786, and died in
Paris on October 2, 1853. He was educated at the Polytechnic
school, Paris, and we gather from his autobiography that
however distinguished were the professors of that institution
they were remarkably incapable of imparting their knowledge
or maintaining discipline.
In 1804 Arago was made secretary to the observatory at
Paris, and from 1806 to 1809 he was engaged in measuring a
meridian arc in order to determine the exact length of a metre.
He was then appointed to a leading post in the observatory,
given a residence there, and made a professor at the Polytechnic
school, where he enjoyed a marked success as a lecturer. He
subsequently gave popular lectures on astronomy, which were
both lucid and accurate — a combination of qualities which was
rarer then than now. He reorganized the national observatory,
the management of which had long been inefficient, but in doing
this his want of tact and courtesy raised many unnecessary
difficulties. He remained to the end a consistent republican,
and after the coup d'etat of 1852, though half blind and dying,
he resigned his post as astronomer rather than take the oath of
allegiance. It is to the credit of Napoleon III. that he gave
directions that the old man should be in no way disturbed, and
should be left free to say and do what he liked.
^ Arago's works, which include eloges on many of the leading matheraa-
ticians of the last five or six centuries, have been edited by M. J. A. Barral,
and published in fourteen volumes, Paris, 1856-57. An autobiography is
prefixed to the first volume.
438 AKAGO [ch. xviii
Arago's earliest physical researches were on the pressure of
steam at different temperatures, and the velocity of sound, 1818
to 1822. His magnetic observations mostly took place from
1823 to 1826. He discovered what has been called rotatory
magnetism, and the fact that most bodies could be magnetized ;
these discoveries were completed and explained by Faraday.
He warmly supported Fresnel's optical theories, and the two
philosophers conducted together those experiments on the
polarization of light which led to the inference that the vibra-
tions of the luminiferous ether were transverse to the direction
of motion, and that polarization consisted in a resolution of
rectilinear motion into components at right angles to each other.
The subsequent invention of the polariscope and discovery of
rotatory polarization are due to Arago. The general idea of the
experimental determination of the velocity of light in the
manner subsequently effected by Fizeau and Foucault was
suggested by him in 1838, but his failing eyesight prevented
his arranging the details or making the experiments.
It will be noticed that some of the last members of the
French school were alive at a comparatively recent date, but
nearly all their mathematical work was done before the year
1830. They are the direct successors of the French writers who
flourished at the commencement of the nineteenth century, and
seem to have been out of touch with the great German mathe-
maticians of the early part of it, on whose researches much of
the best work of that century is based; they are thus placed
here, though their writings are in some cases of a later date
than those of Gauss, Abe], and Jacobi.
The introduction of analysis into England.
The complete isolation of the English school and its devotion
to geometrical methods are the most marked features in its
history during the latter half of the eighteenth century ; and
the absence of any considerable contribution to the advancement
CH.xviii] CAMBRIDGE ANALYTICAL SCHOOL 439
of mathematical science was a natural consequence. One result
of this was that the energy of English men of science was
largely devoted to practical physics and practical astronomy,
which were in consequence studied in Britain perhaps more
than elsewhere.
Ivory. Almost the only English mathematician at the
beginning of this century who used analytical methods, and
whose work requires mention here, is Ivory, to whom the cele-
brated theorem in attractions is due. Sir James Ivory was
born in Dundee in 1765, and died on September 21, 1842.
After graduating at St. Andrews he became the managing
partner in a flax-spinning company in Forfarshire, but continued
to devote most of his leisure to mathematics. In 1804 he was
made professor at the Royal Military College at Marlow, which
was subsequently moved to Sandhurst; he was knighted in
1831. He contributed numerous papers to the Philosophical
Transactio7is, the most remarkable being those on attractions.
In one of these, in 1809, he shewed how the attraction of a
homogeneous ellipsoid on an external point is a multiple of that
of another ellipsoid on an internal point : the latter can be
easily obtained. He criticized Laplace's solution of the method
of least squares with unnecessary bitterness, and in terms which
shewed that he had failed to understand it.
The Cambridge Analytical School. Towards the beginning
of the last century the more thoughtful members of the Cambridge
school of mathematics began to recognize that their isolation
from their continental contemporaries was a serious evil. The
earliest attempt in England to explain the notation and methods
of the calculus as used on the continent was due to Woodhouse,
who stands out as the apostle of the new movement. It is
doubtful if he could have brought the analytical methods into
vogue by himself ; but his views were enthusiastically adopted
by three students, Peacock, Babbage, and Herschel, who suc-
ceeded in carrying out the reforms he had suggested. In a
book which will fall into the hands of few but English readers
I may be pardoned for making space for a few remarks on these
440 CAMBRIDGE ANALYTICAL SCHOOL [ch. xviii
four mathematicians, though otherwise a notice of them would
not be required in a work of this kind.^ The original stimulus
came from French sources, and I therefore place these remarks
at the close of my account of the French school ; but I should
add that the English mathematicians of this century at once
struck out a line independent of their French contemporaries.
Woodhouse. Robert Woodhouse was born at Norwich on
April 28, 1773; was educated at Caius College, Cambridge, of
which society he was subsequently a fellow ; was Plumian pro-
fessor in the university ; and -c(5htinued to live at Cambridge till
his death on December 23, 1827.
Woodhouse's earliest work, entitled the Principles of Ana-
lytical Calculation, was published at Cambridge in 1803. In
this he explained the differential notation and strongly pressed
the employment of it; but he severely criticized the methods
used by continental writers, and their constant assumption of
non-evident principles. This was followed in 1809 by a trigono-
metry (plane and spherical), and in 1810 by a historical treatise
on the calculus of variations and isoperimetrical problems. He
next produced an astronomy; of which the first book (usually
bound in two volumes), on practical and descriptive astronomy,
was issued in 1812, and the second book, containing an account
of the treatment of physical astronomy by Laplace and other
continental writers, was issued in 1818. All these works deal
critically with the scientific foundation of the subjects considered
— a point which is not unfrequently neglected in modern text-
books.
A man like Woodhouse, of scrupulous honour, universally
respected, a trained logician, and with a caustic wit, was well
fitted to introduce a new system; and the fact that when he
first called attention to the continental analysis he exposed the
unsoundness of some of the usual methods of establishing it,
more like an opponent than a partisan, was as politic as it
was honest. Woodhouse did not exercise much influence on
^ The following account is condensed from my History of the Study of
Mathematics at Cambridge, Cambridge, 1889.
CH. xviii] CAMBRIDGE ANALYTICAL SCHOOL 441
the majority of his contemporaries, and the movement might
have died away for the time being if it had not been for the
advocacy of Peacock, Babbage, and Herschel, who formed an
Analytical Society, with the object of advocating the general use
in the university of analytical methods and of the differential
notation.
Peacock. George Peacock^ who was the most influential of
the early members of the new school, was born at Denton on
April 9, 1791. He was educated at Trinity College, Cambridge,
of which society he was subsequently a fellow and tutor. The
establishment of the university observatory was mainly due to
his efforts, and in 1836 he was appointed to the Lowndean
professorship of astronomy and geometry. In 1839 he was
made dean of Ely, and resided there till his death on Nov. 8,
1858. Although Peacock's influence on English mathematicians
was considerable, he has left but few memorials of his work ;
but I may note that his report on progress in analysis, 1833,
commenced those valuable summaries of current scientific progress
which enrich many of the annual volumes of the Transactions of
the British Association.
Babbage. Another important member of the Analytical
Society was Charles Babbage, who was born at Totnes on Dec.
26, 1792 ; he entered at Trinity College, Cambridge, in 1810;
subsequently became Lucasian professor in the university ; and
died in London on Oct. 18, 1871. It was he who gave the
name to the Analytical Society, which, he stated, was formed
to advocate " the principles of pure d-i&m. as opposed to the dot-
age of the university." In 1820 the Astronomical Society was
founded mainly through his efforts, and at a later time, 1830 to
1832, he took a prominent part in the foundation of the British
Association. He will be remembered for his mathematical
memoirs on the calculus of functions, and his invention of an
analytical machine which could not only perform the ordinary
processes of arithmetic, but could tabulate the values of any func-
tion and print the results.
Herschel. The third of those who helped to bring analytical
442 CAMBRIDGE ANALYTICA-L SCHOOL [ch. xviii
methods into general use in England was the son of Sir William
Herschel (1738-1822), the most illustrious astronomer of the
latter half of the eighteenth century and the creator of modern
stellar astronomy. Sir John Frederick William Herschel was born
on March 7, 1792, educated at St. John's College, Cambridge,
and died on May 11, 1871. His earliest original work was a
paper on Cotes's theorem, and it was followed by others on
mathematical analysis, but his desire to complete his father's
work led ultimately to his taking up astronomy. His papers
on light and astronomy contain a clear exposition of the
principles which underlie the mathematical treatment of those
subjects.
In 1813 the Analytical Society published a volume of
memoirs, of which the preface and the first paper (on continued
products) are due to Babbage ; and three years later they
issued a translation of Lacroix's Traite elementaire du calcul
differentiel et du calcul integral. In 1817, and again in 1819,
the difi'erential notation was used in the university examinations,
and after 1820 its use was well established. The Analytical
Society followed up this rapid victory by the issue in 1820 of
two volumes of examples illustrative of the new method ; one
by Peacock on the diflferential and integral calculus, and the
other by Herschel on the calculus of finite differences. Since
then English works on the infinitesimal calculus have abandoned
the exclusive use of the fluxional notation. It should be noticed
in passing that Lagrange and Laplace, like the majority of other
modern writers, employ both the fluxional and the differential
notation ; it w^as the exclusive adoption of the former that was
so hampering.
Amongst those who materially assisted in extending the
use of the new analysis were William Whewell (1794-1866)
and George Biddell Airy (1801-1892), both Fellows of Trinity
College, Cambridge. The former issued in 1819 a work on
mechanics, and the latter, who was a pupil of Peacock, published
in 1826 his Tracts, in which the new method was applied with
great success to various physical problems. The efforts of the
CH. xviii] CAMBRIDGE ANALYTICAL SCHOOL 443
society were supplemented by the rapid publication of good
text-books in which analysis was freely used. The employment
of analytical methods spread from Cambridge over the rest of
Britain, and by 1830 these methods had come into general use
there.
444
CHAPTER XIX;
MATHEMATICS OF THE NINETEENTH CENTURY.
The nineteenth century saw the creation of numerous new
departments of pure mathematics — notably of a theory of
numbers, or higher arithmetic ; of theories of forms and
groups, or a higher algebra; of theories of functions of
multiple periodicity, or a higher trigonometry ; and of a
general theory of functions, embracing extensive regions of
higher analysis. Further, the developments of synthetic and
analytical geometry created what practically were new subjects.
The foundations of the subject and underlying assumptions
(notably in arithmetic, geometry, and the calculus) were also
subjected to a rigorous scrutiny. Lastly, the application of
mathematics to physical problems revolutionized the foundations
and treatment of that subject. Numerous Schools, Journals,
and Teaching Posts were established, and the facilities for the
study of mathematics were greatly extended.
Developments, such as these, may be taken as opening a
new period in the history of the subject, and I recognize that in
the future a writer who divides the history of mathematics as I
have done would probably treat the mathematics of the seven-
teenth and eighteenth centuries as forming one period, and
would treat the mathematics of the nineteenth century as
commencing a new period. This, however, would imply a
CH. xix] NINETEENTH CENTURY MATHEMATICS 445
tolerably complete and systematic account of the development
of the subject in the nineteenth century. But evidently it is
impossible for me to discuss adequately the mathematics of a
time so near to us, and the works of mathematicians some of
whom are living and some of whom I have met and known.
Hence I make no attempt to give a complete account of the
mathematics of the nineteenth century, but as a sort of appendix
to the preceding chapters I mention the more striking features
in the history of recent pure mathematics, in which I include
theoretical d^^namics and astronomy ; I do not, however, propose
to discuss in general the recent application of mathematics to
physics.
In only a few cases do I give an account of the life and
works of the mathematicians mentioned ; but I have added brief
notes about some of those to whom the development of any
branch of the subject is chiefly due, and an indication of that
part of it to which they have directed most attention. Even
with these limitations it has been very difficult to put together a
connected account of the mathematics of recent times ; and I
wish to repeat explicitly that I do not suggest, nor do I wish
my readers to suppose, that my notes on a subject give the
names of all the chief writers who have studied it. In fact the
quantity of matter produced has been so enormous that no one
can expect to do more than make himself acquainted with the
works produced in some special branch or branches. As an
illustration of this remark I may add that the committee
appointed by the Royal Society to report on a catalogue of
periodical literature estimated, in 1900, that more than 1500
memoirs on pure mathematics were then issued annually, and
more than 40,000 a year on scientific subjects.
Most histories of mathematics do not treat of the work
produced during this century. The chief exceptions with which
I am acquainted are R. d'Adhemar's VCEuvre mathematique
du xix^ siecle ; K. Fink's Geschichte- der Mathematilc^ Tiibingen,
1890 ; E. J. Gerhardt's Geschichte der Mathematik in Deutsch-
landj Munich, 1877 ; S. Giinther's IWm. Unt. zw Geschichte
446 NINETEENTH CENTURY MATHEMATICS [ch. xix
der mathematischen Wissenschaften, Leipzig, 1876, and Ziele und
Resultate der neueren mathematisch - historischen Forschung,
Erlangen, 1876 ; J. G. Hagen, Synopsis der hoheren Mathematik,
3 volumes, Berlin, 1891, 1893, 1906 ; a short dissertation by H.
Hankel, entitled Die Entwickelung der Mathematik in den letzten
Jahrhunderten, Tiibingen, 1885 ; a Discours on the professors
at the Sorbonne by C. Hermite in the Bulletin des sciences
mathematiques, 1890; F. C. Klein's Lectures on Mathematics,
Evanston Colloquium, New York and London, 1894 ; E. Lampe's
Die reine Mathematik in den Jahren 1884^-1899, Berlin, 1899 ;
the eleventh and twelfth volumes of Marie's Histoire des
sciences, in which are some notes on mathematicians who were
born in the last century; P. Painleve's Les Sciences mathe-
matiques au xix^ siecle ; a chapter by D. E. Smith in Higher
Mathematics, by M. Merriman and R. S. Woodward, New York,
1900; and V. Volterra's lecture at the Rome Congress, 1908,
" On the history of mathematics in Italy during the latter half of
the nineteenth century,"
A few histories of the development of particular subjects
have been written — such as those by Isaac Todhunter on the
theories of attraction and on the calculus of probabilities ; those
by T. Muir on determinants, that by A. von Braunmiihl on
trigonometry, that by R. Reiff on infinite series, that by G.
Loria, II passato ed il presente delle principali teorie geometriche,
and that by F. Engel and P. Stackel on the theory of parallels.
The transactions of some of the scientific societies and academies
also contain reports on the progress in different branches of the
subject, while information on the memoirs by particular mathe-
maticians is given in the invaluable volumes of J. C. Poggendorff's
Biographisch - literarisches Handworterbuch zur Geschichte der
exacten Wissenschaften, Leipzig. The Encyklopddie der mathe-
matischen Wissenschaften, which is now in course of issue, aims
at representing the present state of knowledge in pure and
applied mathematics, and doubtless in some branches of mathe-
matics it will supersede these reports. The French translation
of this encyclopaedia contains numerous and valuable additions.
CH. xix] GAUSS 447
I have found these authorities and these reports useful, and I
have derived further assistance in writing this chapter from the
obituary notices in the proceedings of various learned Societies.
I am also indebted to information kindly furnished me by
various friends, and if I do not further dwell on this, it is only
that I would not seem to make them responsible for my errors
and omissions.
A period of exceptional intellectual activity in any subject
is usually followed by one of comparative stagnation ; and
after the deaths of Lagrange, Laplace, Legendre, and Poisson,
the French school, which had occupied so prominent a position
at the beginning of this century, ceased for some years to
produce much new work. Some of the mathematicians whom
I intend to mention first. Gauss, Abel, and Jacobi, were
contemporaries of the later years of the French mathematicians
just named, but their writings appear to me to belong to a
different school, and thus are properly placed at the beginning
of a fresh chapter.
There is no mathematician of this century whose writings
have had a greater effect than those of Gauss ; nor is it on only
one branch of the science that his influence has left a permanent
mark. I cannot, therefore, commence my account of the
mathematics of recent times better than by describing very
briefly his more important researches.
Gauss. ^ Karl Friedrich Gauss was born at Brunswick on
April 23, 1777, and died at Gottingen on February 23, 1855.
His father was a bricklayer, and Gauss was indebted for a
liberal education (much against the will of his parents, who
wished to profit by his wages as a labourer) to the notice which
his talents procured from the reigning duke. In 1792 he was
sent to the Caroline College, and by 1795 professors and pupils
^ Biographies of Gauss have been published by L. Hanselmann, Leipzig,
1878, and by S. von Walterhausen, Leipzig, 1856. The Royal Society of
Gottingen undertook the issue of a collection of Gauss's works, and nine
volumes are already published. Further additions are expected, and some
bints of what may be expected have been given by F. C. Klein.
448 NINETEENTH CENTURY MATHEMATICS [ch. xix
alike admitted that he knew all that the former could teach
him : it was while therethat^e investigated the method o^f
least squares^ and proved by induction the law of quadratic
reciprocity. Thence he vrenTTo Uottingen, where he studied
under Kastner : many of his discoveries in the theory of
numbers were made while a student here. In 1798 he
returned to Brunswick, wEere Jie "earned a somewhat precarious
livelihood by private, tuition.
In 1799 Gauss published^ajdepaonstration that aseiy iritegral
algebraical function of one variable can be expressed as a product
ofresl linear or quadratic_factorg. Hence every algebraical
equation has_a^^oot_of_the_form_aj:i6i^ a theorem of which he
gave later two other distinct proofs. His Disquisitiones
Arithmeticae appeared in 1801. A large part of this had
been submitted as a memoir to the French Academy in the
preceding year, and had been rejected in a most regrettable
manner ; Gauss was deeply hurt, and his reluctance to publish
his investigations may be partly attributable to this unfortunate
incident.
The next discovery of Gauss was in a totally different
department of mathematics. The absence of any planet in the
space between Mars and Jupiter, where Bode's law would have
led observers to expect one, had been long remarked, but it
was not till 1801 that any one of the numerous group of minor
planets which occupy that space was observed. The discovery
was made by G. Piazzi of Palermo ; and was the more interesting
as its announcement occurred simultaneously with a publication
by Hegel in which he severely criticised astronomers for not
paying more attention to phUoaophy, — a science, said he, which
would at once have shewn them that there could not possibly
be more than seven planets, and a study of which would there-
fore have prevented an absurd waste of time in looking for
what in the nature of things could never be found. The new
planet was named Ceres^^ but it was seen under conditions
which appeared to render it impracticable to forecast its orbit.
The observations were fortunately communicated to Gauss ; he
CH. xix] GAUSS 449
calculated its elements, and his analysis put him in the first
rank of theoretical astronomers.
The attention excited by these investigations procured for
him in 1807 the offer of a chair at Petrograd, which he
declined. In the same year he was appointed director of the
Gottingen Observatory and professor of Astronomy there.
These offices he retained to his death ; and after his appoint-
ment he never slept away from his Observatory except on one
occasion when he attended a scientific congress at Berlin. His
lectures were singularly lucid and perfect in form, and it is
said that he used here to give the analysis by which he had
arrived at his various results, and which is so conspicuously
absent from his published demonstrations; but for fear his
auditors should lose the thread of his discourse, he never
willingly permitted them to take notes.
I have already mentioned Gauss's publications in 1799,
1801, and 1802. For some years after 1807 his time was
mainly occupied by work connected with his Observatory. In
1809 he published at Hamburg his Theoria Motus Corporum
Coelestimn, a treatise which contributed largely to the im-
provement of practical astronomy, -and introduced the principle
of curvilinear triangulation ; and on the same subject, but
connected with observations in general, we have his memoir
Theoria C omhinationis Observationum Erroi'ihus Jlinimis
Obnoxia, with a second part and a supplement.
Somewhat later he took up the subject of geodesy, acting
from 1821 to 1848 as scientific adviser to the Danish and
Hanoverian Governments for the survey then in progress ;
his papers of 1843 and 1866, Ueber Gegenstdnde der hohern
Geodiisiej contain his researches on the subject.
Gauss's researches on electricity and magnetism date from
about the year 1830. His firstpaper on the theory of
magnetism, entitled Intensitas Vis Magneticae Terrestris ad
Mensuram Ahsolutam Revocata, was published in 1833. A few
months afterwards he, together with W. E. Weber, invented
the declination instrument and the bifilar magnetometer;
2g
450 NINETEENTH CENTURY MATHEMATICS [ch. xix
and in the same year they erected at Gottingen a magnetic
observatory free from iron (as Humboldt and Arago had
previously done on a smaller scale) where they made magnetic
observations, and in particular showed that it was practicable
to send telegraphic signals. In connection with this Observa-
tory Gauss founded an association with the object of securing
continuous observations at fixed times. The volumes of their
publications, Resultate aus der BeobachUtngen des magnetischen
Vereins for 1838 and 1839, contain two important memoirs by
Gauss : one on the general theory of earth-magnetism, and the
other on the theory of forces attracting according to the inverse
square of the distance.
Gauss, like Poisson, treated the phenomena in electrostatics
as due to attractions and repulsions between imponderable
particles. Lord Kelvin, then William Thomson (1824-1907),
of Glasgow, shewed in 1846 that the effects might also be
supposed analogous to a flow of heat from various sources of
electricity properly distributed.
In electrodynamics Gauss arrived (in 1835) at a result
equivalent to that given by W. E. Weber of Gottingen in
1846, namely, that the attraction between two electrified
particles e and e', whose distance apart is r, depends on their
relative motion and position according to the formula
eeV-2{l + (rr -i7-2)2c-2}.
Gauss, however, held that no hypothesis was satisfactory which
rested on a formula and was not a consequence of a physical
conjecture, and as he could not frame a plausible physical con-
jecture he abandoned the subject.
Such conjectures were proposed by Riemann in 1858, and by
C. Neumann, now of Leipzig, and E. Betti (1823-1892) of Pisa
in 1868, but Helmholtz in 1870, 1873, and 1874 showed that
they were untenable. A simpler view which regards all electric
and magnetic phenomena as stresses and motions of a material
elastic medium had been outlined by Michael Faraday (1791-
1867), and was elaborated by James Clerk Maxwell (1831-
CH.xix] GAUSS' 451
1879) of Cambridge in 1873 ; the latter, by the use of generalised
co-ordinates, was able to deduce the consequences, and the agree-
ment with experiment is close. Maxwell concluded by showing
that if the medium were the same as the so-called luminiferous
ether, the velocity of light would be equal to the ratio of the
electromagnetic and electrostatic units, and subsequent experi-
ments have tended to confirm this conclusion. The theories
previously current had assumed the existence of a simple elastic
solid or an action between matter and ether.
The above and other electric theories were classified by
J. J. Thomson of Cambridge, in a report to the British
Association in 1885, into those not founded on the principle
of the conservation of energy (such as those of Ampere, Grass-
mann, Stefan, and Korteweg) ; those which rest on assumptions
concerning the velocities and positions of electrified particles
(such as those of Gauss, W. E. Weber, Riemann, and R. J. E.
Clausius) ; those which require the existence of a kind of energy
of which we have no other knowledge (such as the theory of C.
Neumann) ; those which rest on dynamical considerations, but
in which no account is taken of the action of the dielectric (such
as the theory of F. E. Neumann) ; and, finally, those which rest
on dynamical considerations and in which the action of the
dielectric is considered (such as Maxwell's theory). In the
report these theories are described, criticised, and compared with
the results of experiments.
Gauss's researches on optics, and especially on systems
of lenses, were published in 1840 in his Dioptrische Unter-
suchungen.
From this sketch it will be seen that the ground covered
by Gauss's researches was extraordinarily wide, and it may be
added that in many cases his investigations served to initiate
new lines of work. He was, however, the last of the great
mathematicians whose interests were nearly universal : since his
fime the literature ot most branches of mathematics has grown
so fast that mathematicians have been forced to specialise in
some particular department or departments. I will now mention
452 NINETEENTH CENTURY MATHEMATICS [ch. xix
very briefly some of the most important of his discoveries in
pure mathematics.
His most celebrated work in pure mathematics is the Dis-
quisitiones Arithmeticae, which has proved a starting-point for
several valuable inyestigations on the theory of numbers. This
treatise and Legendre's Theorie des 7iom5risnremairr^slandard
works on the theory of numbers ; but, just as in his discussion
of elliptic functions Legendre failed to rise to the conception
of a new subject, and confined himself to regarding their theory
as a chapter in the integral calculus, so he treated the theory of
numbers as a chapter in algebra. Gauss, however, realised that
tlielheofy"ofcliscrete magnitudes or higher arithmetic was of
a different kind from that of continuousmagnitudes^ or_algebra^
and he introduced a^ new uotationj^d new methodsof analysis,
of which subsequent writers have generally availed themselves.
The theory of numbers may be divided into two main^jyvisirmSj
namely, the theory of congruences and lie theory of formg.
Both divisions were discussed by Gauss. In particular the
Disquisitiones Arithmeticae introduced the modern theory of
congruences of the firsthand second orders, and to this Gauss
reduced indeterminate analysis. In it also h» discussed the
solution of binomial equations of the form_£'^=J_: this involves
the celebrated theorem thaLjLjs possible to construct, by
elementary geometry, regular polygons of 2^(2^^+ 1) sides^
\yhere^r and n are integers and 'I"' + 1 is j]prime— a discovery
he had madem 1796. " He developed the tlieory of ternary quad-
ratic formsinvolving~Ewo indeterminates. He also investigated
the theory of determinants, and it was on Gauss's results that
Jacobi based his researches on that subject.
The theory of functions of double periodicity had its origin
in the discoveries of Abel and Jacobi, wiiicii i describe later.
Both these mathematicians arrived at the theta functions, which
play so large a part in the theory of the subject. Gauss, how-
ever, had independently, and indeed .at a far earlier date,
discovered these functions and some of their properties, having
been led to them by certain integrals which occurred in the
CH. xix] GAUSS 453
Determinatio Attractionis, to evaluate which he invented the
transformation now associated with the name of Jacobi. Though
Gauss at a later time conamunicated the fact to jacobi, he did
not publish liis~researches ; they occur in a series of note-books
of a date not later than 1808, and are included in his collected
works.
Of the remaining memoirs in pure mathematics the most
remarkable are those on the theory of biquadratic residues
(wherein the notion of complex numbers of the form a + bi was
first introduced into the theory of numbers), in which are in-
cluded severaltables, and notably one of the nunaber of the
classes of binaryjquadratig forms ; that relating to the proof of
the theorem that every algebraical equation .has a real or
imagmary root ; that on the summation of series ; and, lastly,
one on interpolation^ His introduction of rigorous tests for the
conver^egry of infim'fp series is worthy of attention. Specially
noticeable also aj^ his investigations on hypei'geometric
series ; these contain a discussion of the gamma function.
This subject has since become one ol considerable im-
portance, and has been written on by (among others) Kummer
and Riemann ; later the original conceptions were greatly
extended, and numerous memoirs on it and its extensions
have appeared. I should also mention Gauss's theorems on the
curvature of surfaces, wherein he devised a new and general
method of treatment which has led to many new results.
Finally, we have his im])ortant memoir on the conformal
representation of one surface upon another, in which
the results given by Lagrange for surfaces of revolution are
generalised for all surfaces. It would seem also that Gauss
had discovered some of the properties of ^c^iaternions,
though these investigations were not published until a few
years ago.
In the theory of attractions we have a paper on the attraction
of homogeneous ellipsoids ; the already -mentioned memoir of
1839, on the theory of forces attracting according to the
inverse square of the distance ; and the memoir, Determinatio
454 NINETEENTH CENTURY MATHEMATICS [ch. xix
Attractionis, in which it is shown that the secular variations,
which the eleinents of the orbit of a planet experience from
the attraction of another planet which disturbs it, are the same
as if the mass of the disturbing ]3lanet were distributed oyer
Jts orbit into an elliptic ring in such a manner that equal masses
of the ring would correspond to arcs of the orbit described
in equal times.
The great masters of modern analysis are Lagrange, Laplace,
and Gauss, Avho were contemporaries. It is interesting to note
the marked contrast in their styles. Lagrange is perfect both
in form and matter, he is careful to explain his procedure,
and though his arguments are general they are easy to follow.
Laplace, on the other hand, explains nothing, is indifferent to
style, and, if satisfied that his results are correct, is content
to leave them either with no proof or with a faulty one. Gauss
is as Q2^act and elegant as Lagrange, but even more difficu-lt
to follow than Laplace, for he removes every trace of the
analysis by which he reached his results, and studies to give
a proof wliich, while rigorous, shall be as concise and synthetical
as possible.
Dirichlet.^ One of Gauss's pupils to whom I may here
allude is Lejeune Dirichlet, whose masterly exposition of the
discoveries of Jacobi (who was his father-in-law) and of Gauss
has unduly overshadowed his own original investigations on
similar subjects. Peter Gustav Lejeune Dirichlet was born at
Diiren on February 13, 1805, and died at Gottingen on May 5,
1859. He held successively professorships at Breslau and
Berlin, and on Gauss's death in 1855 was appointed to succeed
him as professor of the higher mathematics at Gottingen. He
intended to finish Gauss's incomplete works, for which he was
^ Dirichlet's works, edited by L. Kronecker, were issued in two volumes,
Berlin, 1889, 1897. His lectures on the theory of numbers were edited by
J. W. R. Dedekind, third edition, Brunswick, 1879-81. His investigations
on the theory of the potential were edited by F. Grube, second edition, Leipzig,
1887. His researches on definite integrate have been edited by G. Arendt,
Brunswick, 1904. There is a note on some of his researches by C. W.
Borchardt in Crelles Journal, vol. Ivii, 1859, pp. 91-92.
CH.xix] DIRICHLET. EISENSTEIN 455
admirably fitted, but his early death prevented this. He pro-
duced, however, several memoirs which have considerably facili-
tated the comprehension of some of Gauss's more abstruse methods.
Of Dirichlet's original researches the most celebrated are those
dealing with the establishment of Fourier's theorem, those in
the theory of numbers on asymptotic laws (that is, laws which
approximate more closely to accuracy as the numbers concerned
become larger), and those on primes.
It is convenient to take Gauss's researches as the starting-
point for the discussion of various subjects. Hence the length
with which I have alluded to them.
The Theory of Numbers, or Higher Arithmetic. The researches
of Gauss on the theory of numbers were continued or supple-
mented by Jacobi, who first proved the law of cubic reciprocity ;
discussed the theory of residues ; and, in his Canon Arithmeticus,
gave a table of residues of prime roots. Dirichlet also paid
some attention to this subject.
Eisenstein.^ The subject was next taken up by Ferdinand
Gotthold Eisensteiii, a professor at the University of Berlin, who
was born at Berlin on April 16, 1823, and died there on
October 11, 1852. The solution of the problem of the re-
presentation of numbers by binary quadratic forms is one of
the great achievements of Gauss, and the fundamental principles
upon which the treatment of such questions rest were given by
him in the Disquisifiones Arithmeticae. Gauss there added
some results relating to ternary quadratic forms, but the general
extension from two to three indeterminates was the work of
Eisenstein, who, in his memoir JVeue Theoixme der hoheren
Arithmetik, defined the ordinal and generic characters of ternary
quadratic forms of an uneven determinant ; and, in the case of
definite forms, assigned the weight of any order or genus ; but
he did not consider forms of an even determinant, nor give any
demonstrations of his work.
1 For a sketch of Eisenstein's life and researches see A hhandlungen zur
Geschichte der Mathematik, 1895, p. 143 et seq.
456 NINETEENTH CENTURY MATHEMATICS [ch. xix
Eisenstein also considered the theorems relating to the
possibility of representing a number as a sum of squares, and
showed that the general theorem was limited to eight squares.
The solutions in the cases of two, four, and six squares may
be obtained by means of elliptic functions, but the cases in
which the number of squares is uneven involve special pro-
cesses peculiar to the theory of numbers. Eisenstein gave the
solution in the case of three squares. He also left a statement
of the solution he had obtained in the case of five squares ; ^
but his results were published without proofs, and apply only
to numbers which are not divisible by a square.
Henry Smith. ^ One of the most original mathematicians
of the school founded by Gauss was Henry Smith. Henry
John Stephen Smith was born in London on November 2,
1826, and died at Oxford on February 9, 1883. He was
educated at Rugby, and at Balliol College, Oxford, of which
latter society he was a fellow; and in 1861 he was elected
Savilian professor of Geometry at Oxford, where he resided till
his death.
The subject in connection with which Smith's name is specially
associated is the theory of numbers, and to this he devoted the
years from 1854 to 1864. The results of his historical researches
were given in his report published in parts in the Transactions
of the British Association from 1859 to 1865. This report
contains an account of what had been done on the subject to
that time together with some additional matter. The chief
outcome of his own original work on the subject is included
in two memoirs printed in the Philosophical Transactions for
1861 and 1867 ; the first being on linear indeterminate equations
and congruences, and the second on the orders and genera of
ternary quadratic forms. In the latter memoir demonstrations
of Eisenstein's results and their extension to ternary quadratic
^ Crelles Journal, vol. xxxv, 1847, p. 368.
2 Smith's collected mathematical works, edited by J. W. L. Glaisher, and
prefaced by a biographical sketch and other papers, were published in two
volumes, Oxford, 1894. The following account is extracted from the obituary
notice in the monthly notices of the Astronomical Society, 1884, pp. 138-149.
< H. xix] HENRY SMITH 457
forms of an even determinant were supplied, and a complete
classification of ternary quadratic forms was given.
Smith, however, did not confine himself to the case of three
indeterminates, but succeeded in establishing the principles on
which the extension to the general case of n indeterminates
depends, and obtained the general formulae — thus effecting
the greatest advance made in the subject since the publication
of Gauss's work. In the account of his methods and results
which appeared in the Proceedings of the Royal Society, ^ Smith
remarked that the theorems relating to the representation of
numbers by four squares and other simple quadratic forms,
are deducible by a uniform method from the principles there
indicated, as also are the theorems relating to the representation
of numbers by six and eight squares. He then proceeded to
say that as the series of theorems relating to thp representation
of numbers by sums of squares ceases, for the reason assigned
by Eisenstein, when the number of squares surpasses eight, it
was desirable to complete it. The results for even squares were
known. The principal theorems relating to the case of five
squares had been given by Eisenstein, but he had considered only
those numbers which are not divisible by a square, and he had
not considered the case of seven squares. Smith here completed
the enunciation of the theorems for the case of five squares, and
added the corresponding theorems for the case of seven squares.
This paper was the occasion of a dramatic incident in the
history of mathematics. Fourteen years later, in ignorance of
Smith's work, the demonstration and completion of Eisenstein's
theorems for five squares were set by the French Academy as
the subject of their " Grand prix des sciences mathematiques."
Smith wrote out the demonstration of his general theorems so
far as was required to prove the results in the special case of
five squares, and only a month after his death, in March 1883,
the prize was awarded to him, another prize being also awarded
to H. Minkowski of Bonn. No episode could bring out in a
more striking light the extent of Smith's researches than that
1 See vol. xiii, 1864, pp. 199-203, and vol. xvi, 1868, pp. 197-208.
458 NINETEENTH CENTURY MATHEMATICS [ch. xix
a question, of which he had given the solution in 1867, as a
corollary from general formulae which governed the whole class
of investigations to which it belonged, should have been regarded
by the French Academy as one whose solution was of such
difficulty and importance as to be worthy of their great prize.
It has been also a matter of comment that they should have
known so little of contemporary English and German researches
on the subject as to be unaware that the result of the problem
they were proposing was then lying in their own library.
J. W. L. Glaisher of Cambridge has recently extended ^ these
results, and investigated, by the aid of elliptic functions, the
number of representations of a number as the sum of 2n squares
where n is not greater than 9.
Among Smith's other investigations I may specially mention
his geometrical, memoir, Sur quelques problemes cubiques et
biquadratiques, for which in 1868 he was awarded the Steiner
prize of the Berlin Academy. In a paper which he contributed
to the Atti of the Accademia dei Lincei for 1877 he established
a very remarkable analytical relation connecting the modular
equation of order ??, and the theory of binary quadratic forms
belonging to the positive determinant n. In this paper the
modular curve is represented analytically by a curve in such a
manner as to present an actual geometrical image of the complete
systems of the reduced quadratic forms belonging to the deter-
minant, and a geometrical interpretation is given to the ideas of
"class," "equivalence," and " reduced form." He was also the
author of important papers in which he succeeded in extending
to complex quadratic forms many of Gauss's investigations
relating to real quadratic forms. He was led by his researches
on the theory of numbers to the theory of elliptic functions, and
the results he arrived at, especially on the theories of the theta
and omega functions, are of importance.
Kummer. The theory of primes received a somewhat unex-
pected development by E. E. Kummer of Berlin, who was
^ For a summary of his results see his paper in the Proceedings of the
London Mathematical Society^ 1907, vol. v, second series, pp. 479-490.
CH. xix] KUMMER 459
born in 1810 and died in 1893. In particular he treated
higher complex members of the form a + ^4/", where j is a com-
plex root of jP -\= 0, x> being a prime. His theory brought out
the unexpected result that the proposition that a number can be
resolved into the product of powers of primes in one and only
one way is not necessarily true of every complex number. This
led to the theory of ideal primes, a theory which was developed
later by J. W. R. Dedekind. Kummer also extended Gauss's
theorems on quadratic residues to residues of a higher order,
and wrote on the transformations of hypergeometric functions.
The theory of numbers, as treated to-day, may be said to
originate with Gauss. I have already mentioned very briefly
the investigations of Jacobi, Dirichlet, Eisenstein, Heni^y Smithy
and Kummer. I content myself with adding some notes on the
subsequent development of certain branches of the theory.^
The distribution of primes has been discussed in particular by
P. L. Tchehycheff^ (1821-1894) of Petrograd, G. F. B. Riemaim,
and J, J. Sylvester. Riemann's short tract on the number of
primes which lie between two given numbers affords a striking
instance of his analytical powers. Legendre had previously
shown that the number of primes less than n is approximately
nKlogeii - 1 '08366) ; but Riemann went farther, and this tract
and a memoir by Tchebycheff contain nearly all that has been
done yet in connection with a problem of so obvious a character,
that it has suggested itself to all who have considered the theory
of numbers, and yet which overtaxed the powers even of Lagrange
and Gauss. In this paper also Riemann stated that all the roots
of r(Js^- l)(s- l)7r~*/^{(s) are of the form ^ + it where t is
real. It is believed that the theorem is true, but as yet it has
defied all attempts to prove it. Riemann's work in this connection
has proved the starting-point for researches by J. S. Hadamard,
H. C. F. von Mangoldt, and other recent writers.
The partition of numbers, a problem to which Euler had
^ See H. J. S. Smith, Report on the Theory of Numbers in vol. i of his
works, and 0. Stolz, Groessen und Zahlen, Leipzig, 1891.
^ Tchebycheff's collected works, edited by H. Markoff and N. Sonin, have
been published in two volumes. A French translation was issued 1900, 1907.
460 NINETEENTH CENTURY MATHEMATICS [ch. xix
paid considerable attention, has been treated by A. Cayley^
J. J. Sylvester, and P. A. MacMahcn. The representation of
numbers in special forms, the possible divisors of numbers
of specified forms, and general theorems concerned with the
divisors of numbers, have been discussed by J. Liouville
(1809-1882), the editor from 1836 to 1874 of the well-known
mathematical journal, and by J. W. L. Glaisher of Cambridge.
The subject of quadratic binomials has been studied by A. L.
Caucliy; of ternary and quadratic forms by L. KronecUer^
(1823-1891) of Berlin; and of ternary forms by C. Ilermite
of Paris.
The most common text-books are, perhaps, that by O. Stolz
of Innspruck, Leipzig, 1885-6 ; that by G. B. Mathews, Cam-
bridge, 1892; that by E. Lucas, Paris, 1891; and those by
P. Bachmann, Leipzig, 1892-1905. Possibly it may be found
hereafter that the subject is approached better on other lines
than those now usual.
The conception of Number, has also been discussed at
considerable length during the last quarter of the nineteenth
century. Transcendent numbers had formed the subject of two
memoirs by Liouville, but were subsequently treated as a
distinct branch of mathematics, notably by L. Kronecker and
G. Cantor. Irrational numbers and the nature of numbers
have also been treated from first principles, in particular by
K. Weierstrass, J. W. R. Dedekind,^ H. C. R. Meray, G. Cantor,
G. Peano, and B. A. W. Russell. This subject has attracted much
attention of late years, and is now one of the most flourishing
branches of modern mathematics. Transfinite, cardinal, and
ordinal arithmetic, and the theory of sets of points, may be
mentioned as prominent divisions. The theory of aggregates is
related to this subject, and has been treated by G. Cantor, P. du
Bois-Raymond, A. Schonflies, E. Zermelo, and B. A. W. Russell.
^ See the Bulletin of the New York (American) Mathematical Society, vol.
i, 1891-2, pp. 173-184.
2 Dedekind's Essays may serve as an introduction to the subject. They
have been translated into English, Chicago, 1901.
CH.xix] ABEL 461
Elliptic and Ahelian Functions, or Higher Trigonometry.^
The theory of functions of double and multiple periodicity
is another subject to which much attention has been paid during
this century. I have already mentioned that as early as 1808
Gauss had discovered the theta functions and some of their
properties, but his investigations remained for many years con-
cealed in his notebooks ; and it was to the researches made
between 1820 and 1830 by Abel and Jacobi that the modern
development of the subject is due. Their treatment of it has
completely superseded that used by Legendre, and they are
justly reckoned as the creators of this branch of mathematics.
Abel.^ Niels Ilenrick Abel was born at Findoe, in Norway,
on August 5, 1802, and died at Arendal on April 6, 1829, at
the age of twenty- six. His memoirs on elliptic functions,
originally published in Crelle's Journal (of which he was one of
the founders), treat the subject from the point of view of the
theory of equations and algebraic forms, a treatment to which
his researches naturally led him.
The important and very general result known as Abel's
theorem, which was subsequently applied by Riemann to the
theory of transcendental functions, was sent to the French
Academy in 1826, but was not printed until 1841 : its publica-
tion then was due to inquiries made by Jacobi, in consequence
of a statement on the subject by B. Holmboe in his edition of
Abel's works issued in 1839. It is far from easy to state Abel's
theorem intelligently and yet concisely, butUbroadly-^peaking,
it may be described as a theorem for evaluating the sum of a_
number of integrals which have the same integrand, but different
^ See the introduction to ElUptische Functionen, by A. Enneper, second'
edition (ed, by F. Miiller), Halle, 1890 ; and Geschichte der Theorie der ellip-
tischen Transcendenten, by L. Konigsbergei', Leipzig, 1879. On the history
of Abelian functions see the Transactions of the British Association, vol, Ixvii,
London, 1897, pp. 246-286.
2 The life of Abel by C. A. Bjerknes was published at Stockholm in 1880,
and another by L, de Pesloiian at Paris in 1906. Two editions of Aljel's
wijrks have been published, of which the last, edited by Sylow and Lie, and
issued at Christiania in two volumes in 1881, is the more complete. See also
the Abel centenary volume, Christiania, 1902 ; and a memoir by G. Mittag-
Leffler.
462 NINETEENTH CENTURY MATHEMATICS [ch. xix
limits — theseHmits being the roots of an algebraic equation.
The theorem^ives the sum of the integrals in terms ofTthe con-
stants occurring jn fhis ^^^y^tin,., an4 \^ th^ -fnt^granfl We
^nay regard the inverse of the integral of_thisintegrand as a
new transcendental function, and if so^e theorem furnishes a
property of this function. For instance, if Abel's theorem be
applied to the integrand (1 - x^y^^"^ it gives the addition theorem
for the circular (or trigonometrical) functions.
The name of Abelian function has been given to the higher
transcendents of multij)le periodicity which were first discussed
by Abel. The Abelian functions connected with a curve / (x, y)
are of the ioYYi\.fudx where 2* is a rational function of x and y.
The theory of Abelian functions has been studied by a very large
number of modern writers.
Abel criticised the use Qf infinite series, and discovered the
wftlj-l^nnwTi theorem which furnishes atest_for th^ vfllidity n.f
the result obtained by multipb^ing_one infinite_^
another. He also proved^ the binomial theorem for the
expansion of (1 + xY' when x and— j^_ are complex. As
illustrating his fertility of ideas I may, in passing, notice his
celebrated demonstration that it is impossible to express a root
of the general quintic equation in terms of it^s_mpiffir>ip.nt,s by
means ot^a timte number of radicals and rational functiona: this
tEeorem was the naoreimportant since it definitely limited a;field
of mathematics whick had previously attracted numerous writers.
I should add that this theorem had been enunciated as early as
1798 by Paolo Rufiini, an Italian physician practising at
Modena ; but I believe that the proof he gave was deficient in
generality.
Jacobi.2 Carl Gustav Jacob Jacohi, born of Jewish parents
1 See Abel, (Euvres, 1881, vol. i, pp. 219-250; and E. W. Barnes,
Quarterly Journal of Mathematics, vol, xxxviil, 1907, pp. 108-116.
^ See C. J. Gerhardt's Geschichte der Mathematik in Deutschla^id, Munich,
1877. Jacobi's collected works were edited by Dirichlet, three volumes, Berlin,
1846-71, and accompanied by a biography, 1852 ; a new edition, under the
supervision of C. W. Borchardt and K. Weierstrass, was issued at Berlin in
seven volumes, 1881-91. See also L. Konigsberger's C. O. J. Jacobi, Leipzig,
1904.
CH.xix] JACOBI 463
at Potsdam on Dec. 10, 1804, and died at Berlin on Feb. 18,
1851, was educated at the University of Berlin, where he ob-
tained the degree of Doctor of Philosophy in 1825. In 1827 he
became extraordinary professor of Mathematics at Konigsberg,
and in 1829 was promoted to be an ordinary professor. This
chair he occupied till 1842, when the Prussian Government gave
him a pension, and he moved to Berlin, where he continued to
live till his death in 1851. He was the greatest mathematical
teacher of his generation, and his lectures, though somewhat
unsystematic in arrangement, stimulated and influenced the
more able of his pupils to an extent almost unprecedented at
the time.
Jacobi's most celebrated investigations are those on elliptic
functions, the modern notation in which is substantially due to
him, and the theory of which he established simultaneously with
Abel, but independently of him. Jacobi's results are given in
his treatise on elliptic functions, published in 1829, and in some
later papers in Crelle's Journal', they are earlier than Weier-
strass's researches which are mentioned below. The correspond-
ence between Legendre and Jacobi on elliptic functions has been
reprinted in the first volume of Jacobi's collected works. Jacobi,
like Abel, recognised that elliptic functions were not merely a
group of theorems on integration, but that they were types
of a new kind of function, namely, one of double periodicity ;
hence he paid particular attention to the theory of the theta
function. The following passage,^ in which he explains
this view, is sufficiently interesting to deserve textual reproduc-
tion : —
E quo, cum uuiversam, quae fingi potest, amplectatur periodicitatem
analyticam elucet, fuuctiones ellipticas non aliis adnumerari debere
transcendentibus, quae quibusdam gaudent elegantiis, fortasse pluribus
illas aut niaioribus, sed speciem quandam lis inesse perfecti et absoluti.
Among Jacobi's other investigations I may specially single
1 See Jacobi's collected works, vol. i, 1881, p. 87.
464 NINETEENTH CENTURY MATHEMATICS [ch. xix
out his papers on Determinants, which did a great deal to bring
them into general use ; and particularly his introduction of the
Jacobian, that is, of the functional determinant formed by the
71^ partial differential coefficients of the first order of 7i given
functions of 7i independent variables. I ought also to mention
his papers on Abelian transcendents ; his investigations on the
theory of numbers, to which I have already alluded ; his im-
portant memoirs on the theory of differential equations, both
ordinary and partial ; his development of the calculus of varia-
tions ; and his contributions to the problem of three bodies,
and other particular dynamical problems. Most of the results of
the researches last named are included in his Vorlesunyen iiber
Dynamih.
Riemann.^ Georg Friedrich Bernhard Rieimmn was born
at Breselenz on Sept. 17, 1826, and died at Selasca on July 20,
1866. He studied at Gottingen under Gauss, and subsequently
at Berlin under Jacobi, Dirichlet, Steiner, and Eisenstein, all
of whom were professors there at the same time. In spite of
poverty and sickness he struggled to pursue his researches. In
1857 he was made professor at Gottingen, general recognition
of his powers soon followed, but in 1862 his health began to
give way, and four years later he died, working, to the end,
cheerfully and courageously.
Riemann must be esteemed one of the most profound and
brilliant mathematicians of his time ; he was a creative genius.
The amount of matter he produced is small, but its originality
and power are manifest — his investigations on functions and on
geometry, in particular, initiating developments of great im-
portance.
His earliest paper, written in 1850, was on the general theory
of functions of a complex variable. This gave rise to a new method
^ Riemann's collected works, edited by H. Weber and prefaced by an
account of his life by Dedekind, were published at Leipzig, second edition,
1892 ; an important supplement, edited by M. Neither and W. Wirtinger, was
issued in 1902. His lectures on elliptic functions, edited by H. B. L. Stahl,
were published separately, Leipzig, 1899. Another short biography of
Riemann has been written by E. J. Schering, Gottingen, 1867.
CH. xix] EIEMANN 465
of treating the theory of functions. The development of this
method is specially due to the Gottingen school. In 1854
Riemann wrote his celebrated memoir on the hypotheses on
which geometry is founded : to this subject I allude below.
This was succeeded by memoirs on elliptic functions and on
the distribution of primes : these have been already mentioned.
He also investigated the conformal representation of areas, one
on the other : a problem subsequently treated by H. A. Schwarz
and F. H. Schottky, both of Berlin. Lastly, in multiple periodic
functions, it is hardly too much to say that in his memoir in
Borchardt's Journal for 1857, he did for the Abelian functions
what Abel had done for the elliptic functions. A posthumous
fragment on linear differential equations with algebraic coefficients
has served as the foundation of important work by L. Schlesinger.
I have already alluded to the researches of Legendre, Gauss,
Abel, Jacobi, and Riemann on elliptic and Abelian functions.
The subject has been also discussed by (among other writers)
J. G. Bosenhain (1816-1887) of Konigsberg, who wrote (in
1844) on the hyperelliptic, and double theta functions ; A. Gopel
(1812-1847) of Berlin, who discussed ^ hyperelliptic functions;
L. Kronecker ^ of Berlin, who wrote on elliptic functions ; L.
Konigberger^ of Heidelberg and 7^. Brioschi^ (1824-1897) of Milan,
both of whom wrote on elliptic and hyperelliptic functions; Henry
Smith of Oxford, who discussed the transformation theory, the
theta and omega functions, and certain functions of the modulus ;
A. Cayley of Cambridge, who was the first to work out (in 1845)
the theory of doubly infinite products and determine their period-
icity, and Avho has written at length on the connection between
the researches of Legendre and Jacobi ; and C. Hermite of Paris,
whose researches are mostly concerned with the transformation
theory and the higher development of the theta functions.
^ See Crelles Journal, vol. xxxv, 1847, pp. 277-312 ; an obituary notice,
by Jacobi, is given on pp. 313-317.
^ Kronecker's collected works in four volumes, edited by K. Hensel, are
now in course of publication at Leipzig, 1895, &c.
^ See Konigberger's lectures, published at Leipzig in 1874.
^ His collected works were published in two volumes, Milan, 1901, 1902.
2h
466 NINETEENTH CENTURY MATHEMATICS [ch. xix
Weierstrass.i The subject of higher trigonometry was put
on a somewhat different footing by the researches of Weierstrass.
Karl Weierstrass, born in Westphalia on October 31, 1815, and
died at Berlin on February 19, 1897, was one of the greatest
mathematicians of the nineteenth century. He took no part in
public affairs ; his life was uneventful ; and he spent the last
forty years of it at Berlin, where he was professor.
With two branches of pure mathematics — elliptic and Abelian
functions, and the theory of functions — his name is inseparably
connected. His earlier researches on elliptic functions related
to the theta functions, which he treated under a modified form
in which they are expressible in powers of the modulus. At a
later period he developed a method for treating all elliptic func-
tions in a symmetrical manner. Jacobi had shown that a
function of n variables might have In periods. Accordingly
Weierstrass sought the most general expressions for such func-
tions, and showed that they enjoyed properties analogous to
those of the hyperelliptic functions. Hence the properties of
the latter functions could be reduced as particular cases of
general results.
He was naturally led to this method of treating hyperelliptic
functions by his researches on the general theory of functions ;
these co-ordinated and comprised various lines of investigation
previously treated independently. In particular he constructed
a theory of uniform analytic functions. The representation of
functions by infinite products and series also claimed his especial
attention. Besides functions he also wrote or lectured on the
nature of the assumptions made in analysis, on the calculus of
variations, and on the theory of minima surfaces. His methods
are noticeable for their wide -reaching and general character.
Recent investigations on elliptic functions have been largely
based on Weierstrass's method.
Among other prominent mathematicians who have recently
^ Weierstrass's collected works are now in course of issue, Berlin, 1894, &c.
Sketches of his career by G. Mittag-Leffler and H. Poincare are given in Acta
Mathematica, 1897, vol. xxi, pp. 79-82, and 1899, vol. xxii, pp. 1-18.
CH. xix] WEIERSTRASS 467
written on elliptic and hyperelliptic functions, I may mention
the names of G. H. Halphen'^ (1844-1889), an officer in the
French army, whose investigations were largely founded on
Weierstrass's work; F. C. Klein of Gottingen, who has written
on Abelian functions, elliptic modular functions, and hyperelliptic
functions ; H. A. Sckwarz of Berlin ; H. Weber of Strassburg ;
M. Nother of Erlangen ; H. B. L. Stahl of Tiibingen ; F. G.
Frobemus of Berlin ; J. W. L. Glaisher of Cambridge, who has
in particular developed the theory of the zeta function ; and
H. F. Baker of Cambridge.
The usual text -books of to-day on elliptic functions are
those by J. Tannery and J. Molk, 4 volumes, Paris, 1893-
1901; by P. E. Appell and E. Lacour, Paris, 1896; by H.
Weber, Brunswick, 1891 ; and by G. H. Halphen, 3 volumes,
Paris, 1886-1891. To these I may add one by A. G. Greenhill
on the Applications of Elliptic Functions, London, 1892.
The Theory of Functions. I have already mentioned that
the modern theory of functions is largely due to Weierstrass and
H. C. R- Meray. It is a singularly attractive subject, and has
proved an important and far-reaching branch of mathematics.
Historically its modern presentation may be said to have been
initiated by A. Cauchy, who laid the foundations of the theory of
synectic functions of a complex variable. Work on these lines
was continued hyJ. Liouville, who wrote chiefly on doubly periodic
functions. These investigations were extended and connected
in the work by A, Briot (1817-1882), and J. C. Bouquet (1819-
1885), and subsequently were further developed by C. Hermite.
Next I may refer to the researches on the theory of algebraic
functions which have their origin in V. A. Puiseux's memoir of
1851, and G, F. B. Riemann^s papers of 1850 and 1857 ; in con-
tinuation of which H. A. Schivarz of Berlin established accurately
certain theorems of which the proofs given by Riemann were
open to objection. To Riemann also we are indebted • for
^ A sketch of Halphen's life and works is given in LiouviUes Journal for
1889, pp. 345-359, and in the Comptes Rendus, 1890, vol. ex, pp. 489-497.
468 NINETEENTH CENTURY MATHEMATICS [ch. xix
valuable work on nodular functions which has been recently
published in his Nachtrdge. Subsequently F. C. Klein of
Gottingen connected Riemann's theory of functions with the
theory of groups, and wrote on automorphic and modular
functions; //. Poincare of Paris also wrote on automorphic
functions, and on the general theory with special applications to
differential equations. Quite recently K. Ileiisel of Marburg
has written on algebraic functions ; and W. Wirtinger of Vienna
on Abelian functions.
I have already said that the work of Weierstrass shed a new
light on the whole subject. His theory of analytical functions
has been developed by G. Mittag-Leffler of Stockholm ; and
C. Hermite, P. E. A2:>pell, C. E. Picard, E. Goursat, E. N.
Laguerre^ and J, S. Iladamard, all of Paris, have also written
on special branches of the general theory ; while E. Borel,
R. L. Baire, H. L. Lebesgiie, and E. L. Lindellrf have produced
a series of tracts on uniform functions which have had a wide
circulation and influence.
As text-books I may mention the Theory of Functions of
a Complex Variable, by A. R. Forsyth, second edition, Cam-
bridge, 1900; AbeVs Theoremhy H. F. Baker, Cambridge, 1897,
and Multiple Periodic Functions by the same writer, Cambridge,
1907 ; the Theorie des fonctions algebriqttes by P. E. Appell
and E. Goursat, Paris, 1895 ; parts of C. E. Picard's Traite
d^ Analyse, in 3 volumes, Paris, 1891 to 1896 ; the Theory of
Functions by J. Harkness and F. Morley, London, 1893; the
Theory of Functions of a Peal Vai^iable and of Fourier's
Series by E. W. Hobson, Cambridge, 1 907 ; and Die Theorie
des AheVschen Functionen by H. B, L. Stahl, Leipzig, 1896.
Higher Algebra. The theory of numbers may be considered
as a higher arithmetic, and the theory of elliptic and Abelian
functions as a higher trigonometry. The theory of higher
algebra (including the theory of equations) has also attracted
considerable attention, and was a favourite subject of study of
the mathematicians whom I propose to mention next, though
CH.xix] CAUCHY 469
the interests of these writers were by no means limited to this
subject.
Cauchy.^ Attgustin Loids Cauchy, the leading representa-
tive of the French school of analysis in the nineteenth century,
was born at Paris on Aug. 21, 1789, and died at Sceaux on
May 25, 1857. He was educated at the Polytechnic school, the
nursery of so many French mathematicians of that time, and
adopted the profession of a civil engineer. His earliest mathe-
matical paper was one on polyhedra in 1811. Legendre thought
so highly of it that he asked Cauchy to attempt the solution of
an analogous problem which had baffled previous investigators,
and his choice was justified by the success of Cauchy in 1812.
Memoirs on analysis and the theory of numbers, presented in
1813, 1814, and 1815, showed that his ability was not confined
to geometry alone. In one of these papers he generalised some
results which had been established by Gauss and Legendre ; in
another of them he gave a theorem on the number of values
which an algebraical function can assume when the literal
constants it contains are interchanged. It was the latter
theorem that enabled Abel to show that in general an algebraic
equation of a degree higher than the fourth cannot be solved by
the use of a finite number of purely algebraical expressions.
To Abel, Cauchy, and Gauss we owe the scientific treatment
of series which have an infinite number of terms. In particular,
Cauchy established general rules for investigating the con-
vergency and divergency of such series, rules which were extended
by J. L. F. Bertrand (1822-1900) of Paris, Secretary of the
French Academie des Sciences, A. Pringsheim of Munich,
and considerably amplified later by E. Borel, by M. G.
Servant, both of Paris, and by other writers of the modern
French school. In only a few works of an earlier date
is there any discussion as to the limitations of the series
employed. It is said that Laplace, who was present when
^ See La Vie et les travatix de Cauchy by L. Valson, two volumes, Paris,
1868. A complete edition of his works is now being issued by the French
Government.
470 NINETEENTH CENTURY MATHEMATICS [ch. xix
Cauchy read his first paper on the subject, was so im-
pressed by the illustrations of the danger of employing such
series withoui a rigorous investigation of their convergency,
that he put on one side the work on which he was then
engaged and denied himself to all visitors, in order to see
if any of the demonstrations given in the earlier volumes of the
Mecanique celeste were invalid ; and he was fortunate enough to
find that no material errors had been thus introduced. The
treatment of series and of the fundamental conceptions of the
calculus in most of the text-books then current was based on
Euler's works, and was not free from objection. It is one
of the chief merits of Cauchy that he placed these subjects
on a stricter foundation.
On the restoration in 1816 the French Academy was
purged, and, incredible though it may seem, Cauchy accepted
a seat procured for him by the expulsion of Monge. He
was also at the same time made professor at the Polytechnic ;
and his lectures there on algebraic analysis, the calculus, and
the theory of curves, were published as text -books. On the
revolution in 1830 he went into exile, and was first appointed
professor at Turin, whence he soon moved to Prague to
undertake the education of the Comte de Chambord. He
returned to France in 1837; and in 1848, and again in 1851,
by special dispensation of the Emperor was allowed to occupy
a chair of mathematics without taking the oath of allegiance.
His activity was prodigious, and from 1830 to 1859 he
published in the Ti^ansactions of the Academy, or the Comptes
BeTuius, over 600 original memoirs and about 150 reports.
They cover an extraordinarily wide range of subjects, but are of
very unequal merit.
Among the more important of his other researches are those
on the legitimate use of imaginary quantities ; the determination
of the number of real and imaginary roots of any algebraic
equation within a given contour ; his method of calculating
these roots approximately ; his theory of the symmetric functions
of the coefficients of equations of any degree; his a pWoW
\ CH.xix] AKGAND * 471
valuation of a quantity less than the least difference between the
roots of an equation ; his papers on determinants in 1841, which
assisted in bringing them into general use ; and his investiga-
tions on the theory of numbers. Cauchy also did something to
reduce the art of determining definite integrals to a science;
the rule for finding the principal values of integrals was
enunciated by him. The calculus of residues was his invention.
His proof of Taylor's theorem seems to have originated from a
discussion of the double periodicity of elliptic functions. The
means of showing a connection between different branches of a
subject by giving complex values to independent variables is
largely due to him.
He also gave a direct analytical method for determining
planetary inequalities of long period. To physics he con-
tributed memoirs on waves and on the quantity of light
reflected from the surfaces of metals, as well as other papers
on optics.
Argand. I may mention here the name of Jean Robert
Argandj who was born at Geneva on July 18, 1768, and
died at Paris on August 13, 1822. In his Essai, issued
in 1806, he gave a geometrical representation of a complex
number, and applied it to show that every algebraic equation
has a root. This was prior to the memoirs of Gauss and
Cauchy on the same subject, but the essay did not attract
much attention when it was first published. An even
earlier demonstration that ^( - 1 ) may be interpreted to
indicate perpendicularity in two-dimensional space, and even
the extension of the idea to three-dimensional space by a
method foreshadowing the use of quaternions, had been given
in a memoir by C. Wessel, presented to the Copenhagen
Academy of Sciences in March 1797; other memoirs on the
same subject had been published in the Philosophical
Transactions for 1806, and by H. Kiihn in the Transactions
for 1750 of the Petrograd Academy. ^
^ See W. W. Beman in the Proceedings of tlie American Association for
the Advancement of Science, vol. xlvi, 1897.
472 NINETEENTH CENTURY MATHEMATICS [ch. xix
I have already said that the idea of a simple complex number
like a + hi where i^ = 0 was extended by Kummer. The general
theory has been discussed by K. Weierstrass, H. A. Schwarz
of Berlin, J. W. K. Dedekind, H. Poincare, and other writers.
Hamilton.^ In the opinion of some writers the theory
of quaternions will be ultimately esteemed one of the great
discoveries of the nineteenth century in pure mathematics. That
discovery is due to Sir William Eoivan Hamilton, who was
born in Dublin on August 4, 1805, and died there on September
2, 1865. His education, which was carried on at home, seems
to have been singularly discursive. Under the influence of an
uncle who was a good linguist, he first devoted himself to
linguistic studies ; by the time he was seven he could read
Latin, Greek, French, and German with facility; and when
thirteen he was able to boast that he was familiar with as many
languages as he had lived years. It was about this time that
he came across a copy of Newton's Universal Arithmetic. This
was his introduction to modern analysis, and he soon mastered
'the elements of analytical geometry and the calculus. He next
read the Principia and the four volumes then published of
Laplace's Me'canique celeste. In the latter he detected a mistake,
and his paper on the subject, written in 1823, attracted con-
siderable attention. In the following year he entered at Trinity
College, Dublin. His university career is unique, for the chair of
Astronomy becoming vacant in 1827, while he was yet an under-
graduate, he was asked by the electors to stand for it, and was
elected unanimously, it being understood that he should be left
free to pursue his own line of study.
His earliest paper on optics, begun in 1823, was pub-
lished in 1828 under the title of a Theory of Systems of
Rays, to which two supplements were afterwards added; in
the latter of these the phenomenon of conical refraction is pre-
dicted. This was followed by a paper in 1827 on the principle
^ See the life of Hamilton (with a bibliography of his Avritings) by E. P.
Graves, three volumes, Dublin, 1882-89 ; the leading facts are given in an
article in the North British lievieto for 1886.
CH.xix] HAMILTON. GRASSMANN 473
of Varying Action, and in 1834 and 1835 by memoirs on
a General Method in Dynamics — the subject of theoretical
dynamics being properly treated as a branch of pure mathe-
matics. His lectures on Quaternions were published in 1852.
Some of his results on this subject would seem to have
been previously discovered by Gauss, but these were unknown
and unpublished until long after Hamilton's death. Amongst
his other papers, I may specially mention one on the
form of the solution of the general algebraic equation of the
fifth degree, which confirmed Abel's conclusion that it cannot
be expressed by a finite number of purely algebraical ex-
pressions ; one on fluctuating functions ; one on the hodograph ;
and, lastly, one on the numerical solution of differential
equations. His Elements of Quaternions was issued in
1866 : of this a competent authority says that the methods
of analysis there given show as great an advance over those of
analytical geometry, as the latter showed over those of Euclidean
geometry. In more recent times the subject has been further
developed by P. G. Tait (1831-1901) of Edinburgh, by A.
Macfarlane of America, and by C. J. Joly in his Manual of
Quaternions, London, 1905.
Hamilton w^as painfully fastidious on what he published, and
he left a large collection of manuscripts which are now in the
library of Trinity College, Dublin, some of which it is to be
hoped will be ultimately printed.
Grassmann.i The idea of non-commutative algebras and of
quaternions seems to have occurred to Grassmann and Boole at
about the same time as to Hamilton. Herinanii Gunther Grass-
mann was born in Stettin on April 15, 1809, and died there in
1877. He was professor at the gymnasium at Stettin. His
researches on non-commutative algebras are contained in his
Ausdehnungslehre, first published in 1844 and enlarged in 1862.
This work has had great influence, especially on the continent,
where Grassmann's methods have generally been followed in
1 Grassmann's collected works in three volumes, edited by P. Engel, are
now in course of issue at Leipzig, 1894, &c.
474 NINETEENTH CENTURY MATHEMATICS [ch. xix
preference to Hamilton's. Grassmann's researches have been
continued and extended, notably by S. F. V. Schlegel and G.
Peano.
The scientific treatment of the fundamental principles of
algebra -initiated by Hamilton and Grassmann was continued by
De Morgan and Boole in England, and was further developed
by H. Hankel (1839-1873) in Germany in his work on com-
plexes, 1867, and, on somewhat different lines, by G. Cantor in
his memoirs on the theory of irrationals, 1871 ; the discussion
is, however, so technical that I am unable to do more than allude
to it. Of Boole and De Morgan I say a word or two in passing.
Boole. George Boole, born at Lincoln on November 2, 1815,
and died at Cork on December 8, 1864, independently invented a
system of non-commutative algebra, and was one of the creators
of symbolic or mathematical logic. ^ From his memoirs on
linear transformations part of the theory of invariants has
developed. His Finite Differences remains a standard work on
that subject.
De Morgan.^ Augustus de Morgan, born in Madura
(Madras) in June 1806, and died in London on March 18,
1871, was educated at Trinity College, Cambridge. In 1828
he became professor at the then newly-established University
of London (University College). There, through his works
arid pupils, he exercised a wide influence on English mathe-
maticians. He was deeply read in the philosophy and
history of mathematics, but the results are given in scattered
articles; of these I have made considerable use in this book.
His memoirs on the foundation of algebra ; his treatise on the
differential calculus published in 1842, a work of great ability,
and noticeable for his treatment of infinite series ; and his
articles on the calculus of functions and on the theory of
probabilities, are worthy of special note. The article on the
^ On the history of mathematical logic, see P. E. B. Jourdain, Quarterly
Journal of Mathematics, vol. xliii, 1912, pp. 219-314.
^ De Morgan's life was written by his widow, S. E. de Morgan, Jjondon,
1882.
CH.xix] DE MORGAN. GALOIS. CAYLEY 475
calculus of functions contains an investigation of tlie principles
of symbolic reasoning, but the applications deal with the solution
of functional equations rather than mth the general theory of
functions.
Galois.^ A new development of algebra — the theory of
groups of substitutions — was suggested by Evariste Galois, who
promised to be one of the most original mathematicians of the
nineteenth century, born at Paris on October 26, 1811, and
killed in a duel on May 30, 1832, at the early age of 20.
The theory of groups, and of subgroups or invariants, has
profoundly modified the treatment of the theory of equations.
An immense literature has grown up on the subject. The
modern theory of groups originated with the treatment by
Galois, Cauchy, and J. A. Serret (1819-1885), professor at
Paris ; their work is mainly concerned with finite discontinuous
substitution groups. This line of investigation has been
pursued by C. Jordan of Paris and E. Netto of Strassburg.
The problem of operations with discontinuous groups, with
applications to the theory of functions, has been further taken
up by (among others) F. G. Frobenius of Berlin, F. C. Klein
of Gottingen, and W. Burnside formerly of Cambridge and now
of Greenwich.
Cayley.2 Another Englishman whom we may reckon
among the great mathematicians of this prolific century was
Arthur Cayley. Cay ley was born in Surrey, on Aug. 16, 1821,
and after education at Trinity College, Cambridge, was called
to the bar. But his interests centred on mathematics ; in 1863
he was elected Sadlerian Professor at Cambridge, and he spent
there the rest of his life. He died on Jan. 26, 1895.
Cayley's writings deal with considerable parts of modern
pure mathematics. I have already mentioned his writings on
the partition of numbers and on elliptic functions treated from
Jacobi's point of view; his later writings on elliptic func-
^ On Galois's investigations, see the edition of his works with an intro-
duction by E. Picard, Paris, 1897.
2 Cayley's collected works in thirteen volumes were issued at Cam-
bridge, 1889-1898.
476 NINETEENTH CENTURY MATHEMATICS [ch. xix
tions dealt mainly with the theory of transformation and
the modular equation. It is, however, by his investigations
in analytical geon:ietry and on higher algebra that he will be
best remembered.
In analytical geometry the conception of what is called
(perhaps, not very happily) the absolute is due to Cayley. As
stated by himself, the "theory, in effect, is that the metrical
properties of a figure are not the properties of the figure
considered ^?er se . . . but its properties when considered in
connection with another figure, namely, the conic termed the
absolute"; hence metric properties can be subjected to de-
scriptive treatment. He contributed largely to the general
theory of curves and surfaces, his work resting on the
assumption of the necessarily close connection between alge-
braical and geometrical operations.
In higher algebra the theory of- invariants is due to Cayley ;
his ten classical memoirs on binary and ternary forms, and his
researches on matrices and non-commutative algebras, mark an
epoch in the development of the subject.
Sylvester.^ Another teacher of the same time was James
Joseph Sylvester, born in London on Sept. 3, 1814, and died on
March 15, 1897. He too Avas educated at Cambridge, and
while there formed "a lifelong friendship with Cayley. Like
Cayley he was called to the bar, and yet preserved all his
interests in mathematics. He held professorships successively
at Woolwich, Baltimore, and Oxford. He had a strong
personality and was a stimulating teacher, but it is difficult
to describe his writings, for they are numerous, disconnected,
and discursive.
On the theory of numbers Sylvester wrote valuable papers
on the distribution of primes and on the partition of numbers.
On analysis he wrote on the calculus and on differential
equations. But perhaps his favourite study was higher
algebra, and from his numerous memoirs on this subject I
^ Sylvester's collected works, edited by H. F. Baker, are in course of
publication at Cambridge ; 2 volumes are already issued.
CH.xix] SYLVESTER. LIE 477
may in particular single out those on canonical forms, on the
theory of contravariants, on reciprocants or differential in-
variants, and on the theory of equations, notably on Newton's
rule. I may also add that he created the language and
notation of considerable parts of those subjects on which he
wrote.
The writings of Cayley and Sylvester stand in marked
contrast : Cayley's are methodical, precise, formal, and com-
plete; Sylvester's are impetuous, unfinished, but none the
less vigorous and stimulating. Both mathematicians found
the greatest attraction in higher algebra, and to both that
subject in its modern form is deeply indebted.
Lie.^ Among the great analysts of the nineteenth century
to whom I must allude here, is Mai^ius Sophus Lie, born on
Dec. 12, 1842, and died on Feb. 18, 1899. Lie was educated
at Christiania, whence he obtained a travelling scholarship,
and in the course of his journeys made the acquaintance of
Klein, Darboux, and Jordan, to whose influence his subse-
quent career is largely due.
In 1870 he discovered the transformation by which a sphere
can be made to correspond to a straight line, and, by the use
of which theorems on aggregates of lines can be translated into
theorems on aggregates of spheres. This was followed by a
thesis on the theory of tangential transformations for space.
In 1872 he became professor at Christiania. His earliest
researches here were on the relations between differential equa-
tions and infinitesimal transformations. This naturally led him
to the general theory of finite continuous groups of substitutions ;
the results of his investigations on this subject are embodied in
his Theorie der Transformationsgruppen, Leipzig, three volumes,
1888-1893. He proceeded next to consider the theory of
infinite continuous groups, and his conclusions, edited by
G. SchefFers, were published in 1893. About 1879 Lie
turned his attention to differential geometry; a systematic
1 See the obituary notice by A. R. Forsyth in the Year-Book of the
RoyoX Society, Loudon, 1901.
478 NINETEENTH CENTURY MATHEMATICS [ch.xix
exposition of this is in course of issue in his Geometrie der
Beriih^ungstransfonnationen.
Lie seems to have been disappointed and soured by the
absence of any general recognition of the value of his results.
Reputation came, but it came slowly. In 1886 he moved to
Leipzig, and in 1898 back to Christiania, where a post had
been created for him. He brooded, however, over what he
deemed was the undue neglect of the past, and the happiness
of the last decade of his life was much affected by it.
Hermite.i Another great algebraist of the century was
Charles liermite^ born in Lorraine on December 24, 1822, and
died at Paris, January 14, 1901. From 1869 he was professor at
the Sorbonne, and through his pupils exercised a profound in-
fluence on the mathematicians of to-day.
While yet a student he wrote to Jacobi on Abelian functions,
and the latter embodied the results in his works. Hermite's
earlier papers were largely on the transformation of these
functions, a problem which he finally effected by the use of
modular functions. He applied elliptic functions to find solutions
of the quintic equation and of Lame's differential equation.
Later he took up the subject of algebraic continued fractions,
and this led to his celebrated proof, given in 1873, that e cannot
be the root of an algebraic equation, from which it follows that
e is a transcendental number. F. Lindemann showed in a
similar way in 1882 that ir is transcendental. The proofs have
been subsequently improved and simplified by K. Weierstrass,
D. Hilbert, and F. C. Klein.2
To the end of his life Hermite maintained his creative
interest in the subjects of the integral calculus and the theory
of functions. He also discussed the theory of associated co-
variants in binary quantics and the theory of ternary quantics.
1 Hermite's collected works, edited by E. Picard, are being issued in four
volumes; vol. i, 1905, vol. ii, 1908, vol. iii, 1912.
2 Tj^g value of tt was calculated to 707 places of decimals by W. Shanks
in 1873 ; see Proceedings of the Royal Society, vol. xxi, p. 318, vol. xxii, p. 45.
The value of e was calculated to 225 places of decimals by F. Tichanek ;
see F. J. Studnicka, Vortrdge ilber monoperiodische Functionem, Prague,
1892, and L' Inter mediare des Mathimaticiens, Paris, 1912, vol. xix, p. 247.
CH. xix] HEUMITE 479
So many other writers have treated the subject of Higher
Algebra (including therein the theory of forms and the theory
of equations) that it is difficult to summarise their conclusions.
The convergency of series has been discussed by J. L. Raahe
(1801-1859) of Zurich, J. L. F. Bertrmid, the secretary of the
French Academy ; E. E. Kummer of Berlin ; U. Dini of Pisa ;
A. Pringsheim of Munich ; ^ and Sir George Gabriel Stokes
(1819-1903) of Cambridge,^ to whom the well-known theorem on
the critical values of the sums of periodic series is due. The last-
named writer introduced the important conception of non-uniform
convergence ; a subject subsequently treated by P. L. Seidel.
Perhaps here, too, I may allude in passing to the work of
G. F. B. Riemann, G. G. Stokes, H. Hankel, and G. Darboux
on asymptotic expansions ; of H. Poincare on the application
of such expansions to differential equations ; and of E. Borel
and E. Cesar o on divergent series.
On the theory of groups of substitutions I have already
mentioned the work, on the one hand, of Galois, Cauchy, Serret,
Jordan, and Netto, and, on the other hand, of Frobenius, Klein,
and Burnside in connection with discontinuous groups, and that
of Lie in connection with continuous groups.
I may also mention the following writers : C. W. Borchardt ^
(1817-1880) of Berlin, who in particular discussed generating
functions in the theory of equations, and arithmetic-geometric
means. C. Hermite, to whose work I have alluded above.
Enrico Betti of Pisa and F. Briqschi of Milan,, both of whom
discussed binary quantics ; the latter applied hyperelliptic func-
tions to give a general solution of a sextic equation. S. H.
Aronhold (1819-1884) of Berlin, who developed symbolic
methods in connection with the invariant theory of quantics.
^ On the researches of Raabe, Bertraiid, Kummer, Dini, and Pringsheim,
see the Bulletin of the New York (American) Mathematical Society, vol, ii,
1892-3, pp. 1-10.
^ Stokes's collected mathematical and physical papers in five volumes, and
Lis memoir and scientific correspondence in two volumes, were issued at
Cambridge, 1880 to 1907.
^ A collected edition of Borchardt's w'orks, edited by G. Hettner, was
issued at Berlin in 1888.
480 NINETEENTH CENTURY MATHEMATICS [ch. xix
P. A. G or dan ^ of Erlangen, wlio has written on the theory of
equations, the theories of groups and forms, and shown that there
are only a finite number of concomitants of quantics. R. F. A.
Clehsch^ (1833-1872) of Gottingen, who independently investi-
gated the theory of binary forms in some papers collected and
published in 1871 ; he also wrote on Abelian functions. P. A.
MacMahon, formerly an officer in the British army, who has
written on the connection of symmetric functions, invariants and
covariants, the concomitants of binary forms, and combinatory
analysis. F. C. Klein of Gottingen, who, in addition to his
researches, already mentioned, on functions and on finite dis-
continuous groups, has written on differential equations. A. R.
Forsyth of Cambridge, who has developed the theory of invariants
and the general theory of differential equations, ternariants, and
quaternariants. P. Painleve of Paris, who has written on the
theory of differential equations. And, lastly, J). Hilbert of
Gottingen, who has treated the theory of homogeneous forms.
No account of contemporary writings on higher algebra
would be complete without a reference to the admirable Higher
Algeh^a by G. Salmon (1819-1904), provost of Trinity College,
Dublin, and the Cours dialgebre snperiem^e by J. A. Serret, in
which the chief discoveries of their resjDective authors are
embodied. An admirable historical summary of the theory of
the complex variable is given in the Vorlesungen uber die
complexen Zahlen, Leipzig, 1867, by H. Hankel, of Tiibingen.
Analytical Geometry. It will be convenient next to call
attention to another division of pure mathematics — analytical
geometry — which has been greatly developed in recent years.
It has been studied by a host of modern writers, but I do not
propose to describe their investigations, and I shall content
^ An edition of Gordan's work on invariants (determinants and binary
forms), edited by G. Kerschensteiner, was issued at Leipzig in three volumes,
1885, 1887, 1908.
2 An account of Clebsch's life and works is printed in the Mathematische
Annalen, 1873, vol. vi, pp. 197-202, and 1874, vol. vii, pp. 1-55.
CH.XIX] ANALYTICAL GEOMETRY 481
myself by merely mentioning the names of the following
mathematicians.
James ^oo^A i (1806-1878) and James MacCullagh'^ (1809-
1846), both of Dublin, were two of the earliest British writers
in this century to take up the subject of analytical geometry,
but they worked mainly on lines already studied by others.
Fresh developments were introduced by Julius Plucker'^ (1801-
1868) of Bonn, who devoted himself especially to the study of
algebraic curves, of a geometry in which the line is the element
in space, and to the theory of congruences and complexes ; his
equations connecting the singularities of curves are well known; in
1847 he exchanged his chair for one of physics, and subsequently
gave up most of his time to researches on spectra and magnetism.
The majority of the memoirs on analytical geometry by
A. Cayley and by Henry Smith deal with the theory of curves
and surfaces ; the most remarkable of those of L. 0. Hesse
(1811-1874) of Munich are on the plane geometry of curves;
of those of J. G. Darboux of Paris are on the geometry of
surfaces; of those of G. H. Halphen (1844-1889) of Paris are
on the singularities of surfaces and on tortuous curves ; and of
those of P. 0. Bonnet are on ruled surfaces, curvature, and
torsion. The singularities of curves and surfaces have also been
considered by H. G. Zeuthen of Copenhagen, and by H. C. H.
Schubert^ of Hamburg. The theory of tortuous curves has
been discussed by M. N other of Erlangen ; and R.F.A. Clebsch ^
of Gottingen has applied Abel's theorem to geometry.
Among more recent text-books on analytical geometry are
J. G. Darboux's Theorie generate des surfaces^ and Les Systemes
orthogonaim et les coordonnees curvilignes ; P. F. A. Clebsch's
Vorlesu/ngen iiber Geometrie, edited by F. Lindemann ; and
^ See Booth's Treatise on some neiv Geo^metrical Methods, London, 1873.
2 See MacCullagh's collected works edited by Jellett and Haughton,
Dublin, 1880.
2 Pliicker's collected works in two volumes, edited by A. Schoenflies and
F. Pockels, were published at Leipzig, 1875, 1896.
* Schubert's lectures were published at Leipzig, 1879.
^ Clebsch's lectures have been published by F. Lindemann, two volumes,
Leipzig, 1875, 1891.
2 I
482 NINETEENTH CENTURY MATHEMATICS [ch. xix
G. Salmon's Conic Sections, Geometry of Three Dimensions, and
Higher Plane Curves; in whicli tjie chief discoveries of these
writers are embodied.
Pliicker suggested in 1846 that the straight line should be
taken as the element of space. This formed the subject of investi-
gations by G. Battaglini (1826-1892) of Rome, F. C. Klein, and
S. Lie} Recent works on it are R. Sturm's Die Gebilde ersten unci
zweiten Grades der Liniengeometrie, 3 volumes, Leipzig, 1892,
1893, 1896, and C. M. Jessop's Treatise on the Line Complex,
Cambridge, 1903.
Finally, I may allude to the extension of the subject-matter
of analytical geometry in the writings of A. Cayley in 1844,
H. G. Grassmann in 1844 and 1862, G, F. B. Biemann in
1854, whose work was continued by G. Veronese of Padua,
H. C, H. Schubert of Hamburg, C. Segre of Turin, G. Castel-
nuovo of Rome, and others, by the introduction of the idea of
space of n dimensions.
Analysis. Among those who have extended the range of
analysis (including the calculus and differential equations) or
whom it is difficult to place in any of the preceding categories
•are the following, whom I mention in alphabetical order.
P. E. Appell 2 of Paris ; J. L. F. Bertrand of Paris ; G, Boole
of Cork ; A. L. Gauchy of Paris ; J. G. Darboux ^ of Paris ;
A. B. Forsyth of Cambridge ; F. G. Frohenius of Berlin ;
J. Lazarus Fuchs (1833-1902) of Berlin; G. H, Halphen of
Paris ; C. G. J. Jacobi of Berlin ; C. Jordan of Paris ; L. Konigs-
berger of Heidelberg; Sophie Koivalevski^ (1850-1891) of
Stockholm ; M. S. Lie of Leipzig ; E. Picard ^ of Paris ; //.
Poincare^ of Paris; G. F. B. Biemann of Gottingen ; H. A.
Schwarz of Berlin ; J. J. Sylvester ; and K. Weierstrass of Berlin,
who developed the calculus of variations.
The subject of differential equations should perhaps have been
^ On the history of this subject see G. Loria, II passato ed il presente delle
principali tewie geometricJie, Turin, 1st ed. 1887 ; 2n(i ed. 1896.
2 Biographies of Appell, Darboux, Picard, and Poincare, with biblio-
graphies, by E. Lebou, were issued in Paris in 1909, 1910.
^ See the Bulletin des sciences vmthematiques, vol. xv, pp. 212-220.
CH. xix] STEINER 483
separated and treated by itself. But it is so vast that it is
difficult — indeed impossible — to describe recent researches in a
single paragraph. It will perhaps suffice to refer to the admirable
series of treatises, seven volumes, on the subject by A. R.
Forsyth, which give a full presentation of the subjects treated.
A recent development on integral equations, or the inversion
of a definite integral, has attracted considerable attention. It
originated in a single instance given by Abel, and has been
treated by Y. Yolterra of Rome, J. Fredholm of Stockholm, D.
Hilbert of Gottingen, and numerous other recent writers.
Synthetic Geometry. The writers I have mentioned above
mostly concerned themselves with analysis. I will next describe
some of the more important works produced in this century on
synthetic geometry.^
Modern synthetic geometry may be said to have had its
origin in the works of Monge in 1800, Carnot in 1803, and
Poncelet in 1822, but these only foreshadowed the great ex-
tension it was to receive in Germany, of whicTi Steiner and von
Staudt are perhaps the best known exponents.
Steiner.2 Jacob SteiTner^ "the greatest geometrician since
the time of ApoUonius," was born at Utzensdorf on March 18,
1796, and died at Bern on April 1, 1863. His father was a
peasant, and the boy had no opportunity to learn reading and
writing till the age of fourteen. He subsequently went to
Heidelberg and thence to Berlin, supporting himself by giving
lessons. His Systematische Entwickelungen was published in
1832, and at once made his reputation : it contains a full dis-
cussion of the principle of duality, and of the projective and
homographic relations of rows, pencils, &c., based on metrical
^ Tlie Ap&r^u historique sur Vorigine et U devdopj)evient des methodes en
geometrie, by M. Chasles, Paris, second edition, 1875 ; and Die synthetisclie.
Geometrie im Alterthum und in der Neuzeit, by Th. Reye, Strassburg, 1886,
contain interesting summaries of the history of geometry, but Chasles's work
is written from an exclusively French point of view.
2 Steiner's collected works, edited by Weierstrass, were issued in two
volumes, Berlin, 1881-82. A sketch of his life is contained in the Erin-
neming an Steiner, by C. F. Geiser, Schatfhausen, 1874.
484 NINETEENTH CENTURY MATHEMATICS [ch. xix
properties. By the influence of CreUe, Jacobi, and the von
Humboldts, who were impressed by the power of this work,
a chair of geometry was created for Steiner at Berlin, and
he continued to occupy it till his death. The most important
of his other researches are contained in papers which appeared
in Grelle^s Journal : these relate chiefly to properties of algebraic
curves and surfaces, pedals and roulettes, and maxima and
minima : the discussion is purely geometrical. Steiner's works
may be considered as the classical authority on recent synthetic
geometry.
Von Staudt. A system of pure geometry, quite distinct
from that expounded by Steiner, was proposed by Karl Gem-g
Christian von Staudt, born at Rothenburg on Jan. 24, 1798,
and died in 1867, who held the chair of mathematics at
Erlangen. In his Geonietrie der Lage, published in 1847, he
constructed a system of geometry built up without any reference
to number or magnitude, but, in spite of its abstract form, he
succeeded by means of it alone in establishing the non-metrical
projective properties of figures, discussed imaginary points, lines,
and planes, and even obtained a geometrical definition of a
number : these views were further elaborated in his Beitrdge zur
Geometrie der Lage, 1856-1860. This geometry is curious and
brilliant, and has been used by Culmann as the basis of his
graphical statics.
As usual text-books on synthetic geometry I may mention
M. Chasles's Traite de geometrie superieure, 1852; J. Steiner's
Vorlesungen iiher synthetische Geometrie, 1867 ; L. Cremona's
Mements de geometrie projective, English translation by
C. Leudesdorf, Oxford, second edition, 1893; and Th. Reye's
Geometrie der Lage, Hanover, 1866-1868, English translation
by T. F. Holgate, New York, part i, 1898. A good presenta-
tion of the modern treatment of pure geometry is contained in
the Introduzione ad una teoria geometrica delle curve piane,
1862, and its continuation Preliminari di una teoria geometrica
delle superjicie by Luigi Cremona (1830-1903) : his collected
works, in three volumes, may be also consulted.
CH.xix] NON-EUCLIDEAN GEOMETRY 485
The diflFerences in ideas and methods formerly observed in
analytic and synthetic geometries tend to disappear with their
further development.
Non- Euclidean Geonieti^. Here I may fitly add a few words
on recent investigations on the foundations of geometry.
The question of the truth of the assumptions usually
made in our geometry had been considered by J. Saccheri
as long ago as 1733 ; and in more recent times had been
discussed by N. I. Lobatschewsky (1793-1856) of Kasan,
in 1826 and again in 1840; by Gauss, perhaps as early as
1792, certainly in 1831 and in 1846; and by J. Bolyai (1802-
1860) in 1832 in the appendix to the first volume of his
father's Tentamen; but Riemann's memoir of 1854 attracted
general attention to the subject of non- Euclidean geometry,
and the theory has been since extended and simplified by various
writers, notably by A. Cayley of Cambridge, E. Beltrami ^
(1835-1900) of Pavia, by H. L. F. von Helmholtz (1821-1894)
of Berlin, by S. R Tannery (1843-1904) of Paris, by F. C.
Klein of Gottingen, and by A. N. Whitehead of Cambridge in
his Universal Algebra. The subject is so technical that I confine
myself to a bare sketch of the argument ^ from which the idea
is derived.
The Euclidean system of geometry, with which alone most
people are acquainted, rests on a number of independent
axioms and postulates. Those which are necessary for Euclid's
geometry have, within recent years, been investigated and
scheduled. They include not only those explicitly given by
him, but some others which he unconsciously used. If these are
1 Beltrami's collected works are (1908) in course of publication at Milan.
A list of Ms writings is given in the Annali di matemaUca, March 1900.
2 For references see my Mathematical Recreations and Essays, London,
sixth edition, 1914, chaps, xiii, xix. A historical siimmary of the treatment
of non-Euclidean geometry is given in Die Theorie der ParalleUinien by
F. Engeland P. Stackel, Leipzig, 1895, 1899 ; see also J. Frischaufs Elemente
del' absoluten Qeometrie, Leipzig, 1876 ; and a report by G. B. Halsted on
progress in the subject is printed in Science, N.S., vol. x, New York, 1899,
pp. 545-557.
486 NINETEENTH CENTURY MATHEMATICS [ch. xix
varied, or other axioms are assumed, we get a different series
of propositions, and any consistent body of such propositions
constitutes a system of geometry. Hence there is no limit to
the number of possible Non-Euclidean geometries that can be
constructed.
Among Euclid's axioms and postulates is one on parallel
lines, which is usually stated in the form that if a straight
line meets two straight lines, so as to make the sum of the two
interior angles on the same side of it taken together less than
two right angles, then these straight lines being continually
produced will at length meet upon that side on which
are the angles which are less than two right angles. Ex-
pressed in this form the axiom is far from obvious, and from
early times numerous attempts have been made to prove
it.^ All such attempts failed, and it is now known that the
axiom cannot be deduced from the other axioms assumed by
Euclid.
The earliest conception of a body of Non-Euclidean geometry
was due to the discovery, made independently by Saccheri,
Lobatschewsky, and John Bolyai, that a consistent system of
geometry of two dimensions can be produced on the assump-
tion that the axiom on parallels is not true, and that through
a point a number of straight (that is, geodetic) lines can be
drawn parallel to a given straight line. The resulting geometry
is called hyperbolic.
Riemann later distinguished between boundlessness of space
and its infinity, and showed that another consistent system of
geometry of two dimensions can be constructed in which all
straight lines are of a finite length, so that a particle moving
along a straight line will return to its original position. This
leads to a geometry of two dimensions, called elliptic geometry,
analogous to the hyperbolic geometry, but characterised by the
fact that through a point no straight line can be drawn which,
^ Some of the more interesting and plausible attempts have been collected
by T. P. Thompson in his Geometry vnthout Axioms, London, 1833, and later
by J. Richard in his Philosophie de mathematique, Paris, 1903.
CH. xix] NON-EUCLIDEAN GEOMETRY 487
if produced far enough, will not meet any other given straight
line. This can be compared with the geometry of figures drawn
on the surface of a sphere.
Thus according as no straight line, or only one straight line,
or a pencil of straight lines can be drawn through a point
parallel to a given straight line, we have three systems of
geometry of two dimensions known respectively as elliptic,
parabolic or homaloidal or Euclidean, and hyperbolic.
In the parabolic and hyperbolic systems straight lines are
infinitely long. In the elliptic they are finite. In the hyper-
bolic system there are no similar figures of unequal size ; the
area of a triangle can be deduced from the sum of its angles,
which is always less than two right angles ; and there is a finite
maximum to the area of a triangle. In the elliptic system all
straight lines are of the same finite length ; any two lines inter-
sect ; and the sum of the angles of a triangle is greater than
two right angles.
In spite of these and other peculiarities of hyperbolic and
elliptical geometries, it is impossible to prove by observation
that one of them is not true of the space in which we live.
For in measurements in each of these geometries we must
have a unit of distance ; and if we live in a space whose
properties are those of either of these geometries, and such
that the greatest distances with which we are acquainted
{ex. gr. the distances of the fixed stars) are immensely smaller
than any unit, natural to the system, then it may be impossible
for us by our observations to detect the discrepancies between the
three geometries. It might indeed be possible by observations
of the parallaxes of stars to prove that the parabolic system and
either the hyperbolic or elliptic system were false, but never
can it be proved by measurements that Euclidean geometry
is true. Similar difficulties might arise in connection with
excessively minute quantities. In short, though the results of
Euclidean geometry are more exact than present experiments
can verify for finite things, such as those with . which we have
to deal, yet for much larger things or much smaller things or
488 NINETEENTH CENTURY MATHEMATICS [ch. xix
for parts of space at present inaccessible to us they may not
be true.
Other systems of Non- Euclidean geometry might be con-
structed by changing other axioms and assumptions made by
Euclid. Some of these are interesting, but those mentioned
above have a special importance from the somewhat sensational
fact that they lead to no results inconsistent with the properties
of the space in which we live.
We might also approach the subject by remarking that in
order that a space of two dimensions should have the geometrical
properties with which we are familiar, it is necessary that it
should be possible at any place to construct a figure congruent
to a given figure; and this is so only if the product of the
principal radii of curvature at every point of the space or
surface be constant. This product is constant in the case (i)
of spherical surfaces, where it is positive ; (ii) of plane surfaces
(which lead to Euclidean geometry), where it is zero ; and (iii)
of pseudo-spherical surfaces, where it is negative. A tractroid
is an instance of a pseudo-spherical surface ; it is saddle-shaped
at every point. Hence on spheres, planes, and tractroids we
can construct normal systems of geometry. These systems are
respectively examples of hyperbolic, Euclidean, and elliptic
geometries. Moreover, if any surface be bent without dilation
or contraction, the measure of curvature remains unaltered. Thus
these three species of surfaces are types of three kinds on which
congruent figures can be constructed. For instance a plane can
be rolled into a cone, and the system of geometry on a conical
surface is similar to that on a plane.
In the preceding sketch of the foundations of Non-Euclidean
geometry I have assumed tacitly that the measure of a distance
remains the same everywhere.
The above refers only to hyper-space of two dimensions.
Naturally there arises the question whether there are different
kinds of hyper-space of three or more dimensions. Riemann
showed that there are three kinds of hyper -space of three
dimensions having properties analogous to the three kinds of
CH.xix] KINEMATICS. GRAPHICS 489
hyper -space of two dimensions already discussed. These are
differentiated by the test whether at every point no geodetical
surfaces, or one geodetical surface, or a fasciculus of, geodetical
surfaces can be drawn parallel to a given surface ; a geodetical
surface being defined as such that every geodetic line joining
two points on it lies wholly on the surface.
Foundations of Mathematics. Assumptions made in the
Subject. The discussion on the Non - Euclidean geometry
brought into prominence the logical foundations of the subject.
The questions of the principles of and underlying assumptions
made in mathematics have been discussed of late by J. W. R.
Dedekind of Brunswick, G. Cantor of Halle, G. Frege of Jena,
G. Peano of Turin, the Hon. B. A. W. Russell and A. N.
Whitehead, both of Cambridge.
KineTnatics. The theory of kinematics, that is, the investiga-
tion of the properties of motion, displacement, and deformation,
considered independently of force, mass, and other physical con-
ceptions, has been treated by various writers. It is a branch
of pure mathematics, and forms a fitting introduction to the
study of natural philosophy. Here I do no more than allude
to it.
I shall conclude the chapter with a few notes — more or less
discursive - — on branches of mathematics of a less abstract
character and concerned with problems that occur in nature.
I commence by mentioning the subject of MecJianics. The
subject may be treated graphically or analytically.
Graphics. In the science of graphics rules are laid down
for solving various problems by the aid of the drawing-board :
the modes of calculation which are permissible are considered
in modern projective geometry, and the subject is closely
connected with that of modern geometry. This method of
attacking questions has been hitherto applied chiefly to problems
in mechanics, elasticity, and electricity ; it is especially useful in
engineering, and in that subject an average draughtsman ought
490 NINETEENTH CENTURY MATHEMATICS [ch. xix
to be able to obtain approximate solutions of most of the
equations, differential or otherwise, with which he is likely to
be concerned, which will not involve errors greater than would
have to be allowed for in any case in consequence of our imper-
fect knowledge of the structure of the materials employed.
The theory may be said to have originated with Poncelet's
work, but I believe that it is only within the last twenty years
that systematic expositions of it have been published. Among
the best known of such works I may mention the Graphische
Statik, by C. Culmann, Zurich, 1875, recently edited by
W. Ritter ; the Lezioni di statica grafica, by A. Favaro, Padua,
1877 (French translation annotated by R Terrier in 2 volumes,
1879-85); the Calcolo grafico, by L. Cre^nona, Milan, 1879
(English translation by T. H. Beare, Oxford, 1889), which is
largely founded on Mobius's work ; La statique graphique, by
M. Levy, Paris, 4 volumes, 1886-88 ; and La statica grafica, by
G. Sairotti, Milan, 1888.
The general character of these books will be sufficiently
illustrated by the following note on the contents of Culmann's
work. Culmann commences with a description of the geo-
metrical representation of the four fundamental processes of
addition, subtraction, multiplication, and division ; and pro-
ceeds to evolution and involution, the latter being effected by
the use of equiangular spiral. He next shows how the quantities
considered — such as volumes, moments, and moments of inertia
— may be represented by straight lines ; thence deduces the
laws for combining forces, couples, &c. ; and then explains the
construction and use of the ellipse and ellipsoid of inertia,
the neutral axis, and the kern ; the remaining and larger part
of the book is devoted to showing how geometrical drawings,
made on these principles, give the solutions of many practical
problems connected with arches, bridges, frameworks, earth
pressure on walls and tunnels, &c.
The subject has been treated during the last twenty years
by numerous writers, especially in Italy and Germany, and
applied to a large number of problems. But as I stated at the
CH.xix] ANALYTICAL MECHANICS 491
beginning of thivS chapter that I should as far as possible avoid
discussion of the works of living authors I content myself
with a bare mention of the subject.^
Analytical Mechanics. I next turn to the question of
mechanics treated analytically. The knowledge of mathematical
mechanics of solids attained by the great mathematicians of the
last century may be said to be summed up in the admirable
3fecanique analytique by Lagrange and Traite de mecaniqne
by Poisson, and the application of the results to astronomy
forms the subject of Laplace's Mecanique celeste. These works
have been already described. The mechanics of fluids is
more difficult than that of solids and the theory is less
advanced.
Theoretical Statics, especially the theory of the potential
and attractions, has received considerable attention from the
mathematicians of this century.
I have previously mentioned that the introduction of the idea
of the potential is due to Lagrange, and it occurs in a memoir
of a date as early as 1773. The idea was at once grasped by
Laplace, who, in his memoir of 1784, used it freely and to
whom the credit of the invention was formerly, somewhat
unjustly, attributed. In the same memoir Laplace also ex-
tended the idea of zonal harmonic analysis which had been
^ In an English work, I may add here a brief note on Clifford, who was
one of the earliest British mathematicians of later times to advocate the use of
graphical and geometrical methods in preference to analysis. William
Kingdon Clifford, born at Exeter on May 4, 1845, and died at Madeira on
March 3, 1879, was educated at Trinity College, Cambridge, of which society
he was a fellow. In 1871 he was appointed professor of applied mathematics
at University College, Loudon, a post which he retained till his death. His
remarkable felicity of illustration and power of seizing analogies made him
one of the most brilliant expounders of mathematical principles. His health
failed in 1876, when the writer of this book undertook his work for a few
months ; Clifford then went to Algeria and returned at the end of the year,
but only to break down again in 1878. His most important works are his
Theory of Biquaternions, On the Classification of Loci (unfinished), and The
Theory of Graphs (unfinished). His Canonical Dissection of a Riemann's
Surface and the Elevients of Dynamic also contain much interesting matter.
For further details of Clifford's life and work see the authorities quoted in the
article on him in the Dictionary of National Biography, vol. xi.
492 NINETEENTH CENTURY MATHEMATICS [ch. xix
introduced by Legendre in 1783. Of Gauss's work on attractions
I have already spoken. The theory of level surfaces and lines
of force is largely due to Ckasles, who also determined the
attraction of an ellipsoid at any external point. I may also here
mention the Barycentrisches Calcul, published in 1826 by
A. F. Mohius^ (1790-1868), who was one of the best known of
Gauss's pupils. Attention must also be called to the important
memoir, published in 1828, on the potential and its properties,
by G. Green 2 (1793-1841) of Cambridge. Similar results were
independently established, in 1839, by Gauss, to whom their
general dissemination was due.
Theoretical Dynamics, which was cast into its modern form
by Jacobi, has been studied by most of the writers above
mentioned. I may also here repeat that the principle of
"Varying Action" was elaborated by Sir William Hamilton
in 1827, and the "Hamiltonian equations" were given in
1835; and I may further call attention to the dynamical
investigations of J. E. E. Bour (1832-1866), of Liouville, and
• of J. L. F. Bertrand, all of Paris. The use of generalised co-
ordinates, introduced by Lagrange, has now become the custo-
mary means of attacking dynamical (as well as many physical)
problems.
As usual text-books I may mention those on particle and
rigid dynamics by E. J. Routh, Cambridge; Legons sur
Vintegration des equations differentielles de la Tnecanique by
P. Painleve, Paris, 1895, Integration des equations de la
mecanique by J. Graindorge, Brussels, 1889 ; and C. E.
Appell's Traite de mecanique rationnelle, Paris, 2 vols., 1892,
^ Mobius's collected works were published at Leipzig in four volumes, 1885-87.
2 A collected edition of Green's works was published at Cambridge in
1871. Other papers of Green which deserve mention here are those in 1832
and 1833 on the equilibrium of fluids, on attractions in space of n dimensions,
and on the motion of a fluid agitated by the vibrations of a solid ellipsoid ;
and those in 1837 on the motion of waves in a canal, and on the reflexion and
refraction of sound and light. In the last of these, the geometrical laws of
sound and light are deduced by the principle of energy from the undulatory
theory, the phenomenon of total reflexion is explained physically, and certain
properties of the vibrating medium are deduced. Green also discussed the
propagation of light in any crystalline medium.
CH. xix] THEORETICAL ASTRONOMY 493
1896. Allusion to the treatise on Natural Philosophy by Sir
William Thomson (later known as Lord Kelvin) of Glasgow, and
P. G. Tait of Edinburgh, may be also here made.
On the mechanics of fluids, liquids, and gases, apart from
the physical theories on which they rest, I propose to say
nothing, except to refer to the memoirs of Green, Sir George
Stokes, Lord Kelvin, and von Helmholtz. The fascinating but
difficult theory of vortex rings is due to the two writers last
mentioned. One problem in it has been also considered by
J. J. Thomson, of Cambridge, but it is a subject which is as
yet beyond our powers of analysis. The subject of sound
may be treated in connection with hydrodynamics, but on
this I would refer the reader who wishes for further infor-
mation to the work first published at Cambridge in 1877 by
Lord Rayleigh.
Theoretical Astronomy is included in, or at any rate closely
connected with, theoretical dynamics. Among those who in this
century have devoted themselves to the study of theoretical
astronomy the name of Gauss is one of the most prominent ; to
his work T have already alluded.
Bessel.^ The best known of Gauss's contemporaries was
Friedrich Wilhelm Bessel^ who was born at Minden on
July 22, 1784, and died at Konigsberg on March 17, 1846.
Bessel commenced his life as a clerk on board ship, but in
1806 he became an assistant in the observatory at Lilienthal,
and was thence in 1810 promoted to be director of the new
Prussian Observatory at Konigsberg, where he continued to
live during the remainder of his life. Bessel introduced into
pure mathematics those functions which are now called by his
name (this was in 1824, though their use is indicated in a
memoir seven years earlier) ; but his most notable achievements
were the reduction (given in his Fundamenta Astronomia^^
^ See pp. 35-53 of A. M. Gierke's History of A stronoviy, Edinburgh, 1887.
Bessel's collected works and correspondence have been edited by R. Engelmann
and published in four volumes at Leipzig, 1875-82.
494 NINETEENTH CENTURY MATHEMATICS [ch. xix
Konigsberg, 1818) of the Greenwich observations by Bradley
of 3222 stars, and his determination of the' annual parallax
of 61 Cygni. Bradley's observations have been recently reduced
again by A. Auwers of Berlin.
Leverrier.i Among the astronomical events of this century
the discovery of the planet Neptune by Leverrier and Adams is
one of the most striking. Urbain Jean Joseph Leverrier, the
son of a petty Government employe in Normandy, was born at
St. L6 on March 11, 1811, and died at Paris on September 23,
1877. He was educated at the Polytechnic school, and in 1837
was appointed as lecturer on astronomy there. His earliest
researches in astronomy were communicated to the Academy in
1839 : in these he calculated, within much narrower limits
than Laplace had done, the extent within which the inclinations
and eccentricities of the planetary orbits vary. The independent
discovery in 1846 by Leverrier and Adams of the planet
Neptune by means of the disturbance it produced on the orbit
of Uranus attracted general attention to physical astronomy,
and strengthened the opinion as to the universality of gravity.
In 1855 Leverrier succeeded Arago as director of the Paris
observatory, and reorganised it in accordance with " the require-
ments of modern astronomy. Leverrier now set himself the
task of discussing the theoretical investigations of the planetary
motions and of revising all tables which involved them. He
lived just long enough to sign the last proof-sheet of this
work.
Adams. 2 The co-discoverer of Neptune was John Couch
Adams, who was born in Cornwall on June 5, 1819, educated
at St. John's College, Cambridge, subsequently appointed
Lowndean professor in the University, and director of the
Observatory, and who died at Cambridge on January 21, 1892.
1 For further details of his life see Bertrand's Uoge in vol. xli of the
Memoires de Vacademie ; and for an account of his worJc see Adams's
address in vol. xxxvi of the Monthly Notices of the Koyal Astronomical
Society.
2 Adams's collected papers, with a biography, were issued in two volumes,
Cambridge, 1896, 1900.
CH.xix] ADAMS 495
There are three important problems which are specially
associated with the name of Adams. The first of these is his
discovery of the planet Neptune from the perturbations it
produced on the o^bit of Uranus : in point of time this was
slightly earlier than Leverrier's investigation.
The second is his memoir of 1855 on the secular accelera-
tion of the moon's mean motion. Laplace had calculated this
on the hypothesis that it was caused by the eccentricity of
the earth's orbit, and had obtained a result which agreed sub-
stantially with the value deduced from a comparison of the
records of ancient and modern eclipses. Adams shewed that
certain terms in an expression had been neglected, and that
if they were taken into account the result was only about
one -half that found by Laplace. The results agreed with
those obtained later by Delaunay in France and Cayley in
England, but their correctness has been questioned by Plana,
Pontecoulant, and other continental astronomers. The point is
not yet definitely settled.
The third investigation connected with the name of Adams,
is his determination in 1867 of the orbit of the Leonids or
shooting stars which were especially conspicuous in November,
1866, and whose period is about thirty-three years. H. A.
Newton (1830-1896) of Yale, had shewn that there were only
five possible orbits. Adams calculated the disturbance which
would be produced by the planets on the motion of the node
of the orbit of a swarm of meteors in each of these cases, and
found that this disturbance agreed with observation for one of
the possible orbits, but for none of the others. Hence the orbit
was known.
Other well-known astronomers of this century are G. A. A.
Plana (1781-1864), whose work on the motion of the moon
was published in 1832; Count P. G. D. Pontecoulant {11 ^b-
1871); C. E. Delaunay (1816-1872), whose work on the lunar
theory indicates the best method yet suggested for the analytical
investigations of the whole problem, and whose (incomplete)
lunar tables are among the astronomical achievements of this
496 NINETEENTH CENTURY MATHEMATICS [ch. xix
century; P. A. Hansen'^ (1795-1874), head of the observatory
at Gotha, who compiled the lunar tables published in London
in 1857 which are still used in the preparation of the Nautical
Almanack, and elaborated the methods employed for the
determination of lunar and planetary perturbations ; F. F.
Tisserand (1845-1896) of Paris, whose Mecanique celeste is now
a standard authority on dynamical astronomy ; and Simon New-
comb (1835-1909), superintendent of the American Fphemeris,
who re-examined the Greenwich observations from the earliest
times, applied the results to the lunar theory, and revised
Hansen's tables.
Other notable work is associated with the names of Hill,
Darwin, and Poincare. G. W, Hill,'^ until recently on the
staff of the American Ephemeris^ determined the inequalities
of the moon's motion due to the non-spherical figure of the
earth — an investigation which completed Delaunay's lunar
theory.^ Hill also dealt with the secular motion of the moon's
perigee and the motion of a planet's perigee under certain
conditions ; and wrote on the 'analytical theory of the motion
of Jupiter and Saturn, with a view to the preparation of tables
of their positions at any given time. Sir G. H. Darwin (1845-
1912), of Cambridge, wrote on the effect of tides on viscous
spheroids, the development of planetary systems by means of
tidal friction, the mechanics of meteoric swarms, and the
possibility of pear-shaped planetary figures. H. Poincare (1854-
1912), of Paris, discussed the difficult problem of three bodies,
and the form assumed by a mass of fluid under its own attrac-
tion, and is the author of an admirable treatise, the Mecanique
celeste, three volumes. The treatise on the lunar theory by E. W.
Brown, Cambridge, 1896; his memoir on Inequalities in the
Motion of the Moon due to Planetary Action, Cambridge,
1908; and a report (printed in the Report of the British
} For an account of Hansen's numerous memoirs see the Transactions
of the Royal Society of London for 1876-77.
2 G. W. Hill's collected works have been issued in four volumes,
Washington, 1905.
3 On recent development of the lunar theory, see the Transactions of the
British Association, vol. Ixv, London, 1895, p. 614.
CH. xix] SPECTRUM ANALYSIS 497
Association^ London, 1899, vol. lxix, pp. 121-159) by E. T.
Wkittaker on researches connected with the solution of the
problem of three bodies, contain valuable accounts of recent
progress in the lunar and planetg-ry theories.
Within the last half century the results of spectrum analysis
have been applied to determine the constitution of the heavenly
bodies, and their directions of motions to and from the earth.
The early history of spectrum analysis will be always associated
with the names of G. R. Kirchhoff (1824-1887) of Berlin, of
A. J. Angstrom (1814-1874) of Upsala, and of George G. Stokes
of Cambridge, but it pertains to optics rather than to astronomy.
How unexpected was the application to astronomy is illustrated
by the fact that A. Comte in 1842, when discussing the study
of nature, regretted the waste of time due to some astronomers
paying attention to the fixed stars, since, he said, nothing
could possibly be learnt about them ; and indeed a century ago
it would have seemed incredible that we could investigate the
chemical constitution of worlds in distant space.
During the last few years the range of astronomy has
been still further extended by the art of photography. To
what new results this may lead it is as yet impossible to say.
In particular we have been thus enabled to trace the forms of
gigantic spiral nebulae which seem to be the early stages of vast
systems now in process of development.
The constitution of the universe, in which the solar system
is but an insignificant atom, has long attracted the attention of
thoughtful astronomers, and noticeably was studied by William
Herschel. Recently J. C. Kapteyn of Groningen has been able
to shew that all the stars whose proper motions can be detected
belong to one or other of two streams moving in difierent
directions, one with a velocity about three times as great as the
other. The solar system is in the slower stream. These results
have been confirmed by A. S. Eddington and F. W. Dyson.
It would appear likely that we are on the threshold of
wide-reaching discoveries about the constitution of the visible
universe.
2k
498 NINETEENTH CENTURY MATHEMATICS [ch. xix
Mathematical Physics, An account of the history of
mathematics and allied sciences in the last century would be
misleading if there were no reference to the application of
mathematics to numerous problems in heat, elasticity, light,
electricity, and other physical subjects. The history of mathe-
matical physics is, however, so extensive that I could not pretend
to do it justice, even were its consideration properly included in
a history of mathematics. At any rate I consider it outside the
limits I have laid down for myself in this chapter. I abandon
its discussion with regret because the Cambridge school has
played a prominent part in its development, as witness (to
mention only three or four of those concerned) the names
of Sir George G. Stokes, professor from 1849 to 1903, Lord
Kelvin, J. Clerk Maxwell (1831-1879), professor from 1871
to 1879, Lord Rayleigh, professor from 1879 to 1884,
Sir J. J. Thomson, professor from 1884, and Sir Joseph Larmor,
professor from 1 903.
499
INDEX,
Abacus, description of, 123-5
— ref. to, 3, 26, 57, 113, 127, 131,
138, 139, 183
Abd-al-gehl, 161-2
Abel, 461-62
— ref. to, 392, 424, 425, 438, 447,
452, 461, 463, 465, 469, 473
Abel's theorem, 462, 481
Abelian functions, 396, 424, 452,
461, 462, 465, 465-7, 468, 478,
480
Aberration (astronomical), 380
Abu Djefar ; see Alkarismi
Abul-Wafa ; see Albuzjani
Academy, Plato's, 42
— the French, 282, 315, 457-8
— the Berlin, 315, 356
Accademia dei Lincei, 315
Achilles and tortoise, paradox, 31
Action, least, 398, 403, 408
; — varying, 492
Adalbero of Rheims, 137
Adam, C, 268
Adams, J. C, 494-5. ref. to, 494
Addition, processes of, 188
— symbols for, 5, 104, 105, 106,
153, 172, 173, 194, 206-8, 211,
214, 215, 216, 217, 228, 240
Adelhard of Bath, 165
— ref. to, 177
Adheraar, R. d', 445
Africanus, Julius, 114
Agrippa, Cornelius, ref. to, 119
^Atog^, 3-8. ref. to, 73, 103
Airy, G. B., 442
Albategni, 161
Alberi on Galileo, 247
Albuzjani, 161
Alcuin, l«-4« — li^'^
Alembert, d' ; see D'Alembert
Alexander the Great, 46, 51
Alexandria, university of, 51, 92,
96, 113, 115
Alexandrian library, 51, 83, 115
— Schools, chapters iv, v
— symbols for numbers, 126-7
Alfarabius, ref. to, 166
Alfonso of Castile, 175
Alfonso's tables, 175
Alfred the Great, ref. to, 133
Algebra. Treated geometrically by
Euclid and his School, 57-60, 102.
Development of rhetorical and
syncopated algebra in the fourth
century after Christ, 102-10.
Discussed rhetorically by the
Hindoo and Arab mathemati-
cians, chapter ix ; by the early
Italian writers, chapter x ; and
Pacioli, 210. Introduction of
syncopated algebra by Bhaskara,
153, 154 ; Jordanus, 171-3
Regiomontanus, 202-5 ; Record
214; Stifel, 215-17; Cardan
223-5 ; Bombelli, 228 ; and Ste
vinus, 228. Introduction of sym
bolic algebra by Vieta, 230-34
Girard, 235 ; and Harriot, 238
Developed by (amongst others)
Descartes, 275-6 ; WaUis, 292-3
500
INDEX
Newton, 331-2; and Euler,
396-8. Eecent extensions of,
468-80
Algebra, definitions of, 183
— earliest problems in, 102
— earliest theorem in, 95-6
— higher, 468-80
— historical development, 102-3
— histories of, 50, 292
— origin of term, 156
Algebra, symbols in, 239-43
Algebraic equations ; see Simple
equations, Quadratic equations,
&c.
Algebrista, 170
Algorism, 158, 166, 174, 178, 183,
188, 219
Alhazen, 161-2. ref. to, 166
Alhossein, 160
Alkarismi, 155-8
— ref. to, 167, 183, 224
Alkarki, 159-60
Alkayami, 159
Al-Khwarizmi ; see Alkarismi
Allman, G. J., ref. to, 13, 14, 19,
24, 28, 29, 35, 41
Almagest, the, 96-8
— ref. to, 81, 86, 111, 146, 156,
158, 160, 162, 164, 165, 166, 171,
176, 177, 179, 180, 201, 227
Al Mamun, Caliph, ref. to, 145,
156
Almanacks, 178, 186-7
Al Mansur, Caliph, ref. to, 146
Alphonso of Castile, 168
Alphonso's tables, 169
Al Raschid, Caliph, ref. to, 145
Amasis of Egypt, ref. to, 16
America, discovery of, 200
Ampere, 436. ref. to, 451
Amthor, A., 72
Amyclas of Athens, 46
Analysis, Cambridge School, 438-43
— higher, 482
— in synthetic geometry, 43
Analytical geometry, origin of, 264,
272-5, 298 ; on development of,
see chapters xv-xix
Anaxagoras of Clazomenae, 34
Anaximander, 18
Anaximenes, 18
Anchor ring, 46, 86
Anderson on Vieta, 231
Angle, sexagesimal division, 4, 243
— trisection of, 34, 37, 85, 234,
316
Angstrom, 495
Angular coefficient, 312
Anharmonic or Cross ratios ; see
Geometry (modern synthetic)
Anthology, Palatine, 61, 102
Antioch, Greek School at, 145
Antipho, 38
Apian on Jordanus, 171
Apices, 125, 138
Apogee, sun's, 161
Apollonius, 77-83
— ref. to, 52, 89, 112, 146, 158,
161, 164, 171, 227, 230, 234,
274, 293, 311, 316, 350, 380,
483
Appell, P. E., 467, 468, 482
Appell, C. E., 492
Apse, motion of lunar, 374, 389
Arabic numerals, 117, 128, 147,
- 152, 155, 158, 166, 168, 169,
184-7
— origin of, 184, 185
Arabs, Mathematics of, chapter ix
— introduced into China, 9
— introduced into Europe, chap, x
Arago, 437-8
— ref. to, 91, 400, 433, 450, 494
Aratus, 46, 86
Arbogast, 401. ref. to, 400
Archimedean mirrors, 65
— screw, 65
Archimedes, 64-8
— ref. to, 52, 62, 79, 81, 82, 85,
86, 91, 101, 102, 112, 146, 158,
164, 171, 227, 244, 259, 288,
310, 311, 367, 387
Archippus, 28
Archytas, 28-30
— ref. to, 26, 36, 42, 44
Area of triangle, 89-90
Areas, conservation of, 256
Arendt, G., on Dirichlet, 454
Argand, J. R., 471
Argyrus, 118
Aristaeus, 48
— ref. to, 46, 57, 77, 78, 316
Aristarchus, 62-4. ref. to, 86, 227
Aristotle, 48-9
INDEX
501
Aristotle, ref. to, 13, 14, 25, 52,
133, 145, 227
Aristoxenus, 21
Arithmetic. Primitive, chapter vii
Pre-Hellenic, 2-5. Pythagorean,
24-8. Practical Greek, 58, 101,
112, 127, 128. Theory of, treated
geometrically by most of theGreek
mathematicians to the end of the
first Alexandrian School, 58 ; and
thenceforward treated empirically
(Boethian arithmetic) by most of
the Greek and European mathe-
maticians to the end of the fom--
teenth century after Christ, 95,
127-8, 182-3. Algoristic arith-
metic invented by the Hindoos,
152 ; adopted by the Arabs, 154,
158 ; and used since the four-
teenth century in Europe, 165,
168, 184-7 ; development of
European arithmetic, 1300-1637,
chapter xi
Arithmetic, higher ; see Numbers,
theory of
Arithmetical machine, 282, 354, 441
— problems, 61, 72, 73
— progressions, 27, 69, 151
— triangle, 219, 231, 284-5
' ApiO ixriTiKTi, signification of, 57
Aronhold, S. H., 479
Arts, Bachelor of, 142
— Master of, 142-3
Arya-Bhata, 147-8
— ref. to, 150, 152, 154, 161
Aryan invasion of India, 146
Arzachel, 165
Assumption, rule of false, 151, 170,
208, 209
Assumptions, 489
Assurance, life, 389
Astrology, 152, 179-80, 255
Astronomical Society, London, 441,
474
Astronomy. Descriptive astronomy
outside range of work, vi. Early
Greek theories of, 17, 18, 34, 46,
61, 62, 76, 83. Scientific astro-
nomy founded by Hipparchus,
86-7 ; and developed by Ptolemy
in the Almagest, 96-8. Studied
by Hindoos and Arabs, 147, 148,
150, 151, 160-61, 165. Modern
theory of, created by Copernicus,
213 ; Galileo, 249, 250 ; and
Kepler, 256-7. Physical astro-
nomy created by Newton, chap-
ter XVI. Developed by (amongst
others) Clairaut, 373-4 ; La-
grange, 405, 406-7 ; Laplace,
414-18 ; and in recent times by
Gauss and others, chapter xix
Asymptotes, theory of, 340
Athens, School of, chapter iii
— second School of, 111-13
Athos, Mount, 118
Atomic theory in chemistry, 431
Atomistic School, 31
Attains, 77
Attic symbols for numbers, 126-7
Attraction, theories of, 321-3, 330,
333-5, 373, 387, 406, 413, 422,
436, 439, 446, 453, 491, 492
Australia, map of, 254
Autolycus, 61
Auwers, A., 494
Avery's steam-engine, 91
Babbage, 441. ref. to, 439, 442
Babylonians, mathematics of, 5, 6
Bachelor of Arts, degree of, 142
Bachet, 305-6
— ref. to, 221, 297, 298
Bachmann, P., 460
Bacon, Francis, 252. ref. to, 298
Bacon, Roger, 174-7
— ref. to, 165, 167, 169
Baily, R. F., on Flamsteed, 338
Baize, R. L., 468
Baker, H. F., 468
— ref. to, 467, 475.
Baldi on Arab mathematics, 155
Ball, W. W. R., ref. to, 37, 118, 141,
214, 223, 236, 238, 253, 288, 295
305, 306, 319, 336, 339, 440, 485
Barlaam, 117-18
Barnes, E. W., 462
Barometer, invention of, 282-3, 308
Barral on Arago, 437
Barrow, 309-12
— ref. to, 52, 92, 237, 241, 275,
299, 321, 323, 324, 328, 341, 342,
347, 362, 394 ,
Bastien on D'Alembert, 374
502
INDEX
Battaglini, G., 482
Beare, T. H., on graphics, 490
Beaune, De, ref. to, 276
Bede on finger symbolism, 113
Beeckman, I., ref. to, 269-70
Beldomandi, 180
Beltrami, E., 485
Beman, W. W., 471
Benedictine monasteries, 131, 135
Ben Ezra, 166. ref. to, 168
Berkeley, G., 386, 428
Berlet on Riese, 215
Berlin Academy, 315, 356
Bernelinus, 139
Bernhardy on Eratosthenes, 83
Bernoulli, Daniel, 377-8
Bernoulli, Daniel, ref. to, 368, 393
Bernoulli, James, 366-7
— ref. to, 243, 316, 365
Bernoulli, James II., 369
Bernoulli, John, 367-8
— ref. to, 224, 243, 350, 359, 363,
365, 368, 369, 379, 391, 393, 394,
396
Bernoulli, John 11. , 368
Bernoulli, John III., 369
Bernoulli, Nicholas, 368
— ref. to, 341, 367, 393
Bernoulli's numbers, 367
Bernoullis, the younger, 368-9
Bertrand, 281, 374, 479, 482, 492,
494
Berulle, Cardinal, ref. to, 270
Bessel, 493-4
Bessel's functions, 493
Beta function, 396, 424
Betti, E., 450, 479
Bevis and Hutton on Simpson, 388
B^zout, 401
Bhaskara, 150-54
— ref. to, 147, 154, 162
Bija Ganita, 150, 153-4
Binomial equations, 405, 411, 452
Binomial theorem, 217, 219, 327-8,
341, 397, 462
Biot, 437. ref. to, 352, 417
Biquadratic equation, 159, 223,
226, 233
Biquadratic reciprocity, 424
Biquadratic residues, 453
Bjerknes on Abel, 461
Bobyuiu on Ahmes, 3. ref. to, 391
Bbckh on Babylonian measures, 2
Bode's law, 416, 448
Boethian arithmetic ; see Arith-
metic
Boethius, 132-3
— ref. to, 95, 114, 135, 136, 138,
142, 175, 182
Boetius ; see Boethius
Bois-Raymond, P. du, 460
Bologna, university of, 139, 140, 180
Bolyai, J., 485, 486
Bombelli, 228, 313
— ref. to, 224, 226, 232, 242
Bonacci ; see Leonardo of Pisa
Boncompagni, ref. to, 9, 155, 156,
166, 167, 206
Bonnet, P. 0., 481
Book-keeping, 187, 209, 245
Boole, G., 474. ref. to, 473, 474,
482
Booth, J., 481
Borchardt, 479. ref. to, 454, 462
Borel, E., 469, 479
Borrel, J., 226
Boscovich, 100
Bossut on Clairaut, 374
Bougainville, De, 370
Bouguer, P., 242
Bouquet, Briot and, 467
Bour, J. E. E., 492
Boyle, 314, 315, 378
Brachistochrone, the, 350, 363,
368, 370, 396
Brackets, introduction of, 235, 242
Bradley, 380. ref. to, 494
Bradwardine, 177-8
Brahmagupta, 148-50
— ref. to, 147, 151, 152, 154, 155,
161, 188, 204, 312
Branker, 316
Braunmiihl, A. von, 446
Braunmiihl, V. von, 391
Breitschwert on Kepler, 254
Bretschneider, ref. to, 13, 33, 41, 57
Brewer on Roger Bacon, 174
Brewster, D., ref. to, 319, 339
Briggs, 236-7. ref. to, 196, 197,
198
Brioschi, F., 465, 479
Briot and Bouquet, 467
British Association, 441
Brocard, H., on Monge, 426
INDEX
503
Brouncker, Lord, 312-13
— ref. to, 149, 314
Brown, E. W., 496
s Brunnel on Gamma function, 396
Bryso, 30, 36
■ Bubnov on Gerbert, 137
Budan, 433
Buffon on Archimedes, 65
Bull problem, ^he, 72-3
Biirgi, J., 196, 197
Burnell on numerals, 184
Burnet on Newton, 349
Burnside, W., 475, 479
Byzantine School, chapter vi
Cajori, F., 391
Calculating machine, 282, 354, 441
Calculation ; see Arithmetic
Calculus, infinitesimal, 265, 342-7,
356-63, 366, 369-73, 380, 386,
395-6, 410
Calculus of operations, 381, 401
— of variations, 396, 402, 403,
464, 482
Calendars, 17, 83, 178, 186-7, 205
Cambridge, university of, 179,
439-43, 498
Campanus, 177. ref. to, 177, 179
Campbell, 332
Cantor, G., 460, 474, 489
Cantor, M., ref. to, vii, 3, 6, 7, 9,
13, 14, 19, 26, 28, 33, 38, 50, 52,
64, 88, 104, 113, 121, 131, 134,
' 144, 167, 171, 184, 199, 201,
208, 215, 254, 313, 353, 356,
360, 368, 371, 387, 391
Capet, Hugh, ref. to, 137
Capillarity, 380, 381, 419, 435
Carcavi, 298
Cardan, 221-5
— ref. to, 60, 212, 218, 225, 226,
227
Careil on Descartes, 269
— ref. to, 276
Carnot, Lazare, 428
— ref. to, 88, 392, 426, 483
Carnot, Sadi, 433
Cartes, Des ; see Descartes
Cartesian vortices, 277, 278, 323,
335, 337
Cassiodorus, 133. ref. to, 114
Castelnuovo, G., 482
Castillon on Pappus's problem,
100
Catacaustics, 317
Cataldi, 236, 313
Catenary, 363-4, 366, 382
Cathedral Schools, the, 134-9
Cauchy, 469-71
— ref. to, 342, 410, 429, 467, 471,
475, 479, 482
Caustics are rectifiable, 317
Cavalieri, 278-81
— ref. to, 235, 237, 256, 268, 289,
299, 344, 347
Cavendish, H., 429-30
Cayley, 475-6
— ref. to, 460, 465, 481, 482, 485,
495
Censo di censo, 211
Census, 203, 211, 217, 232
Centres of mass, 73, 74, 100, 101,
253, 278, 292, 299
Centrifugal force, 302
Ceres, the planet, 448
Cesare, E., 479
Ceulen, van, 236
Chaldean mathematics, 2, 8
Cham bo rd, Comte de, ref. to, 470
Champollion, ref to, 431
Chancellor of a university, 140
Chardin, Sir John, ref. to, 189
Charles the Great, 134, 135
Charles I. of England, ref. to, 288
Charles II. of England, ref. to, 310
Charles V. of France, ref. to, 178
Charles VI. of France, ref. to, 178
Charles, E., on Roger Bacon, 174
Chasles, M., ref. to, 60, 82, 254,
257, 483, 484, 492
Chaucer, ref. to, 183
Chinese, early mathematics, 8-10
Chios, School of, 30
Christians (Eastern Church) op-
posed to Greek science, 111, 112,
115
Chuquet, 205-6. ref. to, 242
Cicero, ref. to, QQ
Ciphers ; see Numerals
Ciphers, discoveries of, 230, 288
Circle, quadrature of (or squaring
the), 24, 29, 34, 37 ; also see w
Circular harmonics, 422
504
INDEX
Cissoid, 85
Clairaut 373-4
— ref. to, 341, 387, 389, 390, 392
Clausius, R. J. E., 451
Clavius, 234
Clebsch, R. F. A., 480, 481
Clement, ref. to, 134
Clement IV. of Rome, ref. to, 134,
176
Clerk Maxwell ; see Maxwell
Gierke, A. M., 493
Clifford, W. K., 491
Clocks, 248, 302, 303
Cocker's arithmetic, 389
Coefficient, angular, 312
Colebrooke, ref. to, 148, 150, 154
Colla, 218, 226
Collins, J., 315-16
— ref. to, 323-4, 328, 342, 349,
354, 358
Collision of bodies, 292, 302, 314
Colours, theory of, 321, 324, 325
Colson on Newton's fluxions, 343,
344, 345, 346, 348
Comets, 374
Commandino, 227. ref. to, 62
Commensurables, Euclid on, 59
Commercium Epistolieum, 359
Complex numbers, 224, 453, 472,
481
Complex variables, 224
Comte, A., ref. to, 497
Conchoid, 85
Condorcet, 377. ref. to, 374
Cone, sections of, 46
— surface of, 70, 150
— volume of, 44, 70, 150
Congruences, 452, 456, 481
Conic Sections (Geometrical). Dis-
cussed by most of the Greek geo-
metricians after Menaechmus,
46 ; especially by Euclid, 60 ; and
Apollonius, 77-80 ; interest in,
revived by writings of Kepler,
256 ; and Desargues, 257 ; and
subsequently by Pascal, 284 ; and
Maclaurin, 385. Treatment of,
by modern synthetic geometry,
425-9, 482-5
Conies (Analytical). Invention of,
by Descartes, 272-6, and by For-
mat, 298 ; treated by Wallis, 289,
and Euler, 395 ; recent exten-
sions of, 482
Conicoids, 69, 70, 71, 395, 406
Conon of Alexandria, 64, 69
Conservation of energy, 378, 403,
428, 451
Constantino VII., the Emperor,
117
Constantinople, fall of, 120
Constitution of the universe, 497
Conti, 358, 360
Continued fractions, 236, 313, 411,
419
Continuity, principle of, 256, 331,
362, 429
Contravariants, 477
Conventual Schools, 134-9
Convergency, 313, 342, 364, 370,
386, 387, 394, 453, 469, 470, 479
Co-ordinates, 272-3, 363
— generalized, 404, 407, 451, 492
Copernicus, 213
— ref. to, 88, 97, 201, 228, 250
Cordova, School of, 140, 164, 165
Cornelius Agrippa, ref. to, 119
Corpuscular theory of light, 326
Cosa, 211
Cosecant, 243
Cosine, 161, 196, 197, 201, 239,
243
Cos X, series for, 314
Cos^^oj, series for, 314
Cossic art, 211
Cotangent, 89, 161, 196, 197, 243
Cotangents, table of, 161
Cotes, 382-3
— ref. to, 195, 348, 385, 394,
442
Courier on Laplace, 421
Cousin on Descartes, 269
Cramer, G., 371-2
Crelle, ref. to, 483
Cremona, L., 426, 484, 490
Ctesibus, 88
Cuba, 211
Cube, duplication of, 29, 37, 41-2,
44, 46-7, 81, 83-4, 85, 89, 234
— origin of problem, 41
Cubic curves, Newton on, 339-41
Cubic equations, 70, 106, 158-9,
218, 219, 224-5, 229, 232-3
Cubic reciprocity, 424, 455
INDEX
505
Culmann on graphics, 484, 490
Curtze, M., ref. to, 171, 178
Curvature, lines of, 426
Curvature of surfaces, 453
Curve of quickest descent, 350, 363,
368, 370, 396
Curves, areas of; see Quadrature
Curves, classification of, 274, 340,
395
Curves of the third degree, 340-1
Curves, rectification of, 291-2, 313-
14, 316, 328, 341-2, 345
Curves, tortuous, 373, 396, 481
Cusa, Cardinal de, 205
Cycloid, 283-4, 287, 291, 302
Cyzicenus of Athens, 46
Cyzicus, School of, chapter iii
D'Alembert, 374-7
- ref. to, 288, 367, 382, 392, 397,
403, 407
Dalton, J., 431
Damascius, 112
Damascus, Greek School at, 145
Darboux, 402,432, 477, 479, 481, 482
Darwin, G. H., 496
Dasypodius on Theodosius, 92
De Beaune, ref. to, 276
De Berulle, Cardinal, ref. to, 270
De Boucquoy, ref. to, 270
De Bougainville, 370
De Careil on Descartes, 269
Decimal fractions, 197-8, 245
Decimal measures, 197, 245, 409
Decimal numeration, 71-2, 81, 147,
152, 155, 158, 166, 169-70, 184-7
Decimal point, 197-8
De Condorcet, 377
Dedekind, J. W. R., ref. to, 454,
460, 464, 472, 489
Defective numbers, 26
De Fontenelle, ref. to, 366
Degree, length of, 83, 92, 161, 374,
437
Degrees, angular, 4, 85
De Gua, 371
De Kempten, ref. to, 122
De la Hire, 317. ref. to, 308
De Laloubere, 309
Delambre, 86, 87, 96, 97-8, 233,
402
Delaunay, 495. ref. to, 495, 496
De I'Hospital, 369-70. ref. tg,
380
Delian problem ; see Cube
De Halves, 371
De Mere, ref. to, 285
De Meziriac, 305-6
— ref. to, 221, 297, 298
Democritus, 31
Demoivre, 383-4. ref. to, 382, 400
De Montmort, 370-1
De Morgan, A., 474-5
— ref. to, 52, 61, 96, 97, 98, 110,
182, 206, 208, 395, 474
De Morgan, S. E., 474
Demptus for minus, 203-4, 211
Denifle, P. H., ref. to, 139
De Rohan, ref. to, 229
Desargues, 257-8
— ref. to, 255, 268, 269, 284, 317,
425
Descartes, 268-78
— ref. to, 84, 229, 231, 238, 241,
242, 252, 257, 258, 259, 264, 268,
287, 289, 291, 293, 297, 298, 320,
321, 323, 331, 365, 367, 371
— rule-of signs of, 276, 331, 371
Descartes, vortices of; see Cartesian
vortices
De Sluze, 316
— ref. to, 307, 311, 312
Desmaze on Ramus, 227
Destouches, ref. to, 374
Determinants, 365, 401, 406, 419,
452, 455, 464, 471, 480
Devanagari numerals, 184, 185
Devonshire, Earl of, ref. to, 383
Didion and Dupin on Poncelet, 428
Difference between, sign for, 232
Differences, finite, 370, 381, 407,
412, 419
— mixed, 419
Differential calculus ; see Calculus
Differential coefficient, 343
Differential equations, 372, 375-7,
396, 401, 406, 425, 464, 473,
476, 478, 480, 482
Differential triangle, the, 311
Differentials, 329, 410
Diffraction, 304, 317, 431, 436-7
Digby, 295
Dini, U., 479
Dinocrates, 51
506
INDEX
Dinostratus of Cyzicus, 46
Diodes, 85-6. ref. to, 92
Dionysius of Tarentum, 28
Dionysodorus," 92
Diophantiis, 103-10
— ref. to, 26, 71, 84, 117, 146, 147,
150, 202, 226, 228, 294, 297, 298,
306, 412 ^V,
Directrix in conies, 79, 100
Dirichlet, Lejeune, 454-5
— ref. to, 296, 432, 455, 459, 462,
464
Distance of sun, 62
Disturbing forces, 335, 405, 495,
496
Ditton, H., 380
Division, processes of, 191-4, 237
— symbols for, 153, 160, 241
Dodecahedron, discovery of, 20
Dodson on life assurance, 389
Don Quixote, 170
Dositheus, 64, 67, 69, 71
Double entry, book-keeping by,
187, 209, 245
Double tlieta functions ; see Elliptic
functions
Dreydorff on Pascal, 281
Dreyer on Tycho Brahe, 256
Duillier, 359
Dupin, ref. to, 428
Duplication of cube ; see Cube
Dupuis on Theon, 95
D'Urban on Aristarchus, 62
Durer, 213. ref. to, 120
Dynamics ; see Mechanics
Dyson, F. W., 497
e, symbol for 2-71828..., 394, 478
Eanbald, Archbishop, ref. to, 134
Earth, density of, 430
— dimensions of, 83, 92, 373, 437
Eccentric angle, 256
Eclipse foretold by Thales, 17
EcHptic, obliquity of, 83, 87
Eddington, A. S., 497
Edessa, Greek School at, 145
Edward VI. of England, ref. to, 214
Egbert, Archbishop, ref. to, 134
Egyptian mathematics, chap, i
Eisenlohr, ref. to, 3, 6, 7
Eisenstein, 455-6
— ref. to. 456, 457, 459, 464
Elastic string, tension of, 315
Elastica, 366
Eleatic School, 30-31
Electricity, 435, 449-51
Elements of Euclid ; see Euclid
Elimination, theory of, 401, 405
Elizabeth of England, ref. to, 238
Ellipse, area of, 69
— rectification of, 372
Elliptic functions, 396, 424, 452,
456, 458, 461-7, 468, 471, 475-6,
479
Elliptic geometry, 486, 487, 488
Elliptic orbits of planets, 165, 256,
330 333
Ellis, G., on Rumford, 430
Ellis, R. L., on Fr. Bacon, 252
Ely on Bernoulli's numbers, 367
Emesa, Greek School at, 145
Emission theory of light, 326
Energy, conservation of, 378, 403,
407-8, 428, 451
Enestrom, ref. to, 276
Engel, F., on Grassmann, 473
— ref. to, 446, 485
Engelmann on Bessel, 493
Enneper, A., ref. to, 461
Envelopes, 302, 317, 363
Epicharmus, 28
Epicurus, 31
Epicycles, 87, 97
Epicycloids, 317, 371
Equality, symbols for, 5, 105, 195,
211, 214, 232, 241
— origin of symbol, 214
= , meanings of, 214, 232, 241
Equations ; see Simple equations,
Quadratic equations, &c.
Equations, difterential, 372, 375-7,
396, 401, 406, 425, 464, 473, 476,
478, 480, 482
— indeterminate, 106, 107, 147,
149, 318, 405
— integral, 483
— number of roots, 448, 469
— position of roots, 224, 317, 331-2.
371, 411, 433 _
— roots of imaginary, 223
— roots of negative, 223
— theory of, 234, 330-32, 394, 410,
468, 475, 477, 479
Equiangular spiral, 367, 490
INDEX
607
Erastothenes, 83-4
— ref. to, 42, 86, 86, 92
Errors, theory of, 383, 389, 405,
418, 422, 439, 448
Ersch and Gruber on Descartes,
269
Essex, ref. to, 287
Ether, luminiferous, 304, 326, 451
Euclid, 52-62
— ref. to, 42, 66, 76, 77, 91, 101,
146, 158, 161, 164, 171, 274, 310;
see also below
Euclid's Elements, 53-60
— ref. to. 111, 112, 114, 133, 146,
158, 161, 164, 165, 166, 169, l7l,
175, 177, 178, 179, 180, 226, 227,
282, 310, 320, 321, 423, 485, 486,
487, 488
Euc. post. 12, Ptolemy's proof of, 98
Euc. I, 5-. ref. to, 15, 175
— I, 12. ref. to, 30
— I, 13. ref. to, 22
— I, 15. ref. to, 15
— I, 23. ref. to, 30
— I, 26. ref. to, 15
— I, 29. ref. to, 22
— I, 32. ref. to, 16, 17, 22,
282
— I, 44. ref. to, 24
— I, 45. ref. to, 24
— I, 47. ref. to, 7, 10, 22,
23-4, 26, 39, 149
— I, 48. ref. to, 7, 22, 26
— II, 2. ref. to, 24
— II, 3. ref. to, 104
— II, 5. ref. to, 58
— II, 6. ref. to, 58
— II, 8. ref. to, 104
— II, 11. ref. to, 44, 58
— II, 14. ref. to, 24, 58
— Ill, 18. ref. to, 29
— Ill, 31. ref. to, 16, 39
— Ill, 35. ref. to, 29
— V. ref. to, 44
— VI, 2. ref. to, 15
— VI, 4. ref. to, 15, 24
— VI, 17. ref. to, 24
— VI, 25. ref. to, 24
— VI, 28. ref. to, 58, 102
— VI, 29. ref. to, 58, 102
— VI, D. ref. to, 88
— IX, 36. ref. to, 397
Euc. X. ref. to, 48, 81
— X, 1. ref. to, 45
— X, 9. ref. to, 48
— X, 117. ref. to, 59
— XI, 19. ref. to, 29
— XII, 2. ref. to, 39, 45
— XII, 7. ref. to, 45
— XII, 10. ref. to, 45
— XIII, 1-5. ref. to, 45, 57
— XIII, 6-12. ref. to, 57
— XIII, 13-18. ref. to, 57
— x:iv. ref, to, 85
— XV. ref. to, 112
Eudemus, 13, 16, 19, 43, 77, 78
Eudoxus, 44-6
— ref. to, 36, 42, 54, 58, 86
Euler, 393-400
— ref. to, 100, 195, 224, 239, 242, 243,
276, 294, 295, 301, 313, 332, 339,
361, 368, 378, 387, 392, 402, 403,
407, 408, 425, 426, 436, 459, 470
Eurytas of Metapontmn, 42
Eutocius, 112. ref. to, 78, 128
Evection, 87
Evolutes, 302 ^^^^
Excentrics, 87, 97 . "^
Excessive numbers, 26
Exchequer, Court of, 183
Exhaustions, method of, 45, 82,
278
Expansion of binomial, 327, 342,
397
— of cos {A±B), 227
— of cos X, 314
— of cos~^ », 314
— ofe^, 364
— oif{x + h), 381
— of /(«), 386
— of log {1-Vx), 308, 309, 364
— of sin(^±^), 227
— of sin a;, 314, 327, 364
— of sin-i X, 314, 327
— of tan""i ic, 314, 364
— of vers X, 364
Expansion in series, 341-2, 364,
370-1, 381, 386-7, 394, 453, 461,
462, 469, 474
Experiments, necessity of, 21, 76,'
176, 251, 252, 431
Exponential calculus, 368
Exponents, 154, 178, 228, 232-3,
238, 242, 245, 276, 289, 341, 394
508
INDEX
Faber Stapulensis on Jordanus, 171
Fabricius on Archytas, 28
Facility, law of, 423
Fagnano, 372
Fahie, J. J., 247
False assumption, rule of, 151, 170,
208, 209
Faraday, ref. to, 438, 450
Faugere on Pascal, 281
Favaro, A., ref. to, 3, 278, 313,
488
Fermat, 293-301
— ref. to, 81, 149, 217, 268, 275,
282, 283, 285, 292, 302, 311,
312, 347, 351, 397, 403, 406, 412
Ferrari, 225-6. ref. to, 22 5, 233
Ferro, 218
Fibonacci ; see Leonardo of Pisa
Figurate numbers, 284
Finck, 243
Finger symbolism, 113, 118, 121,
125, 126
Finite differences, 381, 407, 412,
419, 474
Fink, K., 445
Fiore, 218, 222
Fire engine invented by Hero, 91
Five, things counted by,121-2, 122-3
Fizeau, ref. to, 438
Flamsteed, 338
— ref. to, 379
Florido, 218, 219, 222
Fluents, 321, 328, 337, 338, 343,
344-7, 380, 386
Fluxional calculus, 265, 343-8, 386
— controversy, 347, 348, 356-62
Fluxions. 321, 328, 337, 338, 343,
344-8, 380, 386
Focus of a conic, 79, 256
Fontana ; see Tartaglia
Fontenelle, de, ref. to, 366
Force, component of, in a given
direction, 246-7
Forces, parallelogram of, 48, 246,
370
— triangle of, 213, 245, 370
Forms in algebra, 478-80
— in theory of numbers, 452, 455-
60
Forsyth, A. R., 468, 477, 480, 482
Foucault, ref. to, 438
Fourier, 432-3
Fourier, ref. to, 392, 421, 429, 435
Fourier's theorem, 432, 455
Fractions, continued, 236, 313, 411,
419
— symbols for, 153, 160, 178, 241
— treatment of, 3, 4, 73, 197, 198
Francis I. of France, ref. to, 212
Frederick II. of Germany, 170-71
— ref. to, 169
Frederick the Great of Prussia,
ref. to, 375, 393, 404, 408
Fredholm, J., 483
Frege, G., 489
French Academy, 282, 315, 457-8
Frenicle, 308-9. ref. to, 298
Fresnel, 436-7. ref. to, 304, 438
Friedlein, G., ref. to, 81, 88, 104,
112, 121, 133
Frisch on Kepler, 254
Frischauf on absolute geometry, 485
Frisi on Cavalieri, 278
Frobenius, 467, 475, 479, 482
Fuchs, 482
Functions, notation for, 368
— theory of, 465, 466, 467-8, 475
Fuss, ref. to, 100, 393
Galande, the, 312
Gale on Archytas, 28
Galen, ref. to, 145
Galileo, 247-51
— ref. to, 76, 214, 244, 255, 259,
268, 269, 287, 316, 364
Galley system of division, 192-4
Galois, 475, 479
Gamma function, 396, 424, 453
Garth, ref. to, 188
Gassendi, ref. to, 201, 205
Gauss, 447-54
— ref. to, 224, 342, 352, 392, 418,
419, 423, 425, 438, 447, 454,
455, 456, 457, 458, 459, 461,
464, 465, 469, 471, 473, 485,
491, 492, 493
Geber ibn Aphla, 165
Geiser on Steiner, 483
Gelon of Syracuse, 71
Geminus, ref. to, 13
Generalized co-ordinates, 404, 407,
451, 492
Generating lines, 314
Geodesies, 368, 396, 422
INDEX
509
Geodesy, 254, 449
Geometrical progressions, 27, 59,
69, 72, 151
Geometry. Egyptian geometry,
5-8. Classical synthetic geo-
metry, discussed or used by
nearly all the mathematicians
considered in the first period,
chapters ii-v ; also by Newton
and his School, chapters xvi,
XVII. Arab and medieval geo-
metry, founded on Greek works,
chapters viii, ix, x. Geometry
of the renaissance ; characterized
by a free use of algebra and trigo-
nometry, chapters xii, xiii. Ana-
lytical geometry, 264, 272-4 ;
discussed or used by nearly all
the mathematicians considered
in the third period, chapters
xiv-xix. Modern synthetic geo-
metry, originated with Desargues,
257-8 ; continued by Pascal,
284 ; Maclaurin, 385 ; Monge,
Carnot, and Poncelet, 425-9 ;
recent development of, 483-5.
Non-Euclidean geometry, origin-
ated with Saccheri, Lobatschew-
sky, and John Bolyai, 486
Geometry, origin of, 5-6
— elliptic, 486, 487, 488
— hyperbolic, 486, 487, 488
— line, 482
George I. of England, ref. to, 356
Gerard, 166. ref. to, 165, 168
Gerbert (Sylvester II.), 137-9
Gerhardt, ref. to, 117, 353, 356,
357, 358, 445, 462
Germain, S., 296
Gesta Romanorum, 138
Ghetaldi on ApoUonius, 80
Gibson on origin of calculus, 356
Giesing on Leonardo, 167
Giordano on Pappus's problem, 100
Girard, 234-5. ref. to, 239, 242, 243
Glaisher, 334, 456, 458, 460, 467
Globes, 137
Gnomon or style, 18
Gnomons or odd numbers, 25
Gobar numerals, 138, 184, 185
Goldbach, 371, 395
Golden section, the, 44, 45, 57
Gonzaga, Cardinal, ref. to, 225
Gopel, A., 465
Gordan, P. A., 480
Gothals on Stevinus, 245
Goursat, E., on functions, 468
Gow, ref. to, 3, 6, 13, 50, 52, 77
Graindorge, J., ref/ to, 492
Grammar, students in, 142
Granada, School of, 164
Graphical methods, 58, 336, 489-91
Grassmann, 473-4. ref. to, 451, 482
Graves on Hamilton, 472
Gravesande, s', on Huygens, 301
Gravity, centres of, 73, 74, 100,
101, 253, 278, 292, 299
— law of, 314, 321-3, 330, 332-5,
373-4
— symbol for, 368
Gray on Newton's writings, 319
Greater than, symbol for, 238,
241-2
Greatest common measure, 59
Greek science, 21-2, 49
Green, 492, 493
Greenhill, A. G., on elliptic func-
tions, 467
Greenwood on Hero, 88
Gregory XIII. of Rome, 222
Gregory, David, 379. ref. to, 316
Gregory, James, 313-14
— ref. to, 325, 327, 364
Gresham, Sir Thos., ref. to, 237
Grosseteste, Bishop, ref. to, 175
Groups, theories of, 475, 477
Grube on Dirichlet, 454
Gua, de, 371
Guhrauer on Leibnitz, 353
Guldinus, 252-3. ref. to, 256, 279
Gunpowder, invention of, 176-7
Gunter, E., 196, 243
Giinther, S., 118, 131, 287, 313,
391, 400, 445
Hadamard, J. S., 459, 468
Hadley, ref. to, 325
Hagan, J. G., 393, 446
Haldane, E. S., on Descartes, 268
Halley, 379-80
— ref. to, 77, 80, 94, 314, 332, 333,
337, 339, 374, 383, 402
Halma, M., ref. to, 96, 111
Halphen, G. H., 467, 481, 482
510
INDEX
Halsted, G. B., on hyper-geometry,
485
Hamilton, Sir Wm., 472-3
— ref. to, 183, 408, 473, 474, 492
Hand used to denote five, 122,
126
Hank el, ref. to, 13, 19, 33, 60,
103, 113, 121, 144, 446, 474, 479,
480
Hanselmann, L., on Gauss, 447
Hansen, 496. ref. to, 496
Harkness, J., on functions, 468
Harmonic analysis, 413, 422, 491
Harmonic ratios ; see Geometry
(modern synthetic)
Harmonic series, 27, 432
Haroun Al Raschid. ref. to, 145
Harriot, 3SSe ^^9^ S .
— ref. to, 229, 241, 242, 276
Hastie on Kant, 416
Haughton on MacCullagh, 481
Hauksbee on capillarity, 419
Heap for unknown number, 5, 105,
121-2
Heat, theory of, 432, 483, 435,
498
Heath, D. D., on Bacon, 252
Heath, Sir T. L., 52, 64, 103
Hegel, ref. to, 448
Heiberg, ref. to, 31, 52, 64, 77, 79,
94, 96, 177
Helix, 309
Helmholtz, von, ref. to, 450, 485,
493
Henry IV. of France, ref. to, 229
Henry of Wales, ref. to, 253
Henry 0., ref. to, 101, 214, 239,
293 374
Henry, W. C, on Dalton, 431
Hensel, K., 465, 468
Heracleides, 78
Herigonus, 242
Hermite, 478
— ref. to, 446, 465, 467, 468, 479
Hermotimus of Athens, 46
Hero of Alexandria, 88-9
— ref. to, 102, 128, 150, 227
Hero of Constantinople, 117
Herodotus, ref. to, 3, 5
Herschel, Sir John, 442
— ref. to, 439
Herschel, Sir William, 442, 497
Hesse, 481
Hettner on Borchardt, 479
Heuraet, van, 291, 292
Hiero of Syracuse, 64, 65, 75
Hieroglyphics, Egyptian, 431
Hilbert, D., 478, 480, 483
Hill, G. W., 496
Hiller on Eratosthenes, 83
Hindoo mathematics, 146-55
Hipparchus, 86-8
— ref. to, 67, 84, 88, 89, 96, 98,
160, 161
Hippasus, 20, 28
Hippias, 34-5
Hippocrates of Chios, 37-42
— ref. to, 36, 54
Hippocrates of Cos, 36, 145
Hire, De la, 317. ref. to, 308
Historical methods, 264
Hobson, E. W., 468
Hoche on Nicomachus, 94
Hochheim on Alkarki, 159
Hodograph, 473
Hoecke, G. V., 195, 216
Hoefer, ref. to, 19
Holgate on Reye, 484
Holmboe on Abel, 461
Holy wood, 174. ref. to, 179
Homogeneity, Vieta on, 231,
232
Homology, 258
Honein ibn Ishak, 145
Hooke, 315
— ref. to, 304, 329, 332, 349,
436
Horsley on Newton, 319
Hospital, r, 369-70. ref. to, 380
Huber on Lambert, 400
Hudde, 308. ref. to, 307, 311
Hugens ; see Huygens
Hultsch, ref. to, 61, 88, 89, 99
Humboldt, 450, 483-4
Hutton, ref. to, 229, 388
Huygens, 301-5, 313
— ref. to, 265, 266, 268, 292, 307,
308, 309, 314, 319, 332, 354,
436
Huyghens ; see Huygens
Hydrodynamics. Developed by
Newton, 351-2 ; D'Alembert,
375 ; Maclaurin, 387 ; Euler,
398-9 ; and Laplace, 419
INDEX
511
Hydrostatics. Developed by Ar-
chimedes, 74-5 ; by Stevinus,
245-6 ; by Galileo, 248, 249 ; by
Pascal, 283 ; by Newton, 352 ;
and by Euler, 399
Hypatia, 111 ; ref. to, 112
Hyperbolic geometry, 486, 487, 488
Hyperbolic trigonometry, 400
Hyperboloid of one sheet, 314
Hyper -elliptic functions ; see El-
liptic functions
Hyper-geometric functions, 459
Hyper-geometric series, 453
Hyper-geometry, 485-9
Hypsicles, 85
lamblichus, 110-11. ref. to, 19,
28, 126
Imaginary numbers, 223-4, 228,
470, 471
Imaginary quantities, 470
Inconimensurables, 24, 30, 48, 59,
60
Indeterminate coefficients, 364, 365
Indeterminate forms, 370
Indian mathematics, chapter ix
Indian numerals, 117, 128, 147,
152, 154-5, 158, 166, 168, 169,
184-7
— origin of, 184-5
Indices, 153-4, 178, 228, 232-3, 238,
242, 245, 276, 289, 341, 394
Indivisible College, 314-15
Indivisibles, method of, 256, 278-
81, 307
Inductive arithmetic, 95, 127-8,
182-3
Inductive geometry, 7-8, 10
Infinite series, difficulties in con-
nection with, 31, 313, 342, 364,
370, 386, 394, 453, 462, 469, 474
Infinite series, quadrature of curves
in, 290, 313, 314, 327-8, 341-3
Infinitesimal calculus ; see Calculus
Infinitesimals, use of, 256, 410
Infinity, symbol for, 243
Instruments, mathemeitical, 28, 35,
43
Integral calculus ; see Calculus
Integral equations, 483
Interference, principle of, 304, 326,
431, 436
Interpolation, method of, 290-1,
327-8, 343, 381, 407, 412
Invariants, 475, 476, 477, 479, 480
Involutes, 302
Involution ; see Geometry (modern
synthetic)
Ionian School, the, 1, 14-19, 34
Irrational numbers, 24-5, 30, 48,
59-60
Ishak ibn Honein, 145
Isidorus of Athens, 112
Isidorus of Seville, 133-4. ref, to,
142
Isochronous curve, 363, 366
Isoperimetrical problem, 86, 366-7,
367, 389, 402
Ivory, 439
Jacobi, 462-4
— ref. to, 410, 424, 425, 438, 452,
453, 454, 455, 459, 461, 464, 465,
466, 475, 478, 482, 483, 492
Jacobians, 464
James I, of England, ref. to, 253
James II. of England, ref. to, 338
Jellett on MacCullagh, 481
Jerome on finger symbolism, 114
Jessop, C. M., 482
Jews, science of, 6, 166, 170
John of Palermo, 169
John Hispalensis, 166-7. ref. to,
168
Joly, C. J., on quaternions, 473
Jones, AVm., 380, 394
Jordan, C, 475, 477, 479, 482
Jordanus, 171-4
— ref. to, 167, 205, 208, 211, 216,
231, 240
Jourdain, P. E. B., 474
Julian calendar, 83, 205
Justinian, the Emperor, 112
Kastner, 448
Kant, ref. to, 414, 416
Kapteyn, J. C, 497
Kauffmann (or Mercator), 309, 328
Keill, 356
Kelvin, Lord, 419, 450, 493, 498
Kempten, de, 122
Kepler, 254-7
— ref. to 183, 237,250, 257, 258, 268,
278, 279, 299, 321, 322, 332, 347
512
INDEX
Kepler's laws, 250, 256-7, 278, 322,
332
Kern on Arya-Bhata, 147
Kerschensteiner on Gordan, 480
Kearoi, 114
Kinckhuysen, ref. to, 323, 342
Kinematics, 489
KirchhofF, 497
Klein, F. C, 446, 447, 467, 468,
475, 477, 478, 479, 480, 482,
485
Knoche on Proclus, 111
Koramercll, V., 391
Konigsberger, L., 461, 462, 465,
482
Korteweg, 451
Kowalevski, S., 482
Kremer on Arab science, 144
Kronecker, L., 454, 460
Krumbiegel, B., 72
Ktihn, 471
Kummer, 458-9
— ref. to, 296, 424, 453, 459, 472,
479
Kiinssberg on Eudoxus, 44
Lacour on elliptic functions, 467
Lacroix, 442
Lagrange, 401-12
—ref. to, 100, 266, 275, 295, 350,
352, 361, 368, 378, 387, 392,
396-7, 418, 425, 428, 429, 432,
434, 435, 436, 442, 447, 453, 454,
459, 491, 492
Laguerre, E. N., 468
Lahire, 317. ref. to, 308
Laloubere, 309
Lambert, 400-1. ref. to, 384
'Lame, 296, 478
Lampe, ref. to, 446
Landen, 396, 410
Laplace, 412-21
— ref. to, 266, 339, 352, 361, 378,
392, 411, 421, 422, 423, 425, 429,
434, 436, 439, 440, 442, 447, 454,
469, 472, 491, 494, 495
Laplace's coefficients, 413, 422
Larmor, Sir J., 498
Latitude, introduction of, 18, 88
Lavoisier, 420
Law, faculty of, 142
Lazzarini, V., 167
Least action, 398, 403, 408
Least common multiple, 59
Least squares, 418, 422, 423, 439, 448
Lebesgue, 296, 468
Lebon, E., 482
Legendre, 421-5
— ref. to, 296, 392, 408, 413, 418,
421, 425, 429, 434, 447, 452, 459,
461, 463, 465, 469, 491
Legendre's coefficients, 413, 422
Leibnitz, 353-65
— ref. to, 241, 256, 275, 316, 327,
b29, 343, 345, 346, 347, 348, 349,
350, 366, 367, 369, 370, 379
Leipzig, university of, 179, 180
Lejeune Dirichlet ; see Dirichlet
Lenses, construction of, 249, 277,
303, 311, 325
Leo VI. of Constantinople, 117
Leo X. of Rome, Stifel on, 215
Leodamas of Athens, 46
Leon of Athens, 46
Leonardo da Vinci, 212-13
— ref to, 245
Leonardo of Pisa, 167-70
— ref. to, 60, 209, 210-11
Leonids (shooting stars), 495
Le Paige, 207, 316
Leslie on arithmetic, 121, 185
Less than, symbol for, 238, 241-2
Letters in diagrams, 38
— to indicate magnitudes, 48,
153-4, 172, 216, 231, 232
Leucippus, 31
Leudesdorf on Cremona, 484
Lever, principle of, 61, 74
Leverrier, 494. ref to, 407
Levy on graphics, 490
Lexell on Pappus's problem, 100
L'Hospital, 369-70. ref. to, 380
Lhulier, 100
Libration of moon, 403, 436
Libri, ref. to, 199, 208, 211
Lie, 477-8
— ref. to, 461, 479, 482
Life assurance, 389
Light, physical theories of, 61,
277, 303-4, 326, 399, 431, 436-7,
492
— velocity of, 277, 317, 438, 451
Lilavati, the, 150-4
Limiting values, 370
INDEX
613
Limits, method of, 280, 281
Lindelof, E. L., 468
Lindemann, 37, 478, 481
Lines of curvature, 426
Lintearia, 366
Linus of Liege, 326
Liouville, J,, 460, 467, 492
Lippershey, 249
Lobatschewsky, 54, 485
Lockyer, Sir Norman, 416
Logarithms, 195-7, 216, 235-7, 279
London Mathematical Society, 474
Longitude, 88, 347-8, 380
Lorentz on Alcuin, 134
Loria, ref. to, 13, 14, 19, 33, 50,
88, 308, 391, 446, 482
Louis XIV. of France, ref. to, 302,
303, 354
Louis XYL of France, ref. to, 408
Lucas di Burgo ; see Pacioli
Lucian, ref. to, 26
Lunes, quadrature of, 39-41
Luther, ref. to, 215, 216
Lysis, 28
MacCullagh, 481
Macdonald on Napier, 235
Macfarlane, A., 473
Maclaurin, 384-8
— ref. to, 275, 332, 373, 374- 378,
391, 406
MacMahon. P. A., 460, 480
Magic squares, 118-19. 308, 317
Magnetism, 435-6. 438, 449-51, 481
Mairan, 380
MalveSj de, 371
Mamercus, 18
Mandryatus, 18
Mangoldt, H. C. F. von, 459
Manitius on Hipparchus, 86
Mansion on the calculus, 356
Maps, 238, 253-4
Marcellus, 66, 76
Marie, ref. to, 64, 278, 446
Marinus of Athens, 112
Mariotte, 378
Markoff on Tchebycheflf, 459
Marolois, 235
Marre on Chuquet, 206
Martin, ref. to, 88, 121
Mary of England, ref to, 214
Mascheroni, 56
Mass, centres of, 73, 74, 100-1,
253, 278, 292, 299
Master, degree of, 142
Mastlin, 255
Mathematici Veteres, the, 114
Mathews, G. B., on numbers, 460
Matter, constitution of, 267
Matthiessen, 50
Maupertuis, P.L.M., 398, 408
Maurice of Orange, ref. to, 245, 269
Maurolycus, 226
Maxima and minima, determination
of, 299, 304, 345, 362, 387, 484
Maximilian L of Germany, 202
Maxwell, J. C, 430, 450, 451, 498
Mayer, F. C, 394, 400
jVIayer, J. T., 399
Mechanics. Discussed by Archy-
tas, 28 ; Aristotle, 48 ; Archi-
medes, 73 ; and Pappus, 100-1.
Development of, by Stevinus and
Galileo, 245-9 ; and by Huygens,
302-3. Treated dynamically by
Newton, 334 ef seq. Subsequently
extended by (among others)
D'Alembert, Maclaurin, Euler,
Lagrange, Laplace, and Poisson,
chapters xvii, xviii. Recent
work on, 489-93
Medicine, Greek practitioners, 145
Medieval universities, 139-43
Melanchthon, ref. to, 201, 216
Melissus, 31
Menaechmian triads, 46-7
Menaechmus, 46-7
— ref. to, 36, 53, 77, 78
Menelaus, 94. ref. to, 380
Menge on Euclid, 52
Menou, General, ref. to, 432
Meray, H. C. K., 460, 467
Mercantile arithmetic, 155, 168-9,
182-94, 206, 209
M creator, G., 253
Mercator, N., 309, ref. to, 328
Mercator's projection, 253
Mere, de, ref. to, 285
Merriman, M., 446
Mersenne, 306-7
— ref. to, 269, 282, 398
Meteoric hypothesis, 415-16
Meton, 34
Metrodorus, 102
2l
5U
INDEX
Meziriac, 305-6
— ref. to, 221, 297, 298
Microscope, 249-50, 325
Mill's Logic, ref. to, 43
Milo of Tarentum, 20
Minkowski, H., 408, 457
Minos, King, ref. to, 42
Minus ; see Subtraction
— symbols for, 5, 104, 105, 106,
153, 194-5, 206-8, 211, 214, 215,
216, 217, 240
— origin of symbol, 206-8
Mitchell, J., 430
Mittag-Leffler, 461, 466, 468
Mobius, 492. ref. to, 490
Mohammed, ref. to, 115
Mohammed ibn Musa ; see Alka-
rismi
Moivre, de, 383-4. ref. to, 382, 400
Molk on elliptic functions, 467
Moments in theory of fluxions, 346
Monastic mathematics, 131-6
Monge, 426-8
— ref. to, 392, 470, 483
Montmort, de, 370-71
Montucla, 221
— ref. to, 253, 308, 314, 366, 367
Moon, secular acceleration of, 411-
12, 495
Moors, mathematics of, 164-9
Morgan, A. de ; see De Morgan
Morley, F., on functions, 468
Morley on Cardan, 221
Moschopulus, 118-20
— ref. to, 317
Motion, laws of, 249, 277
Mouton, 354
Muir, T., 446
Miiller ; see Regiomontanus
Miiller, F., 461
Mullinger, ref. to, 134, 139
Multiple points, 341, 371
Multiplication, processes of, 4, 105,
127-8, 128, 188-92
— symbols for, 241
Murdoch, 341
Murr on Regiomontanus, 201, 205
Music, in the quadrivium, 21, 114,
131-6
Musical progression, 27
Mutawakkil, Caliph, ref. to, 145
Mydorge, ref. to, 269, 282
Napier of Merchiston, 235-6
— ref. to, 194, 195, 196, 197, 198, 347
Napier, Mark, ref. to, 235
Napier's i2dSiJ:89^91
Naples, university of, 141, 170
Napoleon I., 354, 409, 417-18, 420,
427, 428, 432
Napoleon III., 437, 470
Naucrates, 78
Navier on Fourier, 433
Navigation, science of, 253, 254
Nebular hypothesis, 415-16
Negative sign, 5, 104, 105, 106,
153, 194-5, 206-8, 211, 214, 215,
216, 217, 240
— geometrical interpretation, 235
Neil, 291
Nelts, E., 391
Neocleides of Athens, 46
Neptune, the planet, 494, 494-5
Nesselmann, ref. to, 50, 59, 103
Netto, E., 475, 479
Neumann, C, 419, 450, 451
Neumann, F. E., 451
Newcomb, S., 496
Newton, H. A., of Yale, 495
Newton, Isaac, chapter xvi (see
table of contents)
— ref. to, 76, 82, 100, 195, 231,
233, 235, 237, 241, 243, 249, 256,
259, 266, 274, 275, 293, 303, 304,
305, 310, 314, 353, 356, 357, 358,
359, 360, 361, 362, 363, 364, 370,
371, 372, 373, 374, 375, 378, 379,
380, 381, 383, 384, 385, 388, 389,
392, 394, 401, 403, 417, 419-20,
432, 472, 477
Newton's Principia, 333-8, 348
— ref. to, 249, 266, 278, 293, 303,
333, 364, 370, 374, 375, 379-80,
382, 383, 389, 392, 403, 417, 419-
20, 472
Nicholas IV. of Rome, ref. to, 177
Nicholas, Paul, ref. to, 143
Nicholas Rhabdas of Smyrna, 118
Nicole, 371. ref. to, 341
Nicomachus, 94-5
— ref. to, 113, 114, 118, 133
Nicomedes, 85
Nicoteles of Alexandria, 64
Nieuwentyt, 362
Nines, casting out the, 160, 188
INDEX
515
Nizze, ref. to, 62, 92
Nonante for ninety, 122-3
Non-Euclidean geometry, 485-9
Nother, M., 464, 467, 481
Number, simple complex, 472
Numbers, defective, 26 -'
— excessive, 26
— figurate, 284
— irrational, 460 ^
— perfect, 26, 59, 306-7
— polygonal, 26, 104
— transcendent, 46J^^
Numbers, theory ' of. Treatment
I of, by Pythagoras, 24-7 ; by
Euclid, 5^-60 ; by Diophantus,
109-10 ; by Fermat, 294-8 ; by
Euler, 397-8 ; by Lagrange, 403,
406 ; by Legendre, 423-4 ; by
Gauss and other mathematicians
of recent times, 448, 452-3, 455-
460, 468, 469, 471, 475, 476
Numerals, symbols for, 121-8, 138,
152, 155, 168, 169, 182-7 ,
Numeration, systems of, 71-2, 81,
chapters vii, xi
Nutation, 380
Octante for eighty, 122
Oenopides of Chios, 30
Offa, ref. to, 134
Oldenburg, 327, 354, 358
Olleris on Gerbert, 136, 139
Omar, Caliph, ref. to, 115
Omega function, 458, 465
Operations, calculus of, 216, 381,
401
Oppert, ref. to, 6
Optics (geometrical). Discussed by
(among others) Euclid, 61 ; Pap-
I pus, 100; Alhazen, 162; Roger
\ Bacon, 176 ; Snell, 254 ; Descar-
tes, 277 ; Barrow, 311 ; Newton,
324-5 ; Gauss, 451 ; and Sir
William Hamilton, 472
— (physical), 61, 277, 303-5, 325-6,
399, 43], 436-7, 492
Orderic Vitalis, ref to, 138
Oresmus, 178. ref. to, 242
Orientation of Egyptian temples, 6
Orleans, university of, 141
Orrery, 46, 76, 253
Oscillation, centre of, 302, 381
Osculating circle, 363
Otho, 226
Oughtred, 238-9
— ref. to, 196, 241, 242, 243, 320, 394
Oxford, university of, 179, 180
Ozanam, 221
TT, value of, 6, 7, 67, 97, 148, 149-
150, 151, 234, 236, 290-91, 313
— incommensurability of, 37, 313,
400, 423
— introduction of symbol, 394-5
— transcendental, 478
Pachymeres, 118
Pacioli, 208-12
— ref to, 187, 188, 194, 212, '215
220, 240
Paciolus ; see Pacioli
Padua, university of, 141, 180, 186
Painleve, P., 446, 480, 492
Palatine Anthology, 61, 102
Pappus, 99-101
— ref. to, 52, 56, 60, 61, 74, 77, 78,
81, 84, 104, 252-3, 273-4, 279,
350
Parabola, evolute of, 302
— quadrature of, 67-9, 280-81,
289-90, 299
— rectification of, 291-2
Parallel lines, 98-9, 256, 423,
486-7
Parallelogram of forces, 48-9, 246,
370
Parent, 371
Paris, university of, 139, 140, 141,
179, 180
Parmenides, 31
Pascal, 281-8
— ref. to, 231, 257, 258, 268, 269,
300, 301, 305, 347, 351, 352,
385, 386, 425
Pavia, university of, 141
Peacock, 441
— ref. to, 121, 168, 182, 430, 439,
442
Peano, G., 460, 474, 489
Pedals, 385, 484
Peletier, 227
Pell, 316. ref. to, 241
Pemberton, ref. to, 323, 348
Pendulum, motion of, 248, 251,
301-2, 315, 434
516
INDEX
Pepin on Fr^nicle's problem, 309
Perfect numbers, 26, 59, 306-7, 397,
398
Perier on Pascal, 281
Perseus, 86
Perspective, 245, 257, 258, 382
Pesloiian, L. de, 461
Peter the Hermit, ref. to, 137
Petrarch, 118, 179
Petri on Cusa, 205
Pfaff, 425
Phalereus, 51
Pherecydes of Syros, 19
Philip II. of Spain, ref. to, 230
Philippus of Athens, 46
Philolaus, 20, 28
Philonides, 78
Philoponus, 41
Philosophy, treatment of, 271-2
Phoenician mathematics, 1-8
Physics, mathematical, 266-7, 497-
498 ; also see headings of
subjects
Piazzi of Palermo, 448
Picard, G. E., 468
Picard, E., 475, 478, 482
Picard, J., 330
Pihan on numerals, 184
Piola on Cavalieri, 278
Pisa, university of, 180
Pitiscus, 233. ref. to, 227
Plana, 495. ref. to, 495
Planetary motions, 46, 62, 81, 87,
97, 165, 213, 250, 256-7, 277-8,
364, 407, 414-17, 448, 449, 454,
494-7
— stability, 407, 414, 435-6
Planets, astrological, 119
Planudes, 117. ref. to, 187
Platina, ref. to, 138
Plato, 42-4
— ref. to, 20, 26, 28, 35, 57, 64
Pliny, ref. to, 92
Pliicker, 481, 482
Plus ; see Addition
— symbols for, 5, 104, 105, 153,
173, 194, 206-8, 211, 214, 215,
217, 240
— origin of symbol + , 206-8
Plutarch, ref. to, 16
Pockels on Pliicker, 481
Poggendorflf, J. C, 446
Poincare, H., 415, 466, 468, 472, 479,
482, 496
Poinsot, 435
Point, Pythagorean def. of, 22
Poisson, 433-6
— ref. to, 392, 411, 429, 447, 450,
491
Polar triangle, 235, 254
Polarization of light, 304, 437,
438
Poles and polars ; see Geometry
(modern synthetic)
Polygonal numbers, 26, 104
Polygons, regular, 452
Polyhedrons, regular, 20, 24, 57,
85, 112
— semi-regular, 71
Poncelet, 428-9
— ref. to, 100, 392, 426, 483, 490
Pontecoulant, 495. ref. to, 495
Porisms of Euclid, 60
— of Diophantus, 110
Port-Royal, society of, 283-4
Potential, the, 417-18, 413-14, 422,
436, 454, 491, 492
Poudra on Desargues, 257
Power, origin of terra, 38
Powers ; see Exponents
Prague, university of, 141, 179,
180
Predari on Cavalieri, 278
Pretender, the Young, ref. to, 384
Prime and ultimate ratios, 410
Primes, 59, 60, 306-7, 455
— distribution of, 423-4, 458-9
465, 476
Pringsheim, 469, 479
Printing, invention of, 199, 200
Probabilities, theory of, 285-7, 300,
302, 367, 383, 384, 389, 401, 403,
405, 418-19, 422, 439, 448, 474
Proclus, 112
— ref. to, 13, 15, 19, 21, 54
Product, symbols for, 241
Progressions, arithmetical, 27, 69
151
— geometrical, 27, 59, 69, 72, 151
— musical, 27
Projectiles, 219, 249
Proportion, symbols for, 239, 241
— treatment by Euclid, 58
Psellus, 117. ref. to, 226
INDEX
517
* Pseudo-spherical space, 488
r Ptolemies, dynasty of, 51, 92, 114
' Ptolemy, 96-9
— ref. to, 67, 81, 84, 86, 88, 146,
156, 158, 160, 161, 164, 165, 166,
171, 176, 177, 179, 180, 201, 227;
also see Almagest
Puiseux, V. A., 467
Pulley, theory of, 28, 74
Purbach, 205. ref. to, 201
Puzzles, 31, 61-2, 220-1, 305
Pyramid, surface of, 70, 150
— volume of, 45, 70, 150
Pythagoras, 19-28
— ref. to, 3, 60
(Pythagorean School, the, 19-30.
ref. to, 42, 53, 110
Quadratic equations, 58, 89, 102,
106, 148-9, 157-8, 210
Quadratic reciprocity, 423, 448
Quadratic residues, 423-4, 459
Quadratrix, 34, 35, 46
Quadrature of circle ; see Circle,
also see %
— cone, 70, 150
Quadrature of curves, 256, 290,
299, 308, 327-8, 341-3
— ellipse, 69
— lunes, 39-41
— parabolas, 67-9, 280-1, 289-90,
299
— sphere, 67, 70
Quadrics, 71, 395, 406
Quadrilateral, area of, 149
Quadrivium, 21, 114, 117, 133,
133-4, 136, 142, 179, 180
Quantics, 479
Quartic equation, 159, 223, 226,
232, 233
Quaternions, 453, 471, 472, 473
Quetelet, ref. to, 245, 307
Quintic equation, 462, 469, 473,
478
Quipus ; see Abacus
Quotient ; see Division
— symbols for, 153, 160, 241
Raabe on convergency, 479
Rahdologia, the, 191, 236
Radical, symbols for, 154, 206, 215,
242, 289
Rahn, 241
Rainbow, explanation of, 176, 277,
311, 324, 325
Raleigh, Sir Walter, ref. to, 237
Ramus, 227-8
Rashdall, ref. to, 139
Ratdolt on Campanus, 177
Ratio, symbols for, 239, 241
Rational numbers, Euclid on, 59
Rayleigh, Lord, 493, 498
Recent mathematics, chapter xix
Reciprocants, 477
Record, 214-15
— ref. to, 125, 185, 195, 241
Recreations, mathematical, 220-1,
305
Rectification of curves, 291-2, 313,
317, 328, 341, 342, 345
Recurring series, 384, 403
Reductio ad absurdum, 39
Reduction in geometry, 39
Reformation, the, 200
Refraction, 176, 254, 276-7, 304,
311, 325, 338-9, 380, 451, 472,
492
— atmospheric, 162
Regiomontanus, 201-5
— ref. to, 161, 211, 212, 228, 243
Regula ignavi, 188-9
Reiff, R., 446
Renaissance, the mathematics of,
chapters xii, xiii
Res used for unknown quantity,
157, 203, 211, 217
Residues, theory of, 423-4, 453,
455
Resistance, solid of least, 370
Reversion of series, 327, 329
Reye on modern geometry, 483,
484
Rhabdas, 118
Rheticus, 226. ref. to, 236, 243
Rhetorical algebra, 102-3, 105, 148,
167, 172-3, 203, 210
Rhind papyrus, the, 3-8
— ref. to, 10, 103
Rhonius, 316
Riccati, 372. ref. to, 378
Ricci, 248
Richard, J., 486
Riemann, 464-5
— ref. to, 54, 450, 451, 453, 459,
518
INDEX
461,. 465, 467, 468, 479, 482, 485,
486, 488
Kiese, 215
Rigaud, ref. to, 238, 316
Ritter on Culmann, 490
Roberval, 307. ref. to, 275, 282, 287
Rodet, ref. to, 3, 147
Rods, Napier's, 189-91, 236
Roemer, 317
Rohan, ref. to, 229
Rolle, 317-18
Roman mathematics, 113-15
— symbols for numbers, 126
Romanus of Lou vain, 227
— ref. to, 229-30
Rome, mathematics at, 113-15
Rome Congress, 446
Roots of equations, imaginary, 223-
24, 470
— negative, 223
— number of, 448, 470
— origin of term, 157
— position of, 276, 317, 331-2, 372,
411, 433
— symmetrical functions of, 331,
401, 470
Roots, square, cube, &c., 154, 206,
215, 242, 289-90
Rope-fasteners, Egyptian, 6
Rosen on Alkarismi, 156
Rosenhain, J. G., 465
Routh on mechanics, 492
Royal Institution of London, 430
Royal Society of London, 314-15
Rudolff, 215. ref. to, 217
Rudolph IL of Germany, ref. to, 255
Ruffini, 462
Rumford, Count, 430
Russell, B. A. W., 460, 489
Saccheri, 485
Saint-Mesme ; see L'Hospital
Saint- Vincent, 307-8
— ref. to, 301, 309
Sairotti on graphics, 490
Salerno, university of, 140
Salmon 480, 482
Sanderson's Logic, 320
Sardou on Cardan, 221
Saunderson of Cambridge, 330
Saurin, 371.
Savile, Sir Hen., 237
Scaliger, 234
Scharptf on Cusa, 205
Schering, ref. to, 464
Schlegel S. F. V., 474
Schlesinger, L., 465
Schneider on Roger Bacon, 174
Schoner on Jordanus, 171
Schbnflies A. , 460, 481
Schools of Charles, 134-9
Schooten, van, 307
— ref. to, 231, 233, 276, 321
Schottky, F. H., 465
Schroeder, 147
Schubert, H. C. H., 481, 482
Schure, E., ref. to, 19
Schwarz, H. A., 465, 467, 472, 482
Scores, things counted by, 122
Scratch system of division, 192-4
Screw, the Archimedean, 65
Secant, 161, 235, 243, 389, 394
Section, the golden, 44, 45, 57
Secular lunar acceleration, 495
Sedillot, ref. to, 9, 144, 161
Segre, C, ^82
Seidel, P. L., 479
Septante for seventy, 122
Serenus, 94. ref. to, 380
Series ; see Expansion
— reversion of, 327, 329
Serret, 402, 475, 479, 480
Servant, M. G., 469
Seville, School of, 164
Sexagesimal angles, 4, 243
Sexagesimal fractions, 97, 169
Sextant, invention of, 325
Sextic Equation, 479
Sforza, ref. to, 208
s'Gravesande on Huygens, 301
Shakespeare, ref. to, 183
Shanks, W., 478
Signs, rule of, 105-6
Simple equations, 106
Simplicius, ref. to, 41
Simpson, Thomas, 388-90
— ref. to, 391, 394
Simson, Robert, 53, ref. to, 80, 81
Sin X, series for, 314, 327, 364
Sin-la;, series for, 314, 327
Sine, 88. 94, 96, 147-8, 150, 161
201, 235, 239, 243, 389, 394
Sines, table of, 67
Sixtus lY. of Rome, ref. to, 202
INDEX
519
Slee on Alcuin, 134
Slide-rule, 196
Sloman on calculus, 356, 358
Slusius : see Sluze, de
Sluze, de, 316
— ref. to, 307, 311, 312 .
Smith, D. E., 446
Smith, Henry, 456-8
— ref. to, 459, 465, 481
Smith, H. J. S., 459
Smith, R. A., on Dalton, 431
Snell, 254. ref. to, 245, 277
Socrates, ref. to, 42
Solar system, 497
Solid of least resistance, 370
Solids ; see Polyhedrons
Sonin on Tchebycheff, 459
Sophists, the, 34
Sound, velocity of, 403, 411, 419-
20
Spanish mathematics, 164-9
Spedding on Francis Bacon, 252
Speidell on logarithms, 197
Sphere, surface and volume of, 66
Spheres, volumes of, 45
Spherical excess, 235
Spherical harmonics, 413, 422
Spherical space, 488-9
Spherical trigonometry, 161, 279
Spheroids, Archimedes on, 69, 70
Spinoza and Leibnitz, 355
Spiral of Archimedes, 69
Spiral, the equiangular, 367, 490
Sponius on Cardan, 221
Square root, symbols for, 154, 206,
215, 242, 289
Squares, table of, 2
Squaring the circle ; see Circle
Stackel, P., 446, 485
Stahl, H. B. L., 464, 467, 468
Staigmuller, ref. to, 208, 213
Stapulensis on Jordanus, 171
Stars, lists of, 88, 97, 254, 493-4
Statics ; see Mechanics
Staudt, von, 484. ref. to, 426,
483
Steam-engine, Hero's, 91
Stefan, 451
Steichen on Stevinus, 245
Steiner, 483-4
— ref. to, 426, 464, 483, 484
Steinschneider on Arzachel, 165
Stevinus, 244-7
— ref. to, 74, 197, 228, 232, 242, 382
Stewart, Matthew, 388
Stifel, 215-17
— ref. to, 194, 207, 226, 227, 228,
231, 232, 276
StifFelius ; see Stifel
Stirling, 341, 386
Stobaeus, ref. to, 53
Stokes, G. G., 479, 493, 497, 498
Stolz, 0., 459, 460
Strabo, ref. to, 2, 42
String, vibrating, theory of, 376-7,
378, 381-2, 403
Studium generale, 141
Studnicka, F. J., 478
Sturm, ref. to, 433, 482
Style or gnomon, 18
Subtangent, 299, 308, 311, 316
— constant, 329, 362
Subtraction, processes of, 188
— symbols for, 5, 1C4, 105, 153,
194-5, 206-8, 211, 214, 215, 216,
240
Suidas, ref. to, 18
Sun, distance and radius of, 34, 62
Sun-dials, 18
Supplemental triangle, 235, 254
Surds, symbols for, 154, 206, 215,
242, 289
Suter on Dionysodorus, 92, 144
Swan-pan ; see Abacus
Sylow and Lie on Abel, 461
Sylvester, 476-7
— ref. to, 332, 397, 459, 482
Sylvester IL, 136-9
Symbolic algebra, 103
Symbolic and mathematical logic,
474
Symbols, algebraical, 239-43
•— trigonometrical, 243
Symmetrical functions of roots of
an equation, 331, 401, 470
Syncopated algebra, 103, 104
Synthetic geometry ; see Geometry
Tabit ibn Korra, 158-9. ref. to, 145
Tait, 473, 493
Tangent (geometrical), 274-5, 307,
311-12
Tangent (trigonometrical), 161, 235,
243, 389, 394
520
INDEX
Tan-^ic, series for, 314, 364
Tanner, P., 268
Tannery, J., on elliptic functions,
467
Tannery, S. P., ref. to, 19, 24, 33,
50, 86, 88, 96, 109, 110, 118,
293, 485
Tartaglia, 217-21
— ref. to, 188, 192-3, 209, 222-3,
224, 226, 231, 240
Tartalea ; see Tartaglia
Tautochronous curve, 302
Taylor (Brook), 380-2
— ref. to, 378, 403
Taylor, C, on conies, 257
Taylor, Is., on numerals, 184, 185
Taylor, T., on Pythagoras, 28
Taylor's theorem, 381, 386, 410,
471
Tchebycheff, 459
Telescopes, 249, 301, 303, 305, 313,
325
Ten as radix ; see Decimal
Tension of elastic string, 315
Terquem on Ben Ezra, 166
Terrier on graphics, 490
ThalM, 14-17 ; ref. to, 3
Thasus of Athens, 46
Theaetetus, 48 ; ref. to, 46, 54, 57
Theano, ref. to, 19
Theodorus of Gyrene, 30. ref. to,
36, 42, 48
Theodosius, 91-2. ref. to, 311
Theonof Alexandria, 111
— ref. to, 55, 128
Theon of Smyrna, 95
Thermodynamics, 433
Thermometer, invention of, 249
Theta functions, 452, 458, 461,
463, 465
Theudius of Athens, 46
Thibaut, G., 147
Thompson, T. P., 486
Thomson, Sir Benjamin, 430
Thomson, Sir J. J., 451, 493, 498
Thomson, Sir William ; see Kelvin
Three bodies, problem of, 399, 405,
464, 496, 496-7
Thurston on Carnot, 433
Thymaridas, 95-6 ; ref. to, 102
Tichanek, F., 478
Tidal friction, 416, 496
Tides, theory of, 250, 378, 387, 417
Timaeus of Locri, 30, 42
Tisserand, 417, 496
Titius of Wittemberg, 416
Todhunter, ref. to, 422, 446
Tonstall, 185
Torricelli, 308
— ref. to, 251, 282, 291, 316
Tortuous curves, 373, 395-6, 481
Toschi, 372
Trajectories, 350, 368
Transversals, 94
Trembley, 401
Treutlein, ref. to, 171, 182, 206
Triangle, area of, 89, 91
— arithmetical, 219, 231, 284-5
Triangle of forces, 213, 245, 246, 370
Triangular numbers, 26
Trigonometrical functions, 88, 94,
96, 147, 148, 150, 161-2, 201-2,
234-5, 239, 243, 368, 389, 394, 462
Trigonometrical symbols, origin of,
243, 389, 394
Trigonometry. Ideas of, in Rhind
papyrus, 7-8. Created by Hip-
parchus, 88 ; and by Ptolemy,
96. Considered a part of as-
tronomy, and treated as such by
the Greeks and Arabs, 161.
Hindoo works on, 147-8, 150,
154. Treated by most of the
mathematicians of the renais-
sance, chapters xii, xiii. De-
velopment of, by John Bernoulli,
368 ; Demoivre, 383 - 4 ; Euler,
394 ; and Lambert, 400
Trigonometry, addition formulae,
88, 227, 462
Trigonometry, higher ; see Elliptic
functions
Trisection of angle, 34, 37, 85, 234,
316
Trivium, the, 114, 133, 136, 141-2
Tschirnhausen, 317. ref. to, 357-8
Tschotii ; see Abacus
Tycho Brahe, 195, 255, 256
Tylor, E. B., ref. to, 121
Ubaldi, 382
Ujein, 150
Undulatory theory (optics), 303-4,
399, 431, 436
INDEX
521
Universe, constitution of the, 497
Universities, medieval, 139-41
— curriculum at, 141-3, 177-81
Universities of renaissance, 200
Unknown quantity, word or symbol
for, 5, 105, 121, 153-4, 157, 203,
211, 216, 217, 228, 231, 232,
276
Urban, d', on Aristarchus, 62
Valson, ref. to, 436, 469
Van Ceulen, 236
Vandermonde, 397, 419
Yan Heuraet, 291, 292
Vanishing points, 382
Van Schooten, 307
— ref. to, 231, 233, 276, 321
Variations, calculus of, 396, 402,
403, 435, 464, 467, 482
Varignon, 370. ref. to, 246
Velaria, 366
Venturi on Leonardo da Vinci, 212
Veronese, G., 482
Vers X, series for, 364
Verulam, Lord, 252. ref. to, 298
Vibrating string, 376-7, 378, 381-2,
403
Vienna, university of, 141, 179
Vieta, 229-34
— ref. to, 80, 195, 217, 226, 228,
229, 236, 238, 240, 242, 307, 321
Viga Ganita, 150-1, 153-4
Vince, ref. to, 346
Vinci, Leonardo da, 212-13
— ref. to, 245
Vinculum, introduction of, 242
Virtual work, 378, 403, 406-7, 428
Vis mortua, 364
Vis viva, 364
Vitalis, ref. to, 138
Vitruvius, ref. to, 74-5
Vivanti, G., 391
Viviani, 316
Vlacq, 197
Vogt, 147
Voltaire on Newton, 337
Volterra, V., ref. to, 446, 483
Von Breitschwert on Kepler, 254
, Von Helmholtz, 450, 485, 493
Von Humboldt, 450, 483-4
Von Murr, ref. to, 201, 205
Von Staudt, 484. ref. to, 426, 483
Vortices, Cartesian, 277, 323, 335,
337
Waddington on Ramus, 227
Wagner, 206
Wallis, 288-93
— ref. to, 62, 149, 238, 242, 268,
281, 295, 299, .302, 309, 313,
314, 316, 319, 321, 324, 327,
337, 338, 342, 347
Wallner, 0. R., 391
Walterhausen, S. von, 447
Wappler on Rudolff, 215
Watches, invention of, 303, 315
Watt, ref. to, 91
Wave theory (optics), 303-4, 399,
431, 436
Weber, H. , 464, 467
Weber, W. E., 449, 450, 451
Weierstrass, K., 466
— ref. to, 410, 460, 462, 468, 472,
478, 482, 483
Weissenborn, ref. to, 131, 136
Werner, ref. to, 134, 137
Wessel, 471
Weyr, ref. to, 3, 6
Whewell, W., 442
Whiston, 330. ref. to, 323, 347
Whitehead, A. N., 485, 489
Whittaker, E. T., 496
Widman, 206. ref. to, 194, 240
Wilkinson on Bhaskara, 150
William of Malmesbury, ref. to, 138
Williamson on Euclid, 52
Wilson on Cavendish, 429
Wilson's theorem, 406
Wingate, E., 237
Wirtinger, W., 464, 468
Witt, 198
Woepcke, ref. to, 61, 144, 159,
169, 170, 184
Wolf, 254
Woodcroft on Hero, 88
Woodhouse, 440-1. ref. to, 439
Woodward, R. S., 446
Work, virtual, 378, 403, 406-7, 428
Wren, 291, 314
~ ref. to, 291, 292, 302, 314, 332
Wright, 253-4
Xenophanes, 30
Xylander, 226
— ref. to, 110, 117, 207, 217, 241
522
INDEX
Year, duration of, 17, 83, 87
Young, Thos., 430-31
— ref. to, 304, 422, 429, 436
Young, Sir Wm., on Taylor, 380
Zangmeister, ref. to, 208
Zeno, 31
Zenodorus, 86
Zensus, 203, 211, 217 232
Zermelo, E., 460
Zero, symbol for, 184-5
Zeta function, 467
Zeuthen, 50, 64, 77, 78, 481
Zeuxippus, 64
Ziegler on Regiomontanus, 201
Zonal harmonics, 422
THE END
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