MacTutor
Henri Lebesgue: the Scientist, the Professor, the Man
The paper Henri Lebesgue: le Savant, le Professeur, l'Homme was written by Arnaud Denjoy, Lucienne Felix and Paul Montel and appeared in L'Enseignement Mathématique (2) 3 (1) (1957), 1-18. We give below an English version based on this paper.
Henri Lebesgue: the Scientist, the Professor, the Man.
The scientist
Today, in all the universities of the world, the measurement of sets in any space, with the fundamental conditions that it must fulfil and the arbitrariness that it admits is studied. Then, this measurement being acquired, the integral of a point function, element of this space, these two linked doctrines are taught to the student and learned by him without offering either the master or the disciple the appearance of the least difficulty. Now these clarities, in their complementary births, sprang from the brain of Lebesgue. Borel had solved the problem of the Euclidean measurement of linear sets, that is to say carried by a straight line. He had introduced, with other principles, the most important of all, namely the complete additivity of measure: when a set is formed by the union of a countable infinity (that is to say, not more abundant than that of the natural numbers) of disjoint measurable sets, this total set is measurable and its measure is the sum of the series formed by the measures of the constituents. The Borel metric, moreover, immediately extensible to spaces of any nature or to non-Euclidean evaluations in Cartesian spaces, this magnificent conception offered the possibility of giving the answer to an enigma vainly posed by Cantor, Jordan, Minkowski and many others. But, without the application of linear measure to the integration of functions of a variable, analysis had little help to expect from Borel's discovery.
Until the doctoral thesis in which Lebesgue, in 1902, published his definition of the integral, only Riemann's method was available to perform this operation on functions defined in an interval. Only a few types of discontinuous functions, mainly functions with bounded variation, were integrable in this way. The primitives of certain bounded derivatives of a very simple nature could not be obtained by this calculation. Fourier's formulas, giving the coefficients of an everywhere convergent trigonometric series whose sum is a known function, these formulas consisting of integrals whose coefficient is proportional to this sum-function, also had no meaning, even in fairly elementary cases where this sum is bounded.
This definition of the integral, published by Riemann, clarified by Darboux, is of profound interest. It is a very beautiful logical construction. But it offered to analysis an instrument of very little power. General theorems on the integration of convergent sequences were only demonstrated under very restrictive conditions, excessively limiting their field of application. The fundamental problems, we have given two examples above, whose progress was stopped by the inadequacy of the integration tool, were numerous. Their solution was only obtained in particular cases, for continuous data or data affected by very simple discontinuities. Like water retained by a mountain scree obstructing a pass, the questions of general statement, remaining unresolved, accumulated in front of this barrier of the deficient integral. And just as, under the weight of the liquid, the obstacle having given way, the lake formed upstream empties promptly, similarly when the Lebesgue integral was applied to the problems hitherto resolved for continuous arguments alone, and by the Riemann integral, the legitimacy of the results previously acquired was manifested each time that the new integral replaced the old one. And the field of validity of the results increased disproportionately in a host of subjects.
Many images have been found to grasp the difference separating the two conceptions. For example, to calculate the average height of a river above a low water mark, over the period of a year (not a leap year), when the daily height, measured in centimetres, is noted. With Riemann, to any fraction of a year constituted by a particular day, that is 1/365, we will apply a factor, constituting the weight attributed to this fraction, and this weight will be the height of the waters on that day. We will add the three hundred and sixty-five products. With Lebesgue, we will collate the various distinct heights observed between the lowest and the highest. To each of these heights we will apply a weight factor which will be the fraction of a year (number of days divided by 365) where the level was registered at this height. We add the results.
In both ways we obtained the average height sought, but that of Lebesgue offers an immense possibility of extension. He divides into parcels the field described by the various values of the function, this being a number. He poses the principle that if the function is between and on a set of measure , on this same set the function will contribute to the integral a quantity between and . This statement, understood by Lebesgue for functions of a real variable and the Euclidean measure of linear sets, can be adopted for functions defined at points of any space, and for any metric on sets in this space.
* * *
Lebesgue introduced the species of measurable functions. The progress was immense. This is because the passage to the limit, exact limit, greatest limit, smallest limit, of a sequence of measurable functions still gives a measurable function. Previously, the passage to the limit destroyed the nature of the variable function. A convergent sequence of continuous functions is not usually a continuous function. It is a class 1 function, according to Baire. The limit of a convergent sequence of class 1 functions is not normally a class 1 function, etc. But the limit of a convergent sequence of measurable functions is measurable. From then on, all the functions encountered in the problems of analysis are measurable. How many statements were once cluttered with hypotheses to define cases where the term-by-term integration of a convergent series of a given nature was legitimate. Now, if the functions are bounded, their Lebesgue integrals tend towards the integral of the limit function. The generality of these results caused a revolution in the methods of analysis. Until Lebesgue, it was limited to the study of continuous functions. From now on, its field will be formed by summable functions, that is to say, functions whose absolute value is integrable in the sense of Lebesgue.
* * *
But he discovered a new phenomenon, that of "almost everywhere", that is to say, properties that are true everywhere, except on a set of zero measure. From a synthetic argument , does an argument derive by an operation and, knowing is the result of an integral operation inverting , at least when is summable? Therefore, any function for which the operation can be performed will result in an argument from which will be derived "almost everywhere" by , as was derived from . However, if this reciprocal link is not exactly achieved by operation applied to , it will be by an operation that is more general than and such that operation is the inverse of (and not just of ).
Thus is an ordinary derivative summable? Its primitive will be equal to its Lebesgue integral between and . And conversely, any summable function integrated between and gives a function whose derivative is "almost everywhere".
Is the sum of a convergent trigonometric series a summable function? Its coefficients are obtained by Fourier formulas where the integrals are taken in the Lebesgue sense. (The latter had established it simply for bounded developable functions; de La Vallée Poussin demonstrated it for any summable functions). And conversely, if is summable and if Fourier's formulas are applied to this function with Lebesgue integration, we form a trigonometric series whose sum is "almost everywhere", on condition that a generalised summation of the series is carried out if necessary, by the Riemann method, or by that of Poisson.
The expression "almost everywhere" did not have my approval. I proposed another one: "on a plenitude" (and previously: "on a full thickness"). Knowing a function "almost everywhere" does not advance any more than knowing it "almost nowhere" if we have not ensured, by the special properties, generally topological or "descriptive" of this function, that it is indifferent for the goal to be achieved to ignore its values on a set of zero measure. From a topological point of view, that is, for characters invariant by a topological transformation [ continuous and increasing, in the case of a variable], a set of zero measure can be an "almost everywhere", its complement, full metric thickness, being topologically an "almost nowhere".
* * *
Lebesgue and Borel opposed each other in a violent polemic. The former wanted to deny Borel the discovery of measure and, perhaps under this inspiration, he presented measure as a consequence of the integral. But Borel, seeming to doubt his incontestable claims of priority for the definition of measure, and disdainfully leaving this discovery to Lebesgue, wanted moreover to contest the value of the new integral. He claimed to obtain it without using measure, by substituting for the argument function f a sequence of polynomials integrated term by term and providing at the limit the Lebesgue integral. But the recourse to these polynomials is an artifice devoid of any interest. In fact, Borel could say:
1° I learned to measure families of disjoint intervals (open sets), closed sets, sets of zero measure, limit sets of increasing or decreasing sequences of measurable sets. All the ideas involved in the theory of measure belong to me. Perhaps Lebesgue added a little order to them.Was it to depreciate the measure that Lebesgue intended to deduce it from the integral? Thus, according to him, the segment of the real axis between a and b would have length because is the area of the rectangle of height 1 having this segment as base. But would not this area have this value because such is the volume of the parallelepiped of height 1 constructed on this rectangle taken as a base, etc.? And yet F Riesz could have given substance to this intuition of Lebesgue. A linear functional of an argument function (here the integral of summable , on a measurable set) is the Lebesgue integral of when we adopt as measure of a set the value of the functional for on and outside . But F Riesz supposes the integral based on the Lebesgue procedure. Also, in the definition of the integral, not taking as a starting point the Euclidean measure of sets and wanting to draw this from the integral posed a priori, this procedure adopted by Lebesgue is artificial and very questionable. Borel and Lebesgue both seem to have suspected the solidity of their claims to have discovered the Euclidean measure of linear sets. In their definitions of the integral, each wanted to depreciate the role of the measure: Borel by ignoring it, Lebesgue by making it appear incidentally as an example of an integral.
2° If we extend, which presents no difficulty, to plane sets my definition of linear measure, and if f is a positive function given on the interval , the area, in my opinion, of the set [] is the Lebesgue integral . I admire the Lebesgue integral all the more because I can claim it as my own.
Lebesgue codified the principles of the measure of linear sets, principles already implemented by Borel. The latter had not conceived that certain sets could be non-measurable. Lebesgue reserved a place for these. It should be noted that the notions of great generality created by Borel, Baire, Lebesgue, related only to linear sets and functions of a real variable.
Fairness requires that the measure be attributed to Borel and the integral to Lebesgue. Moreover, Borel never took advantage of an integral different from that of Riemann. And even he only had to integrate continuous functions or those with an isolated singularity.
* * *
If the integral and all that its author has drawn from it constitutes the capital and imperishable title of Lebesgue, one must not forget his intervention in many subjects, and always to bring original views of a profound interest.
The trigonometric series with the criteria of convergence, the question of the non-uniformity of this convergence for certain continuous functions, the problem of Dirichlet, with the rectification of Riemann's reasoning for the plane and the singularities offered by the points in the case of space; the definition of the area of surfaces, in their first idea the sieves of Lusin for the definition of analytic sets, the general properties of singular integrals, the Cartesian topology of intertwined varieties, the invariance (attached to a remarkable principle) of the number of dimensions in topological transformations, on all these subjects, Lebesgue threw new and profound ideas, from which the later development of the theories was largely inspired.
The only risk to run for the memory of Lebesgue would be that, modern Analysis having so identified its own principles with the ideas created by him, these having proliferated so much, spread in so many varied directions, the new schools would come to forget from which initial trunk these immense branches came. The din of ephemeral fames could sometimes stifle the noise of his renown. But posterity will know how to give to each his due and the history of mathematics will fix one of its turning points as the appearance of the Lebesgue integral in the second year of this century.
The professor, the teacher
Henri Lebesgue, who taught since leaving the École Normale Supérieure, first at the Lycée de Nancy, then in the Faculties of Rennes, Poitiers, Paris, then at the Collège de France and in the two École Normale Supérieures (Rue d'Ulm and Sèvres), was a very great professor. Perhaps he would not have accepted this praise without reservation: it all depends, in fact, on what one means by it!
At the École de Sèvres, when I was a student there, Lebesgue taught in the second and third years, and Emile Picard in the first year. With Picard, a prestigious master, everything was clear; the essential ideas appeared in a luminous simplicity and the calculation gave the desired results like a well-styled servant, without apparent effort. The easiest paths, skilfully chosen and cleared, led to the summits from which the horizon emerges: "Dazzling effect" Lebesgue told me when we spoke about it a few years later. But he told the students with justice: "There is everything in your Picard course" and, in fact, this initiation, although summary, allowed one to pursue mathematical study in all directions.
Another master was a very great teacher: Paul Langevin: from the most profound ideas to the details of application, everything was in place. Calculations done in one's head by subtle processes, a table so well organised that the duster, erasing the accessory, allowed to reveal the structure of the course in a striking diagram, a chain of ideas without failure, all this without notes by the prodigious power of attention, such was the Langevin course, so often written gown by his listeners, but never published by an increasingly demanding concern of the master. Henri Lebesgue was always happy to hear Langevin praised and that is why I allow myself here to record his memory. Would mathematics lend itself as much as the theory of electricity to such treatment? No, probably not. In any case, the Lebesgue course offered nothing similar.
Lebesgue expressed his point of view clearly: "The only teaching that a professor can, in my opinion, give is to think in front of his students." And, for this thought to remain alive and spontaneous despite the years of teaching on immutable or almost immutable programmes, two conditions: always enrich one's own culture and forget the clarification of the previous year that one could recite, forget what has been honed by teachers for centuries and copied in all the manuals, forget the ready-made phrases that replace thought: it is in this sense that Lebesgue claimed to have always refused to "know" mathematics.
The ideal is therefore not to choose the methods so well, to dose the difficulties so well, that a chain of lemmas, theorems, corollaries unfolds like the steps of a staircase with a ramp on each side: "Let's do this and again this, let's notice this and again that; and after all this disparity, Q.E.D." Everyone is capable of following this path with a little perseverance, even with their eyes closed. "It's putting mathematics in pills" said Lebesgue.
Even by avoiding these faults, it is not enough to achieve, thanks to meticulous preparation, a perfect but fixed presentation that one finds in a book; the teacher would then have no other role than to indicate how and with what rhythm one should read. No. Lebesgue formally advised: "The broad outlines of a course must certainly be decided in advance, but I believe that the details must be improvised." Hence the method to follow: to place oneself frankly face to face with the problem, to become aware of its nature, of its structure, to perceive the unknown terrain to explore in order to reach the goal, to imagine ways to try, relays to set up, usable tools, to find a passage at all costs without being put off by a lack of elegance or by the disparity of the details; then, to understand what one has done, to criticise it in one's mind more than in its form, to improve it and finally arrive at what is simple, beautiful, pure from the point of view of structure, adapted to the level and to the desirable point of view, that is to say to reach what is often called with some naivety "a natural method". And even then, not to be satisfied with this beautiful solution, but to seek why it succeeds, what it really achieves, how it reveals relations between mathematical objects. In research, therefore, all the artificial barriers separating the different branches of mathematics disappear, the boundaries between plane geometry and geometry in space, between synthetic or analytical geometry, between elementary or higher mathematics, ancient or modern methods are erased and science is perceived in its unity. Every problem is then susceptible to multiple solutions according to the aspect in which it is studied. Sometimes one of them seems the best: we believe we still hear the master say "in reality it is about ...", but this is only a pre-eminence of a moment, relative to the current illumination of the question: "It happened to me as to everyone to have the impression that I had just understood as it was necessary to understand, that otherwise one did not understand completely. One feels all the relative of this absolute as soon as one has the opportunity to reflect several times on the same difficult question at some intervals. Each time one ends up understanding and each time in a different way. Since there is no better way to understand, there cannot be a better way to teach."
The goal is therefore achieved if, on this day, at this moment, the agreement of the listeners is obtained, if they are not only convinced by logical means, but also enlightened by a sensation that we know well: that of understanding. "If the cogs of a demonstration are dismantled piece by piece, the students will have nothing to object to, but they will not have reached that inner conviction which would require that one has understood enough for the new truths to connect, to be linked, and in several ways, with those that are already familiar." In short, what is understanding? "A notion is only fully useful at the moment when one has understood it well enough to believe that one has always possessed it and to have become incapable of seeing anything in it other than a banal and immediate remark."
If the research effort has not been concealed, one will not be fooled by this illusion of simplicity and no one will be surprised that the simple is what one sees last: "To the presentations that present the discovery as a divine creation where human reason intervenes only to observe with wonder what the gods wanted, let us not substitute presentations that would make the discovery the culmination of syllogistic operations that impose themselves with necessity and evidence."
If the essential thing is thus, not to expose the content of a completed science, previously existing in the mind of the professor, but to bring its creation to life and to make its structure understood, one is led to often call upon the history of science. It is not a question of making the child or the student pass through the historical stages that the specialist in history himself had so much trouble imagining, without taking advantage of the light now projected on so many obscure areas and of the order that replaces chaos, of forgetting that the student is immersed in modern life and is not a primitive. An example "Decimal notation is not a legacy of the Greeks; this is enough for everything related to this notion to be stuck on Greek teaching and not incorporated into it. Our teaching does not yet fully use this historical fact, perhaps the most important in the history of science: the invention of decimal numeration."
Lebesgue defended himself from making the history of science the goal of his research: "I do not do the history of science. I do science." Only the state of current science and its trends are explained and justified by its past. History serves above all to show how the problem arises: "Beyond their historical interest, the mathematical interest (of these studies) is to allow us to recognise the close relationship that unites research carried out more than twenty centuries apart." The entire book of Constructions géométriques is built on this idea.
The historical facts are not very certain and the proper name that serves as a label for the theorems is rarely justified, but what does it matter? "We might as well stick to the established inaccuracies!" But what matters is the history of ideas. It is still necessary to agree: "The history of the ideas that led to a discovery is in reality only the exposition of the logical reasons, imagined after the fact, for the success of those ideas that were fruitful. So do not misunderstand the value of a history of ideas nor, for that reason, deny this value. On the one hand, it makes one understand the methods and their exact scope, on the other hand and above all, the critical work that such a history exposes is exactly that which our mind, or if we wish our subconscious, must constantly do to choose, in the whirlwind of ideas, those which seem to be able to be used and to bring these and all these to consciousness."
We feel certain that in these lines he speaks as a creative mathematician, who he does not dissociate from the professor. It is for himself that he links the present to the past, that he devotes pages of his works to setting out previous research, it is "to reassure himself" and "to show that his personal ideas are closely linked to those of his predecessors."
To understand Lebesgue as a professor, we are thus led to understand him as a scholar, to see his position as a scientist.
His method of exposition, of research in front of students, is the consequence of his conception as a scholar: "The mathematician must explore the field in which he works, observe the role of the mathematical objects he meets there, watch them live, one might say, in order to discern their qualities and recognise the contributions of each of these qualities." And also: "What I obtained in mathematics was the natural and sometimes immediate result of examining the reasons for such success or the impotence of such and such an idea."
To conclude, if one had to summarise the attitude of Henri Lebesgue the professor, I believe that the word sincerity would be the one to pronounce. He had a horror of the hypocrisy by which one saves the value of a reasoning by means of a word, a restriction, an incident that one knows to be beyond the reach of the student and that one pronounces for oneself alone. If the teacher is up to his task, he has a light that he must transmit. Especially for young students, perfection of form must be the result of a collective effort and preparation must above all focus on substance. It is for this task that our master, in the two Écoles normales, tried to prepare future teachers: "I would like the theorems not to be for any of them trees planted in a row along the road of the programme, a road that would cross nothing. I would like them to see for themselves the dominant peaks of a dense and varied forest, to know this forest crisscrossed by a thousand paths allowing it to be explored just as well, to have ventured into some of them, admiring the most beautiful trees in all their aspects. To be a guide in the forest, shouldn't it be good to know it well and fully feel its beauty?"
To be this guide, no one was more qualified than Lebesgue. Of course, improvisation is not always without danger; sometimes, because of fatigue, a mistake not corrected in time in a calculation had disastrous consequences! The course was difficult to learn, to write, and did not replace books, but it brought a light that no book can give. With boundless patience and untiring interest, he adapted to his audience, repeating and varying the demonstrations until they were not only accepted, but felt.
And here we touch on the essential, a point that I hardly dare to indicate here: with the word sincerity, one must, and even more, write kindness. If the scholar cannot dissociate himself from the professor, even less can the latter be dissociated from the sensitive and generous man, from the paternal friend who took part in all the pains that fate blindly inflicts on so many innocents. The younger and more vulnerable the victim, the more he leaned toward them and, in secret, made the gesture of justice or kindness to give relief. No one will ever know exactly all the good he was able to do in this way, which the person concerned was often unaware of. However, our master, M Lebesgue, who loved his students very much, especially those at the École de Sèvres, knew that in return he was loved by them.
The man
Half a century of close and deep friendship, a constant community of thoughts and often feelings allowed me to take the measure of the greatness of Henri Lebesgue.
We met and became friends at the École Normale Supérieure at the end of the last century (the 19th century). Right away, his profound originality, his rare penetration, his inexorable critical spirit, his unwavering logic were evident to his colleagues.
Bachelor's degree certificates did not yet exist: we were subject to a series of examinations which, after two years, granted us the necessary licenses. First, we took analysis courses. We were taught that any developable surface is applicable to the plane and that, conversely, any surface that can be applied to the plane is a developable surface. And Lebesgue came towards us, holding a crumpled handkerchief in his hand. "Is it applicable to the plane?" he asked. And, on our affirmative answer: "This crumpled handkerchief is therefore a developable surface." His critical mind had discerned that the theorem that we were taught was based on hypotheses of continuity of functions and derivatives that are essential to its accuracy, but which are not always well explained. Lebesgue then began to look for non-developable surfaces that are applicable to the plane and was led to discover the integral that bears his name in order to define them. This discovery, like many others that we owe to him, thus has a geometric origin. Lebesgue loved geometry very much and particularly elementary geometry, in which he was interested until his last days. Very meditative, he read little and, in the presence of a memoir, he sought the general course of thought and, having recognised it, he re-established the demonstrations or imagined new ones.
But we also had to deal with chemistry, which did not appeal to Lebesgue. When, after a very hard winter, this Parisian spring appeared whose charm is so captivating, we set off to explore Paris and the hours of the organic chemistry course were often spent at the prow of Lutèce, in the small garden at the foot of the Pont-Neuf. Lebesgue failed the chemistry exam in July. This disaster had to be repaired at the October session under penalty of severe sanctions going as far as expulsion from the École. Lebesgue cut short his vacation to return to the rue d'Ulm and get to work under the direction of our comrade Langevin. When the date of the examination arrived, Langevin told him: "You are completely useless, but you may have a chance of getting through it. The examiner is very hard of hearing; you can speak without fear, but if you have the misfortune to write a single chemical formula on the board, you are lost." The advice was followed to the letter. During the examination, Lebesgue talked at length, facing the examiner and far from the blackboard. "Write," the examiner said. Lebesgue, continuing to speak, then walked slowly toward the board, took a piece of chalk, raised his arm and suddenly, as if struck by a sudden idea, he returned to the examiner. "Write," the examiner repeated. The scene was repeated and Lebesgue succeeded in not writing a single chemical symbol: he passed.
He had hardly been more diligent in the cosmography course that Wolff taught us. However, in the written examination, his copy, which included an astronomical calculation, obtained a magnificent mark. In the oral exam, the delighted professor turned graciously to one of our classmates who had scrupulously followed all his lessons and called out: "M Lebesgue." He was very surprised to see an unfamiliar face appear.
After his agrégation, Lebesgue prepared a doctorate for which he presented this thesis entitled Intégrale, Longueur, Aire, which brought to mathematical science a tool of extraordinary power. This thesis was followed by other works where the fundamental notions of analysis were examined with a fine-tooth comb, derivatives were often absent, and discontinuous functions were given pride of place. It was the time when the profound research of Dini and Baire had attracted researchers to this field.
The reception of Lebesgue's work by the masters of the time was rather reserved. Many feared seeing a teratology of functions established. Darboux, who one might have thought would be favourable, because of his 1875 thesis on discontinuous functions, was hostile to him. Boussinesq was said to have said: "But a function is only interesting when it had a derivative." He was probably talking about the interest of the person who uses it. Picard alone defended Lebesgue's research and appreciated his qualities.
Lebesgue explained the nature of his integral with a pleasant image that was accessible to all. "I have to pay a certain sum," he said; "I dig into my pockets and take out coins and notes of different values. I pay them to my creditor in the order in which they appear until I reach the total of my debt. This is the Riemann integral. But I can operate differently. Having taken out all my money, I gather the notes of the same value, the similar coins, and I make the payment by giving together the monetary signs of the same value. This is my integral."
This notion is certainly simpler than the first, but it requires a certain maturity of mind and some familiarity in the knowledge of the points where a function takes the same value. One day when some young mathematicians who had gathered at my house were discussing the advisability of teaching Lebesgue's integral from the beginning, he arrived unexpectedly. He was asked the question: "With which integral should we begin in front of young students?" "With Riemann's, of course," replied Lebesgue.
M Denjoy recounted in his article the controversy that opposed Borel and Lebesgue on the subject of the notions of measure and integral as a result of their difficulty in objectively acknowledging each other's share. The two men fought each other but could not help but admire and esteem each other mutually: each of them knew how to appreciate the value of the other. One day, Borel tried to establish a ground for conciliation. "Without doubt," he said, "I have faults. But who does not? Does such-and-such have any?" "Yes," replied Lebesgue. "And another such-and-such?" "Certainly." - So perhaps you too have faults - "I don't think so," replied Lebesgue after a moment's reflection. I smiled when Borel told me the story, then, having looked for a fault in Lebesgue, I was very embarrassed.
This integral also gave rise to a controversy of a philological nature. M Denjoy had, after Lebesgue, given a new and definitive power to the notion of integral by his beautiful creation of totalisation. In his works, he introduced the qualifier "perceived" to designate Lebesgue's integral which became the "perceived" integral. Lebesgue, displeased, wrote him one of those long letters in which he was prodigal, humorous, ironic and biting. "You call my integral perceived," he wrote, "what would you say if I called yours joyful?" "You want to call my integral joyful," replied M Denjoy, "I defy you."
Lesbesgue was not one of those mathematicians who live continually in the abstract, who are surprised by reality, to whom their distractions bring at least as much fame as their work. He had his feet on the ground, he was open to all manifestations of practical or sentimental life. His kindness was inexhaustible. If one of his students needed material, intellectual or moral help, he hastened to come efficiently to his aid.
We were in the habit of accompanying each other home and our separations were preceded by comings and goings fertile in chatter. One evening, we were strolling around his door. Night having fallen, I noticed that our oscillations had an increasingly reduced amplitude. I asked him the reason. Then he showed me a hole in the sidewalk where a distracted walker could risk an accident and added that it was now the time that his wife would come home.
But his kindness did not exclude a taste for biting irony. Concerning Bertrand's theorem establishing that any algebraic curve admitting a rectilinear diameter for each direction is composed of conics and of which I had established an extension, I was led to think that, if we fix the degree of the curve, there must exist an upper limit of the number of rectilinear diameters beyond which the curve is composed of conics and I proposed to him to express this limit as a function of the degree. He resolved the question and gave me a manuscript to publish in the Bulletin of the Mathematical Society of France of which I was then the secretary. The memoir was lost and he had to start writing it again. I then read in the proofs of the article: "A first draft of this work having been lost in the care of M Montel, I had to compose a second one."
Lebesgue brought as much energy to defending the just as he spent in supporting the true. Injustice drove him mad. An election in mathematics to the Academy of Sciences seemed to him to be a denial of justice; he stopped attending the sessions for several years and did not resume his place until the injustice seemed to him to have been repaired. Above all, he placed the feeling of duty. Suffering from the illness that was to carry him off, he nevertheless continued to give his course at the Collège de France. Paris, occupied by the enemy [Germany in World War II], was deprived of ordinary means of transport. He chartered one of those tricycles in the form of sedan chairs that were still in circulation and had himself driven to the place where he was teaching. This was his last course collected by Miss Félix. Shortly afterwards, he dictated from his bed to his student, Miss Beauvallet, the substance of a book he was preparing on conics. After his death, I was able to publish two posthumous works, one on Geometric Constructions, the subject of his last course; the other on Conics, thanks to these notes. We find there this persistent taste for elementary geometry which never abandoned him.
He was a great scholar, an admirable professor, a man of incomparable moral nobility. His influence on the development of mathematics will continue for a long time to be exercised by his own works and by those which he inspired.
Last Updated March 2025