起源

《周髀算經》原名《周髀》，出現於西漢時期，記載相關天文學和數學的發展成果，尤其在數學方面有著突破性的進步，後人認為是經典之作，因此則改稱為《周髀算經》。

內容

「周髀」這個名稱，按該書中的解釋，「周」指的是周代，指從周代傳下來的一些方法，「髀」原意指的是股（大腿）或者股骨，在這裡的意思是「用來測量日影的長八尺之表」。

天文學

天文學方面，《周髀》主要闡述蓋天說和四分曆法。

數學

數學方面，《周髀》主要記載漢代的數學成就，率先提出了幾何學重要的勾股定理，並在測量太陽高遠的方法中給出勾股定理的一般公式。

《周髀》中出現運用重差術繪出的日高圖，不過沒有詳細說明方法，三國時，趙爽、劉徽進一步研究，使之成為中國古代測望理論的核心內容。

《周髀》周就是圓，髀就是股。上面記載周公與商高的談話，其中就有勾股定理的最早文字記錄，即「勾三股四弦五」，亦被稱作商高定理。事實上這一定理在時間上還應往前推移。

地理學

地理學方面，《周髀》明確闡述了極晝和極夜現象。

價值

《周髀算經》的作者已經無法得知，從成書時間來看，並非一人一時之作，而是對先秦數學發展成果的總結。

《周髀算經》是中國流傳至今最早的數學著作，是後世數學的源頭，其算術化傾向決定中國數學的性質，歷代數學家奉為經典。

《周髀算經》的採用最簡便可行的方法確定天文曆法，揭示日月星辰的運行規律，囊括四季更替，氣候變化，包涵南北有極，晝夜相推的道理。給後來者生活作息提供有力的保障，自此以後歷代數學家無不以《周髀算經》為參考，在此基礎上不斷創新和發展。

日高與七衡

Dating

Examples of the gnomon described in the work have been found from as early as 2300 BC and the Duke of Zhou, was an 11th-century BC regent and noble during the first generation of the Zhou dynasty. The Zhoubi was traditionally dated to the Duke of Zhou's own life and considered to be the oldest Chinese mathematical treatise. However, although some passages seem to come from the Warring States period or earlier, the current text of the work mentions Lü Buwei and is believed to have received its current form no earlier than the Eastern Han, during the 1st or 2nd century. The earliest known mention of the text is from a memorial dedicated to the astronomer Cai Yong in 178 AD. It does not appear at all in the Book of Han's account of calendrical, astronomical, and mathematical works, although Joseph Needham allows that this may have been from its current contents having previously been provided in several different works listed in the Han history which are otherwise unknown.

Contents

The Zhoubi is an anonymous collection of 246 problems encountered by the Duke of Zhou and figures in his court, including the astrologer Shang Gao. Each problem includes an answer and a corresponding arithmetic algorithm.

It is an important source on early Chinese cosmology, glossing the ancient idea of a round heaven over a square earth (天圆地方, tiānyuán dìfāng) as similar to the round parasol suspended over some ancient Chinese chariots or a Chinese chessboard. All things measurable were considered variants of the square, while the expansion of a polygon to infinite sides approaches the immeasurable circle. This concept of a 'canopy heaven' (蓋天, gàitiān) had earlier produced the jade bi (璧) and cong objects and myths about Gonggong, Mount Buzhou, Nüwa, and repairing the sky. Although this eventually developed into an idea of a 'spherical heaven' (渾天, hùntiān), the Zhoubi offers numerous explorations of the geometric relationships of simple circles circumscribed by squares and squares circumscribed by circles. A large part of this involves analysis of solar declination in the Northern Hemisphere at various points throughout the year.

At one point during its discussion of the shadows cast by gnomons, the work presents a form of the Pythagorean theorem known as the gougu theorem (勾股定理) from the Chinese names—lit. 'hook' and 'thigh'—of the two sides of the carpenter or try square. In the 3rd century, Zhao Shuang's commentary on the Zhoubi included a diagram effectively proving the theorem for the case of a 3-4-5 triangle, whence it can be generalized to all right triangles. The original text being ambiguous on its own, there is disagreement as to whether this proof was established by Zhao or merely represented an illustration of a previously understood concept earlier than Pythagoras. Shang Gao concludes the gougu problem saying "He who understands the earth is a wise man, and he who understands the heavens is a sage. Knowledge is derived from the shadow line, and the shadow is derived from the gnomon angle. The combination of the gnomon with numbers is what guides and rules the ten thousand things."

Commentaries

The Zhoubi has had a prominent place in Chinese mathematics and was the subject of specific commentaries by Zhao Shuang in the 3rd century, Liu Hui in 263, by Zu Gengzhi in the early 6th century, Li Chunfeng in the 7th century, and Yang Hui in 1270.

Translation

A translation to English was published in 1996 by Christopher Cullen, through the Cambridge University Press, entitled Astronomy and mathematics in ancient China: the Zhou bi suan jing. The work includes a preface attributed to Zhao Shuang, as well as his discussions and diagrams for the gougu theorem, the height of the sun, the seven heng and his gnomon shadow table, restored.