Background behind Pythagorean derivation

At this early point in Chinese history the model of the ancient Chinese equivalent of Heaven, 天 Tian, was symbolized as a circle and the earth was symbolized as a square. In order to make this concept easily understood the adopted symbol of the heavens was the ancient Chinese chariot. The charioteer would stand in the square body of the vehicle and a "canopy", the equivalent of an umbrella, stood next to them. The world was thus likened to the chariot in that the earth, the square, was where the charioteer stood, and heaven, the circle, was suspended above them. The concept has thus been termed "Canopy Heaven", 蓋天 (Gaitian).

Eventually the populace began to turn away from the "Canopy Heaven" concept in favor of the concept termed "Spherical Heaven", 渾天 (Huntian). This was partly due to the fact that the people were having trouble accepting heaven's encompassment of the earth in the fashion of a chariot canopy because the corners of the chariot were themselves relatively uncovered. In contrast, "Spherical Heaven", Huntian, has Heaven, Tian, completely surrounding and containing the Earth and was therefore more appealing. Despite this switch in popularity, supporters of the Gaitian "Canopy Heaven" model continued to delve into the planar relationship between the circle and square as they were significant in symbology. In their investigation of the geometric relationship between circles circumscribed by squares and squares circumscribed by circles the author of the Zhoubi Suanjing deduced one instance of what today is known as the Pythagorean Theorem.

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