Zhoubi suanjing 周髀算經

Zhoubi suanjing 周髀算經 "The mathematical book on gnomons and circular paths" is China's oldest surviving book on the basics of astronomical calculations. Originally titled Zhoubi 周髀 "Trigonometry of circles" or "Trigonometry of the Zhou", its usual name was adopted during the Tang period 唐 (618-907). It belongs to the ten classical book on mathematics in ancient China (Suanjing shishu 算經十書). The author is unknown.

The original preface explains the word zhou 周 as pointing to the Zhou dynasty 周 (11th cent.-221 BCE), saying, "In ancient times, when the Son of Heaven governed Zhou, by observing these numbers, [the beginning of the calendar] started with the Zhou (gu shi tianzi zhi Zhou, ci shu wang zhi cong Zhou 古時天子治周，此數望之從周." However, the word zhou also means "circuit, circle, round", thus referring to astronomical movements. The word bi 髀 means "thigh" (gu 股), referring to an eight-chi 尺-long (see weights and measures) gnomon (biao 表, gui 圭 or 晷). Together with its shadow – referred to as a "hook" (gou 勾) - two sides of a right triangle (or a carpenter's square) were formed (the adjacent, and the opposite). With the help of these two data, the length of the hypotenuse can be calculated. This formula corresponds to the Pythagorean theorem (modern term gougu dingli 勾股定理). In this way, the length of the gnomon and its shadow are used to determine distances. By analyzing variations in the length of the gnomon's shadow, astronomers could estimate the length of a tropical year. The gougu method was also used to determine seasonal changes (i.e., the 24 solar terms, jieqi 節氣), and even to estimate the size and distance of the sun, as well as the structure of the cosmos.

In terms of astronomy, the Zhoubi suanjing primarily expounds on the gaitian 蓋天 cosmology ("canopy Heaven") and the old Four-Part Calendar (sifen li 四分曆). The mention of the movement of the celestial sphere (xuanji 璇璣) in four directions serves as textual evidence that ancient observers had already noticed the displacement of the North Pole, indicating an awareness of the precession of the equinoxes (suicha 歲差). The Canopy Theory is an ancient cosmological model that envisions the sky as a domed cover, like an inverted bowl, while the Earth is depicted as a flat plate beneath it. The sun, moon, and stars move along the celestial canopy. This cosmology had a profound impact on Chinese intellectual history. During the Han period 漢 (206 BCE-220 CE), it coexisted and was debated alongside two other competing cosmological theories, the huntian 渾天 model ("spherical Heaven"), and the xuanye 宣夜 model ("infinite night") model. Together, these were known as the Three Schools of Astronomy (luntian sanjia 論天三家). The Four-Part Calendar was a lunisolar calendar (yinli 陰曆) that regulates seasonal changes (sishi jihou 四時季候) using intercalary months (runyue 閏月). It defined the tropical year (modern term yihui guinian 一回歸年) as 365 ¼ days and follows a Metonic cycle, inserting seven intercalary months every nineteen years to keep the lunar months synchronized with the solar year.

Mathematically, the book covers fractional multiplication and division (modern term fenshu cheng-chu fa 分數乘除法, generally fenshu yunsuan 分數運算), arithmetic sequences (dengcha shulie 等差數列), methods for calculating the circumference of a circle (yuanzhou qiuchang fa 圓周求長法), linear interpolation (yici neicha fa 一次內插法), square root extraction for arbitrary positive numbers (zhengshu kaiping fangfa 正數開平方法), representing decimal fractions using fractions (fenshu biaoshi qiling xiaoshu 分數表示奇零小數), and the earliest recorded use of the Pythagorean theorem. Based on its mathematical content, the book could not have been written earlier than the beginning Warring States period 戰國 (5th cent.-221 BCE).

The Pythagorean theorem of the Zhoubi suanjing was used in a very simplified form, namely a base of 3 units, a height of 4 units, and a diagonal of 5 units (gou guang san, gu xiu si, jing yu wu 勾廣三，股修四，徑隅五) – if emulated to the areas above the three lines, representing the numbers 9+16=25. Based on this specific example, the text provides a general formulation of the gougu theorem in the context of astronomical calculations:

Mathematically expressed, 弦=√(勾²+股²). This method is known as gougu ce wang 勾股測望 "prospection by adjacent and opposite".

Furthermore, through a fictional dialogue between Chenzi 陳子 "Master Chen" and Rong Fang 榮方, the book presents an early perspective on the methodology of learning mathematics. It emphasizes the importance of both breadth and depth in knowledge acquisition. To understand one principle and thereby comprehend myriad things is to truly grasp the Way (yan yue er yong bo zhe, zhi lei zhi ming 言約而用博者，智類之明). The Zhoubi suanjing highlights an early Chinese philosophy of learning by emphasising conceptual understanding, breadth of knowledge, diligent practice, and the importance of collaborative learning.

Although the Zhoubi suanjing contains errors and rough approximations, such as assuming the Earth to be flat and using this flat Earth model to infer celestial phenomena (leading to misconceptions like the idea that a one-inch difference in gnomon shadow length corresponds to a thousand li, or that the ratio of a circle's circumference, Pi, to its diameter is simply three) - it still holds an irreplaceable position in the history of Chinese astronomy.

The version that has been passed down to the present day includes annotations by Zhao Shuang 趙爽 (Zhao Ying 趙嬰 or Zhao Junqing 趙君卿; 3rd cent. CE), Zhen Luan 甄鸞 (6th cent.), and Li Chunfeng 李淳風 (602-670). These commentators trace the system of celestial degrees back to the mythological ruler Fu Xi 伏羲 and believed to have been later formalised by the Duke of Zhou 周公, by whom it was transmitted to grand master Shang Gao 商高.

The book's pioneering use of the gougu theorem and observational measurement techniques greatly influenced later mathematicians, who further developed and applied these methods. From the Tang and Song periods onward, the Zhoubi suanjing was formally included in the imperial curriculum as an official mathematics textbook, contributing significantly to the evolution of Chinese mathematical thought. The book also spread beyond China; during the Tang dynasty, it was introduced to Japan, where it became one of the official mathematical textbooks, further extending its influence in East Asia.

Other statements about the cosmological concept of the Zhoubi suanjing are:

During the Tang period, Li Chunfeng annotated the text, correcting errors in the original method for calculating the sun's height. Li also criticises Zhao Shuang's flawed approach to shadow measurement, and amends misinterpretations by Zhen Luan. The most important commentaries are Mei Wending's 梅文鼎 (1633-1721) Zhoubi buzhu 周髀補注, Wu Lang's 吳烺 (1719-1771) Zhoubi suanjing tuzhu 周髀算經圖注, Feng Jing's 馮經 Zhoubi suanjing shu 周髀算經述, Zou Boqi's 鄒伯奇 (1819-1869) Zhoubi suanjing kaozheng 周髀算經考證, Gu Guanguang's 顧觀光 (1799-1862) Zhoubi suanjing jiaokanji 周髀算經校勘記 and Sun Yirang's 孫詒讓 (1848-1908) Zhayi 札迻.

The oldest surviving print dates from 1085. The text is included in the series Xuejin taoyuan 學津討原, Tang-Song congshu 唐宋叢書, Mice huihan 秘冊匯函, Xuejin taoyuan 津逮秘書, Weiboxie suanjing shishu 微波榭算經十書, Wuyingdian juzhenban congshu 武英殿聚珍版叢書, Sibu congkan 四部叢刊, Sibu beiyao 四部備要, Siku quanshu 四庫全書, Huailu congshu erbian 槐廬叢書二編 and Congshu jicheng chubian.