Qianxiang shu 乾象術

Qianxiang shu 乾象術 "Skill [to calculate] celestial phenomena", also called Qianxiang li 乾象曆 (Qianxiang Calendar), was a book on astronomy from the Later Han period 後漢 (25-220 CE) written by Liu Hong 劉洪 (c. 135-210), courtesy name Yuanzhuo 元卓, from Mengyin 蒙陰 (in today's Shandong province).

Hailing from the house of the princes of Lu 魯, he became in 174 an aide (zhangshi 長史) of the Prince of Changshan 常山, a position in which he could play out his profession in astronomy, and submitted the texts Qiyao shu 七曜術, and Bayuan shu 八元術. In 178, he cooperated with Cai Yong 蔡邕 (132-192) in compiling a public treatise on the calendar. In later years, he was a receptionist (yezhe 謁者) of the Marquis of Gongchengmen 谷城門侯, commandant (duwei 都尉) of Guiji East 會稽東, governor (taishou 太守) of the commandery of Shanyang 山陽 and finally household manager (xiang 相) of the Marquis of Qucheng 曲城侯. The Qianxiang shu was finished in 206. It only survives in fragments assembled in Huang Shi's 黄奭 (1809-1853) series Hanxuetang congshu 漢學堂叢書 and Huangshi yishu kao 黄氏逸書考 (part Zishi gouchen 子史鈎沈).

The Qianxiang shu is the first astronomical calendar in Chinese history to include content on the irregular motion of the Moon (modern term yueng chiji 月行遲疾). Before this, ancient calendars - from the Six Ancient Calendars (gu liushu 古六曆) to the Taichu 太初曆 and Santong 三統曆 calendars from the Former Han, and the Eastern Han's Sifen Calendar 四分曆 — did not account for these variations in lunar motion. As a result, discrepancies between the predicted and actual positions of the Moon persisted, with the most significant practical impact being increasing errors in lunar eclipse predictions.

Astronomers had long recognised the issue and actively investigated it. As early as the Warring States period 戰國 (5th cent.-221 BCE), the astrologer Shi Shen 石申 had an implicit awareness of the Moon’s irregular motion. By 52 BCE, Geng Shouchang 耿壽昌 had explicitly noted that the Moon's speed fluctuated. Over the following century, numerous astronomers devoted significant efforts to studying the Moon's motion, including Li Fan 李梵, Su Tong 蘇統, Zong Kan 宗紺, and Liu Gu 劉固, as well as Liu Hong. The Sifen Calendar even explicitly mentioned the inclination between the ecliptic (huangdao 黄道) and the Moon's orbit (baidao 白道), noting that they were not in the same plane. Later, Jia Kui 賈逵 (30-101 CE) determined that the Moon's perigee (jindidian 近地點) advances approximately three degrees per month, completing an entire cycle in nine years before returning to its original position. Liu Hong's Qianxiang shu synthesised centuries of astronomical research and calendrical studies, establishing a relatively complete set of numerical values and computational methods for lunar motion.

The Qianxiang shu was the first text to explicitly introduce the concept of the nodal month (jiaodian yue 交點月), the time it takes for the Moon to return to the same node, where its orbit intersects the ecliptic. Liu Hong provided an exact value for this period, namely 27 ³³⁰³/₅₉₆₉ days, which is remarkably close to the modern value of 27.55455 days. Liu Hong's method of calculating this number continued to be used in subsequent Chinese calendrical systems.

Liu Hong also conducted extensive observations to determine the Moon's actual daily motion over an entire anomalistic month, creating a table that listed the Moon's accurate motion for each day. By comparing these values to the average speed, he calculated the gain and loss rates (sunyi lü 損益率), which allowed for more accurate predictions of the Moon's position. He determined the Moon's daily mean motion during an anomalistic month to be 3°4'. He estimated the inclination between the lunar orbit and the ecliptic at 6°, which is close to the modern value of 5.9°. Liu Hong introduced improved calculations called true conjunction (dingshuo 定朔) and true opposition (dingwang 定望).

Another significant contribution of the Qianxiang shu was refining the length of the tropical year (huigui nian 回歸年). The Sifen Calendar had defined the length of a tropical year as 365.25 days. Liu Hong realised that the inaccuracy of the Sifen Calendar stemmed from an excessive doufen 斗分, the fractional remainder in the calculation of each tropical year. He introduced a more precise value of 365 ¹⁴⁵/₅₈₉ days, which converts to 365.246179 days.

The Qianxian shu also introduced the concept of eclipse limits (shixian 食限), providing a scientific explanation for why eclipses do not occur at every new and full moon. Liu Hong determined that for a solar eclipse to happen, the Moon must be within 15.5 degrees of the node where the lunar orbit intersects the ecliptic during a conjunction (new moon). This threshold was adopted in subsequent Chinese astronomical systems and remained the standard criterion for predicting solar eclipses. By refining these calculations, he improved the accuracy of eclipse cycles (jiaoshi zhouqi 交食周期), leading to more precise predictions. This advancement became known as the famous Three Lunar Paths Method (yuexing sandao shu 月行三道術), a critical step forward in Chinese lunar theory.