Lixue huitong 曆學會通

Lixue huitong 曆學會通 "Integration of calendric sciences" is a book on astronomy written during the early Qing period 清 (1644-1911) by Xue Fengzuo 薛鳳祚 (1600-1670), courtesy name Yifu 儀甫, from Zichuan 淄川 (modern-day Zibo 淄博, Shandong). In his early years, he studied under Lu Shanji 鹿善繼 (1575-1636) and Sun Qifeng 孫奇逢 (1585-1675), and authored the book Shengxue xinchuan 聖學心傳. Later, he studied astronomy and calendrical science under the traditionalist astronomer Wei Wenkui 魏文魁. In 1646, Xue met the Polish Jesuit Jan Mikołaj Smogulecki (1610–1656; Chinese name Mo Nige 穆尼閣) and began studying Western natural sciences, particularly focusing on astronomy and calendrical calculations. His expertise in these fields grew significantly, and he, alongside Wang Xichan 王錫闡 (1628-1682), was widely recognised as one of the two most distinguished independent astronomers of his time in China. Apart from the Lixue huitong, Xue also authored the books Lianghe qinghui 兩河清彙 and Haiyun 海運.

The Lixue huitong has a total length of 56 juan, consisting of a central collection (Zhengji 正集) of 12 juan, practical investigations (Kaoyan 考驗) of 28 juan, and adaptions (Zhiyong 致用) of 16 juan. The latter includes the chapter Tianbu zhenyuan 天步真原, which is a translation of Smogulecki's teachings. The work was finished in 1664.

The book is a comprehensive work that covers a variety of subjects, including astronomy, mathematics, medicine, physics, hydraulics, and firearms, with a primary emphasis on astronomy and mathematics. The astronomical section contains treatises on the principles governing the motions of the sun and moon (Taiyang taiyin zhu xing fayuan 太陽太陰諸行法原), the motions of Jupiter, Mars, and Saturn (Muxing, huoxing, tuxing jingxing fayuan 木星、火星、土星經行法原), the principles of eclipses (Jiaoshi fayuan 交食法原), calendar era cycles (Linian jiazi 曆年甲子), the determination of the true length of the (tropical) year (Qiu suishi 求歲實), the motions of the Five Planets (Wuxing gaoxing 五星高行), eclipse tables (Jiaoshi biao 交食表), fixed stars and meridian stars (Jingxing zhongxing 經星中星), Islamic astronomical methods (Xiyu huihui shu 西域回回術), Islamic astronomical tables (Xiyu biao 西域表), selected Western methods (Jin Xifa xuanyao 今西法選要), and modern Western tables (Jinfa biao 今法表). This work thus serves as a bridge between Chinese and Western astronomical knowledge, integrating (huitong 會通) foreign methodologies with traditional Chinese calendrical science.

The book establishes its epoch (liyuan 曆元) at the winter solstice of 1655, with all calculations predicated on this starting point. The work utilises key astronomical constants from Smogulecki's Tianbu zhenyuan, including: the tropical year as 365 days, 23 ke 刻 (15 min), 3 minutes, and 57 seconds, variations in the obliquity of the ecliptic, and the precession rate of fixed stars at 52 arcseconds per year.

However, the Lixue huitong also presents unique innovations. One notable feature is the integration of Western logarithmic calculations, including logarithmic trigonometry. Chapter Sixian 四綫 "The four graphs" centres on four fundamental trigonometric functions—sine, cosine, tangent, and cotangent—rather than enumerating all six, reflecting the latest mathematical advancements of the era. To cater to the computational habits of Chinese scholars, Xu Fengzuo modified Western sexagesimal (base-60) notation to decimal (base-10) notation. He also dedicated substantial effort to compiling new logarithmic and trigonometric tables, introducing a logarithm table for numbers ranging from 1 to 20,000 to Chinese readers. These contributions earned high praise from the esteemed calendrical expert Mei Wending 梅文鼎 (1633-1721), illustrating that the Lixue huitong was not simply a direct translation of the Tianbu zhenyuan, but rather a true integration and adaptation of Western astronomical knowledge into the Chinese scientific framework.

The imperial series Siku quanshu 四庫全書 includes the two texts Tianbu zhenyuan 天步真原 and Tianxue huitong 天學會通. Specifically, it includes Xu's calculations of a solar eclipse on the first day of the eighth lunar month in 1596 and a lunar eclipse on the full moon of the third lunar month in 1632.