Li Ye

Li Ye, Wade-Giles Li Yeh, original name Li Zhi, literary name Jingzhai, (born 1192, Luangcheng, Hebei province, China—died 1279, Yuanshi), Chinese mathematician and scholar-official who contributed to the solution of polynomial equations in one variable.

Li passed the mandarin jinshi examination (the highest scholar-official title in imperial China) in prose literature at the late age of 38. He was appointed to the administrative position of prefect but fled when the Mongols invaded his district in 1233. Wandering homeless in Shanxi, Shandong, and Henan provinces, Li perfected his literary, mathematical, and astronomical skills and composed several writings. It was during this period that he composed his main work, Ceyuan haijing (1248; “Sea Mirror of Circle Measurements”), which contains 170 problems based on one geometric diagram of a circular city wall circumscribed by a right-angled triangle. Each problem involves two men walking along various roads within the town in order to see one another or some object, such as a tree, and invariably leads to the same answer for the wall’s diameter. Although the problems are highly contrived, they enabled him to list some 692 algebraic formulas for triangular areas and segment lengths. The polynomial equations involve arbitrary powers of the unknown, or “celestial unknown.” Presumably, Li improved both pedagogical and technical aspects of the subject.

In 1257 Li was invited to an audience with the Mongolian leader Kublai Khan. Although the latter was greatly impressed by Li’s replies on the methods of government and scientific issues, he did not offer him an official position at that time. After his enthronement as emperor in 1260, Kublai sought to appoint Li to the Hanlin Academy to write historical annals of the (“barbarian”) Liao and Jin dynasties. Li declined the new emperor’s invitation twice by claiming ill health. Nevertheless, he did finally accept in 1264 and participated in the compilation of the official histories for several months. Li strongly criticized the political and intellectual climate of his time, though, and soon used ill health as a pretext to retire and live as a hermit. Before dying at the age of 87, he told his son that all his writings should be burned, except for his mathematical treatise on circle measurements.

Nevertheless, other writings survive, including his mathematical treatise Yigu yanduan (1259; “New Steps in Computation”) and a literary work, Jingzhai gu jin tou (“Jingzhai’s Essay on Past and Present”).