Ceyuan haijing

Ceyuan haijing (1248), meaning "Sea Mirror of Circle Measurements," is a significant mathematical work by Chinese mathematician Li Ye. This treatise contains 170 problems derived from a single geometric diagram of a circular city wall circumscribed by a right-angled triangle. The problems, though contrived, led Li Ye to list 692 algebraic formulas for triangular areas and segment lengths.1

The work is notable for its use of the "method of the celestial unknown," a technique for solving polynomial equations in one variable using polynomial arithmetic. This method, which Li Ye's book is the oldest surviving record of, involves representing polynomials with a positional notation where the coefficient's location indicates the power of the unknown. This approach was a significant development in Chinese algebra, though the method was later forgotten and only deciphered again in the 18th century.123