Sea-Mirror of the Circle Measurements

Ceyuan haijing, or "Sea Mirror of Circle Measurements," is a significant mathematical work by Li Ye, composed in 1248. This treatise features 170 problems centered around a geometric diagram of a circular city wall inscribed within a right-angled triangle. Each problem presents a scenario with two individuals walking within the town, aiming to see each other or an object, consistently leading to the calculation of the wall's diameter.1

While the problems are contrived, they allowed Li Ye to compile 692 algebraic formulas for triangular areas and segment lengths. The work utilizes polynomial equations with arbitrary powers of an unknown, referred to as the "celestial unknown". This method, likely flourishing in North China before Li Ye, is the oldest surviving work to explain it, though it was probably not the first to address it. Unfortunately, by the 17th century, this method was completely forgotten in China.123