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Mathematical Treasure: The Precious Mirror of Zhu Shijie

Frank J. Swetz (The Pennsylvania State University)

Zhu Shijie (circa 1260-1320) was the preeminent algebraist of ancient China. His most famous work is his Siyuan yujian (Precious [or Jade] Mirror of the Four Elements) from 1303. In this text, he devised a method for solving a system of higher order polynomial equations containing as many as four unknowns. Here are pages from a 19th century copy of this algebra classic:

On the left page is a matrix as used in computing by rod calculation to solve a system of equations. Using a modern form of unknowns, X, Y, Z, and W, the lettered entries in the cells indicate the placement of the numerical coefficients [rods] for the terms given.


The righthand page contains a diagram, perhaps recognized by a modern western reader as “Pascal’s triangle”; however, the Precious Mirror was written in 1303, long before Pascal lived, and credits this diagram to “ancient techniques.” Indeed, the diagram and mathematical intent can be traced back to the 11th century Chinese mathematician Jia Xian (circa 1010-1070). The diagram passed from China to India, then via Arab sources to Europe by the 16th century.

For a detailed discussion of the mathematics of the Precious Mirror, see: Jock Hoe, The Jade Mirror of the Four Unknowns, Christchurch, NZ: Mingming Bookroom, 2007.

The images shown here were obtained through the courtesy of Wikimedia.











Index to Mathematical Treasures

Frank J. Swetz (The Pennsylvania State University), "Mathematical Treasure: The Precious Mirror of Zhu Shijie," Convergence (June 2018)