Jade Mirror of the Four Unknowns

Illustrations in Jade Mirror of the Four Unknowns
Jia Xian triangle

Jade Mirror of the Four Unknowns,[1] Siyuan yujian (simplified Chinese: 四元玉鉴; traditional Chinese: 四元玉鑒), also referred to as Jade Mirror of the Four Origins,[2] is a 1303 mathematical monograph by Yuan dynasty mathematician Zhu Shijie.[3] Zhu advanced Chinese algebra with this Magnum opus.

The book consists of an introduction and three books, with a total of 288 problems. The first four problems in the introduction illustrate his method of the four unknowns. He showed how to convert a problem stated verbally into a system of polynomial equations (up to the 14th order), by using up to four unknowns: 天 Heaven, 地 Earth, 人 Man, 物 Matter, and then how to reduce the system to a single polynomial equation in one unknown by successive elimination of unknowns. He then solved the high-order equation by Southern Song dynasty mathematician Qin Jiushao's "Ling long kai fang" method published in Shùshū Jiǔzhāng (“Mathematical Treatise in Nine Sections”) in 1247 (more than 570 years before English mathematician William Horner's method using synthetic division). To do this, he makes use of the Pascal triangle, which he labels as the diagram of an ancient method first discovered by Jia Xian before 1050.

Zhu also solved square and cube roots problems by solving quadratic and cubic equations, and added to the understanding of series and progressions, classifying them according to the coefficients of the Pascal triangle. He also showed how to solve systems of linear equations by reducing the matrix of their coefficients to diagonal form. His methods predate Blaise Pascal, William Horner, and modern matrix methods by many centuries. The preface of the book describes how Zhu travelled around China for 20 years as a teacher of mathematics.

Jade Mirror of the Four Unknowns consists of four books, with 24 classes and 288 problems, in which 232 problems deal with Tian yuan shu, 36 problems deal with variable of two variables, 13 problems of three variables, and 7 problems of four variables.

  1. ^ Hoe, John (1978) The Jade Mirror of the Four Unknowns - Some Reflections. Math. Chronicle 7, p. 125-156.
  2. ^ Hart, Roger (2013). Imagined Civilizations China, the West, and Their First Encounter. Baltimore, MD: Johns Hopkins Univ Pr. p. 82. ISBN 978-1421406060.
  3. ^ Elman, Benjamin A. (2005). On their own terms science in China, 1550-1900. Cambridge, Mass.: Harvard University Press. p. 252. ISBN 0674036476.