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The History of the Binomial Coefficients in China

    Three names from China figure prominently in the story of the discovery of the binomial coefficients in that culture.  The first is Chia Hsien (circa 1100 A.D.)  who wrote of a method he  called “the tabulation system for unlocking binomial coefficients” in a book called Shih-so suan-shu.  This book is credited by Yanghui (1261), who listed the coefficients of (a+b)¹ through (a+b)⁶; about forty years later, Chu Shih-chieh (1303) gives credit to Yanghui in his book Precious Mirror of the Four Elements, where he continues the expansions out to (a+b)⁸ in an attractive chart preserved to this day, though as Edwards points out, on this list , where the first is written incorrectly as 34 and the second as 35, though it is hard to make out in the digital reproduction below.  Given the drawing, Edwards states that the Chinese obviously understood the main additive rule of  , but there is no clear cut evidence they understood the main multiplicative rule of .

    In Chinese mathematical literature to this day, they call the array of numbers Yanghui’s Triangle.

                                                   


This image reproduced from A.W.F. Edwards' book Pascal's Arithmetical Triangle by the kind permission of Johns Hopkins University Press.