Yang Hui suan-fa (English spelling)

Japanese: 楊輝算法 - ようきさんぽう（英語表記）Yang Hui suan-fa

A general term for Yang Hui's later mathematical works. The works that fall under this name are Suanbao (Multiplication, Division, and Transformation) (3 volumes: Suanfa Tongbian Honmatsu, Suanbao Tongbian Honmatsu, and Suanfa Toyo Honmatsu; the last volume was co-authored with Shi Zhongrong in 1274), Tianyuan Birui Mutai Shifu (Two volumes, 75), and Zhou Gu Zhiqi Quanfa (Continued Ancient and Eccentric Calculations) (2 volumes, 75). In Suanyuan Birui Mutai Shifu, he uses Liu Yi's Yi Gu Genyuan to present a calculation method similar to Horner's method, introduces the work of his predecessors on numerical equations, and studies problems in finding the sum of series under the name of multiplication. In the "Continued Ancient and Magical Calculations," many squares are dealt with, including the diagram from Hetu Luoshu, the four-order square "Sixteen Flowers," the ten-order square "Hundred Children," and the magic circle square, and linear congruence equations are discussed under the name of "Senkanjutsu." It was from "Yang Hui Suanfa" that Seki Takakazu learned about this type of indeterminate equation along with squares.

Source: Encyclopaedia Britannica Concise Encyclopedia About Encyclopaedia Britannica Concise Encyclopedia Information

Japanese:

楊輝の後期の数学書の総称。この名でまとめられるのは，『乗除通変算宝』 (「算法通変本末」「乗除通変算宝」「法算取用本末」の3巻。最後の巻は史仲栄との合撰。 1274) ，『田畝比類乗除捷法』 (2巻，75) ，『続古摘奇算法』 (2巻，75) である。『田畝比類乗除捷法』では劉益の『議古根源』によってホーナーの方法と似た計算法を述べ，数字方程式に関する先人の研究を紹介し，また級数求和法の問題をだ積 (だせき) 術の名のもとに研究した。『続古摘奇算法』には，河図洛書の図をはじめ，4次の方陣「花十六図」，10次の「百子図」，また円形に配列された方陣 magic circleなど多くの方陣が取扱われ，翦管 (せんかん) 術という名称で1次合同式を論じている。関孝和が方陣とともにこの種の不定方程式を知ったのは『楊輝算法』からであった。

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