• Journal for History of Mathematics
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• v.31 no.6
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• pp.273-289
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• 2018

The KaiFangFa開方法 of traditional mathematics was completed in ${\ll}$JiuZhang SuanShu九章算術${\gg}$ originally, and further organized in Song宋 $Yu{\acute{a}}n$元 dinasities. The former is the ShiSuoKaiFangFa釋鎖開方法 using the coefficients of the polynomial expansion, and the latter is the ZengChengKaiFangFa增乘開方法 obtaining the solution only by some mechanical numerical manipulations. ${\ll}$SanHak JeongEui算學正義${\gg}$ basically used the latter and improved the estimate-value array by referring to the written-calculation in ${\ll}$ShuLi JingYun數理精蘊${\gg}$. As a result, ZengChengKaiFangFa was more refined so that the KaiFangFa algorithm is more consistent.

• Journal for History of Mathematics
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• v.26 no.4
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• pp.213-218
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• 2013

This paper is a sequel to the paper [8], where we discussed the connection between ShiShou KaiFangFa originated from JiuZhang SuanShu and ZengCheng KaiFangFa. Investigating KaiFangShu in a Chosun mathemtics book, SanHak JeongEui and ShuLi JingYun, we show that its authors, Nam ByungGil and Lee SangHyuk clearly understood the connection and gave examples to show that the KaiFangShu in the latter is not exact. We also show that Chosun mathematicians were very much selective when they brought in Chinese mathematics.

• Journal for History of Mathematics
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• v.22 no.4
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• pp.29-40
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• 2009

we know that Zeng Cheng Kai Fang Fa is the generalization of the method of square roots and cube roots of ancient through the investigation of China mathematics. In this paper, we have research on traditional solutions equations of China mathematics and the development solutions of equations used by Chosun mathematicians.

• Journal for History of Mathematics
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• v.24 no.1
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• pp.1-13
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• 2011

In Tian mu bi lei cheng chu jie fa(田畝比類乘除捷法) of Yang Hui suan fa(楊輝算法)), Yang Hui annotated detailed comments on the method to find roots of quadratic equations given by Liu Yi in his Yi gu gen yuan(議古根源) which gave a great influence on Chosun Mathematics. In this paper, we show that 'Zeng cheng kai fang fa'(增乘開方法) evolved from a process of binomial expansions of $(y+{\alpha})^n$ which is independent from the synthetic divisions. We also show that extending the results given by Liu Yi-Yang Hui and those in Suan xue qi meng(算學啓蒙), Chosun mathematican Hong Jung Ha(洪正夏) elucidated perfectly the 'Zeng cheng kai fang fa' as the present synthetic divisions in his Gu il jib(九一集).

• Journal for History of Mathematics
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• v.18 no.3
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• pp.39-54
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• 2005

In Chinese Mathematics, Jia Xian(要憲) introduced Zeng cheng kai fang fa(增乘開方法) to get approximations of solutions of Polynomial equations which is a generalization of square roots and cube roots in Jiu zhang suan shu. The synthetic divisions in Zeng cheng kai fang fa give ise to two concepts of Fan il(飜積) and Yi il(益積) which were extensively used in Chosun Dynasty Mathematics. We first study their history in China and Chosun Dynasty and then investigate the historical fact that Chosun mathematicians Nam Byung Gil(南秉吉) and Lee Sang Hyuk(李尙爀) obtained the sufficient conditions for Fan il and Yi il for quadratic equations and proved them in the middle of 19th century.