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Siyuan Yujian in the Joseon Mathematics
홍성사,홍영희,이승온,Hong, Sung Sa,Hong, Young Hee,Lee, Seung On The Korean Society for History of Mathematics 2017 Journal for history of mathematics Vol.30 No.4
As is well known, the most important development in the history of Chinese mathematics is materialized in Song-Yuan era through tianyuanshu up to siyuanshu for constructing equations and zengcheng kaifangfa for solving them. There are only two authors in the period, Li Ye and Zhu Shijie who left works dealing with them. They were almost forgotten until the late 18th century in China but Zhu's Suanxue Qimeng(1299) had been a main reference for the Joseon mathematics. Commentary by Luo Shilin on Zhu's Siyuan Yujian(1303) was brought into Joseon in the mid-19th century which induced a great attention to Joseon mathematicians with a thorough understanding of Zhu's tianyuanshu. We discuss the history that Joseon mathematicians succeeded to obtain the mathematical structures of Siyuan Yujian based on the Zhu's tianyuanshu.
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TianYuanShu and Numeral Systems in Eastern Asia
홍성사,홍영희,이승온,Hong, Sung Sa,Hong, Young Hee,Lee, Seung On The Korean Society for History of Mathematics 2012 Journal for history of mathematics Vol.25 No.4
In Chinese mathematics, there have been two numeral systems, namely one in spoken language for recording and the other by counting rods for computations. They concerned with problems dealing with practical applications, numbers in them are concrete numbers except in the process of basic operations. Thus they could hardly develop a pure theory of numbers. In Song dynasty, 0 and TianYuanShu were introduced, where the coefficients were denoted by counting rods. We show that in this process, counting rods took over the role of the numeral system in spoken language and hence counting rod numeral system plays the role of that for abstract numbers together with the tool for calculations. Decimal fractions were also understood as denominate numbers but using the notions by counting rods, decimals were also admitted as abstract numbers. Noting that abacus replaced counting rods and TianYuanShu were lost in Ming dynasty, abstract numbers disappeared in Chinese mathematics. Investigating JianJie YiMing SuanFa(簡捷易明算法) written by Shen ShiGui(沈士桂) around 1704, we conclude that Shen noticed repeating decimals and their operations, and also used various rounding methods. 중국의 명수법은 기록은 구어체를 사용하고, 계산은 산대를 사용하는 이중 구조를 가지고 있었다. 또 산서는 실생활의 문제만 다루는 과정에서 수학적 구조를 나타내는 방법을 택하여 계산 과정을 제외하면 이들에서 취급한 수는 모두 명수(名數)들이어서 순수한 수론의 발전을 이룰 수 없었다. 송대에 0의 도입과 함께, 천원술의 표현에서 나타나는 계수를 산대로 표시하는 방법을 통하여, 산대가 계산 도구와 함께 추상수의 기수법(記數法)이 되는 과정을 밝힌다. 수량의 단위를 사용한 소수의 표현도 이 과정에서 산대 표현으로 대치되었다. 그러나 명대에 산대 계산이 주산으로 대치되고 천원술이 잊히게 되어 추상수의 개념도 함께 잊히게 되었다. 청대의 산학자 심사계(沈士桂)가 저서 간첩이명산법(簡捷易明算法)에서 분수의 소수표시와 계산을 하는 과정에서 순환소수를 인지하고 이들의 계산법을 확립한 것도 보인다.
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Zeros of Polynomials in East Asian Mathematics
홍성사,홍영희,김창일,Hong, Sung Sa,Hong, Young Hee,Kim, Chang Il The Korean Society for History of Mathematics 2016 Journal for history of mathematics Vol.29 No.6
Since Jiuzhang Suanshu, mathematical structures in the traditional East Asian mathematics have been revealed by practical problems. Since then, polynomial equations are mostly the type of $p(x)=a_0$ where p(x) has no constant term and $a_0$ is a positive number. This restriction for the polynomial equations hinders the systematic development of theory of equations. Since tianyuanshu (天元術) was introduced in the 11th century, the polynomial equations took the form of p(x) = 0, but it was not universally adopted. In the mean time, East Asian mathematicians were occupied by kaifangfa so that the concept of zeros of polynomials was not materialized. We also show that Suanxue Qimeng inflicted distinct developments of the theory of equations in three countries of East Asia.
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김옥자,김영옥,Jin, Yuzi,Kim, Young Wook 한국수학사학회 2015 Journal for history of mathematics Vol.28 No.3
This survey is the second part of the history of science of Song and Yuan dynasties and will covers the period from Jin to Yuan. Following the first part, we look at the calendrical astronomy, mathematics and medicine. In this survey we again follow Yabuuchi's work on the history of science of Song and Yuan period and Du Shiran's work on the history of science of China. We start from the sciences and mathematics of Jin which inherited those of Northern Song and see how they influenced the whole China including Yuan and Southern Song. As a conclusion the tendency to practical usages in the Southern Song as well as the suppression of Han people in Yuan prevented developments of theoretical sciences in Yuan and Ming later.
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郭书春 한국수학사학회 2016 Journal for history of mathematics Vol.29 No.6
Chinese Mathematics during the period of Jin (1115--1234) and Yuan (1271--1368) is an integral part of the high achievements of traditional mathematics during the Song (962--1279) and Yuan dynasties, which is another peak in the history of Chinese mathematics, following the footsteps of the high accomplishments during the Warring States period (475--221 BCE), the Western Han (206 BCE--24 ADE), Three Kingdoms (220--280 AD), Jin dynasty (265--420 AD), and Southern and Northern Dynasties (420--589 AD). During the Jin-Yuan period, Quanzhen Taoism was a dominating branch in Taoism. It offered certain political protection and religious comforts to many during troubled times; it also provided a relatively stable environment for intellectual development. Li Ye (1192--1279), Zhu Shijie (fl.\ late 13th C to early 14th C) and Zhao Youqin (fl. late 13th C to early 14th C), the major actors and contributors to the Jin-Yuan Mathematics achievements, were either heavily influenced by the philosophy of Quanzhen Taoism, or being its followers. In certain Taoist Classics, Li Ye read the records of the relations of a circle and nine right triangles which has been known as Dongyuan jiurong 洞渊九容 of Quanzhen Taoism. These relations made significant contributions in the study of the circles inscribed in a right triangle, the reasoning of which directly led to the birth of the Method of Celestial Elements (Tianyuan shu天元术), which further developed into the Method of Two Elements (Eryuan shu 二元术), the Method of Three Elements (Sanyuan shu 三元术) and the Method of Four Elements (Siyuan shu 四元术). 金元数学是中国传统数学继战国西汉、魏晋南北朝两个高潮之后宋元数学高潮的重要部分。全真道是金元时期盛行的道教教派,它在乱世对人们起到政治庇护与宗教慰藉的作用,也为数学家提供了相对稳定的学术环境。金元数学成就的创造者、发展者李冶、朱世杰、赵友钦等或者受到全真道的极大影响,或者就是全真道教徒。全真道的洞渊九容,在勾股容圆研究中做出重大贡献,其思想直接催生了天元术,并发展为二元术、三元术和四元术。
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