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Yang Hui's NaYinFa  

Hong, Sung-Sa (Department of Mathematics, Sogang University)
Hong, Young-Hee (Department of Mathematics, Sookmyung Women's University)
Lee, Seung-On (Department of Mathematics, Chungbuk National University)
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Journal for History of Mathematics / v.24, no.3, 2011 , pp. 1-12 More about this Journal
It is well known that the sexagesimal cycle(干支) has been playing very important role in ordinary human affairs including astrology and almanacs and the arts of divination(術數). Yin-Yang school related the cycle with the sixty four hexagrams and the system of five notes(五音) and twelve pitch-pipes(十二律), and the processes to relate them are called respectively NaJia(納甲) and NaYin(納音) and quoted in Shen Kuo's Meng qi bi tan(夢溪筆談, 1095). Yang Hui obtained the process NaYin in the context of mathematics. In this paper we show that Yang Hui introduced the concept and notion of functions and then using congruences and the composite of functions, he could succeed to describe perfectly the process in his Xu gu zhai qi suan fa(續古摘奇算法, 1275). We also note that his concept and notion of functions are the earliest ones in the history of mathematics.
five notes; twelve pitch-pipes; sexagesimal cycle; NaYin(納音); Shen Kuo(沈括); Meng qi bi tan(夢溪筆談, 1095); Yang Hui(楊輝); Xu gu zhai qi suan fa(續古摘奇算法, 1275); functions; congruences;
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