| The power series expansion of trigonometric functions,which also called Shortcut method of cutting circle in the Qing Dynasty,was a hot topic for Chinese mathematicians from the middle of Qing Dynasty to the end of Qing Dynasty.From the middle of 18th century to the middle of 19th century,Chinese mathematicians such as Ming An-tu,Dong You-cheng,Xiang Ming-da,Xu You-ren,Dai Xu and Li Shan-lan were stimulated by the introduction of"Du Shi San Shu"in the early Qing Dynasty,discussed the power series expansion of trigonometric function by using the method of geometrical demonstration,and formed a unique research system.Ming An-tu added six new power series expansions to the"Du Shi San Shu(????)",which were collectively known as"Du Shi Jiu Shu(????)",becoming the source of knowledge for scholars in the late Qing Dynasty to study the power series expansion of trigonometric function.After calculus was introduced into China(1859),Xia Luan-xiang,Yang Zhi-pei,Ling Bu-fang,Huang Qi-ming and other Chinese mathematicians used the calculus method to deduce the previous results about power series expansion of trigonometric function,which promoted the Westernization process of this field.Among them,The"Dudemei Geyuan Jieshu Tong Yi?????????"of Ling Bu-fang(1849-1902)is a systematic and comprehensive work to solve and promote power series expansion of trigonometric functions by using methods of calculus,series inversion,ect.The development of power series expansion of trigonometric function in the late Qing Dynasty is one of the important topics for contemporary scholars to study the history of mathematics in the late Qing Dynasty.However,the current literature has few studies on the development of the power series expansion of trigonometric function by Chinese mathematicians after calculus was introduced into China in 1859.It is also rare to study Ling Bu-fang and his "Dudemei Geyuan Jieshu Tong Yi".Therefore,this paper,using literature method and comparative method,discusses the problem of how Ling Bu-fang used calculus to deduce and promote the "Du Shi Jiu Shu",and compares Ling Bu-fang's achievements with those of Xia Luan-xiang,Yang Zhi-pei and Huang Qi-ming,who used calculus to study the power series expansion of trigonometric function in the late Qing Dynasty.The research of this paper concludes as follows:(1)Ling Bu-fang's "Dudemei Geyuan Jieshu Tong Yi" uses calculus,path borrowing method,series self multiplication and series back seeking method to attribute the "Du Shi Jiu Shu" to "Seeking arc by sine",and puts forward thirty-three new power series expansions of trigonometric functions according to the "Du Shi Jiu Shu",which improves the research system of Du's series.Thirty-three new expansions are mainly based on "Du Shi Jiu Shu",which are derived from series self multiplication and path borrowing methods.They focus on improving the problem of relation between circle and diameter,relation between arc and eight lines,and relation between chords?arrows and square of arc.Among them,the 11 th,27th,28 th,29th,31 st,32nd and 33 rd expansions are wrong.In the expression of power series expansion,the book not only inherits Dong You-cheng's progressive language,but also absorbs Li Shan-lan's Chinese translation symbol system.(2)By comparing Ling Bu-fang with Xia Luan-xiang,Yang Zhi-pei,Huang Qi-ming and others in solving the power series expansion of trigonometric function by calculus,we find that,Xia Luan-xiang mainly used the method of the "Di jia shu" combined with term by term integration.Yang Zhi-pei and Huang Qi-ming transformed Mclaurin formula into a method that conforms to Chinese reading habits,and then used it to solve the power series expansion of trigonometric function.Jiang Shi-dong directly included the relevant contents of "Daiweiji Shiji" and "Weiji Suyuan",without personal opinion and promotion.Ling Bu-fang imitated the calculation process of power series expansion of trigonometric function in the "Weiji Suyuan",and deduced it through term by term integration,Mclaurin formula and Taylor formula.In addition,he also clarified the relationship between the results obtained by formula method(Du's changed expansion)and those obtained by Ming An-tu and other Chinese Mathematicians(Du's original expansion),showing the similarities and differences between Chinese and Western researches on the power series expansion of trigonometric function at that time.(3)Ling Bu-fang has a positive response to the convergence of series.The 9th expansion of "Seeking cosine by arc " in the second volume of "Dudemei Geyuan Jieshu Tong Yi" involves the convergence rate of series,and the 22 nd,23rd and 30 th expansion involve the problem of convergence interval of series.In particular,the 22 nd expansion gives the convergence interval of the power series expansion of arctangent function is [ 1,0],and the30 th expansion gives the convergence interval of the expansion of tangent function is 0?x??/4.In a word,"Dudemei Geyuan Jieshu Tong Yi" is a rare monograph in the late Qing Dynasty,which studies the power series expansion of trigonometric function by combining Chinese and Western methods.In addition,it is also a rare work of Applied Calculus,which is different from most of the works focusing on the interpretation of the "Daiweiji Shiji" and the "Weiji Suyuan" at that time.It not only reflects the degree and level of Ling Bu-fang's application of calculus,but also reflects the characteristics of the integration of Chinese and Western mathematics,which can be used as a typical case to investigate the spread and influence of western modern mathematics in China in the late Qing Dynasty. |
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