Greek Texts and the Rigorization of Analysis:An Inquiry into J.L.Lagrange's Work on the History of Mathematics

This paper offers a study of J.L.Lagrange's research on history of mathematics,aiming to clarify Lagrange's intention in carrying out historical work.To this end,I first document how Lagrange worked with and exerted his influence on other scholars in the translation and diffusion of ancient Greek texts.Second,investigating Lagrange's style in doing and writing history of mathematics,this paper takes a new perspective and elucidates his motivation in these activities.In particular,it focuses on Lagrange's presentation of the history of calculus while he was teaching analysis at the Ecole Polytechnique(1795-1799)so as to clarify the function of history in Lagrange's mathematical works.My thesis is that Lagrange's intention in examining the different methods employed by his predecessors was to find inspiration and useful contents in his search for the proper approach to mathematical problems.I thus argue in this paper that history served as a guide or methodology for Lagrange's mathematics.Meanwhile,through an analysis of his historical writing,this paper points to four epistemological values according to which Lagrange judged various historical methods of differential calculus:generality,simplicity,clarity,and rigor.Lagrange's move to rigorize analysis was connected to his interest in and research of Greek texts;he was attempting to introduce the rigor of the ancient Greeks'demonstration in his works of analysis.

ANCIENT CHINESE ALGORITHM: THE YING BUZU SHU (METHOD OF SURPLUS AND DEFICIENCY)VS NEWTON ITERATION METHOD

Air exploratory discussion of an ancient Chinese algorithm, the Ying Buzu Shu, in about 2nd century BC, known as the rule of double false position in the West is given. In addition to pointing out that the rule of double false position is actually a translation version of the ancient Chinese algorithm, a comparison with well-known Newton iteration method is also made. If derivative is introduced, the ancient Chinese algorithm reduces to the Newton method. A modification of the ancient Chinese algorithm is also proposed, and some of applications to nonlinear oscillators are illustrated.

Preliminary Research on the Mathematical Methods and Order of Problems in the“Fungcheng”Chapter of Yang Hui's Mathematical Methods(1261 CE)

Yang Hui was one of the most important authors of mathematical works during the thirteenth century.Mathematical Methods Explaining in Detail The Nine Chapters(Xiangjie jiuzhang suanfa詳解九章算法,1261 CE)is the earliest extant work attributed to Yang Hui.From the thirteenth to the fifteenth century,this work played a crucial role in the circulation and popularization of The Nine Chapters on Mathematical Procedures(Jiuzhang suanshu九章算术).However,the only surviving printed edition of Mathematical Methods is incomplete and contains many mistakes obstructing contemporary researchers'understanding of this work.The "Fangcheng" chapter of The Nine Chapters deals with problems related to solving what today are known as simultaneous sets of linear equations.However,interpreting the text in this chapter of Mathematical Methods and recovering the mathematical practices relating to fangcheng are difficult.Through detailed textual and mathematical analyses,the author of this paper explains Yang Hufs understanding and practice relating to〃the fangcheng method"and"the method of the positive and the negative".This paper includes an appendix that provides a detailed translation of the ambiguous text relating to"the method of the positive and the negative"and gives reasons supporting the interpretation provided here.Yang Hufs understanding of the concepts of"positive"and"negative"and his practice relating to these two concepts may easily be confused with their apparent counterparts in modem mathematics.Also,careful analysis of the mathematical methods in this work reveal that the order of problems in Yang Hufs Reclassifications of Mathematical Methods Explaining in Detail The Nine Chapters([Xiangjie jiuzhang suanfa zuanlei詳解九章算法纂類],namely,the last section of Mathematical Methods)were rearranged according to commentaries to specific methods that appear in Mathematical Methods.Some textual clues referring to the zzprevious question"(qianwen前問)in certain commentaries of Mathematical Methods indeed reflect the order of problems in Reclassifications.Yang Hui made especially detailed commentaries on the problems that he arranged in a sequence that differs with respect to the original order of problems as they appear in the ancient classic work,The Nine Chapters.All these discoveries reveal and serve to prove a close relationship between Yang Hufs Mathematical Methods and his Reclassifications.