著者

中根 美知代

出版者

サイエンス社

雑誌

数理科学 (ISSN:03862240)

巻号頁・発行日

vol.59, no.5, pp.47-54, 2021-05

著者

中根 美知代

出版者

科学基礎論学会

雑誌

科学基礎論研究 (ISSN:00227668)

巻号頁・発行日

vol.35, no.1, pp.21-28, 2007-12-25 (Released:2010-02-03)

参考文献数

20

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Historians of mathematics often mention that Cauchy's Cours d'Analyse (1821) brought a “rigor revolution” to analysis. Since the notion of rigor occasionally appears in the history of mathematics, it is essential to characterize Cauchy's “rigorous” attitude. When Cauchy encountered special examples that didn't satisfy general rules, theories, or formulas, he modified the latter to accommodate the former. This attitude was quite innovative because eighteenth century mathematicians generally neglected such examples and kept the general theories. After Cauchy, nineteenth century mathematicians refined their arguments when they found counterexamples to their theories. The change of treatment of counterexamples is an essential factor of Cauchy's rigor revolution.

著者

ヴィクター J. カッツ著 中根美知代 [ほか] 翻訳

出版者

共立出版

巻号頁・発行日

2005

著者

中根 美知代

出版者

科学基礎論学会

雑誌

科学基礎論研究 (ISSN:00227668)

巻号頁・発行日

vol.28, no.1, pp.31-36, 2000-12-25 (Released:2009-07-23)

参考文献数

8

著者

中根 美知代

出版者

日本科学史学会

雑誌

科学史研究. 第II期 (ISSN:00227692)

巻号頁・発行日

vol.34, no.193, pp.44-46, 1995-04-07

著者

中根 美知代

出版者

日本科学史学会

雑誌

科学史研究 (ISSN:21887535)

巻号頁・発行日

vol.51, no.261, pp.10-17, 2012 (Released:2021-07-20)

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Introduced by Karl Schwarzschild in 1916, action-angle variables provided effective mathematical tools with which to examine quantum phenomena. No historical work describes clearly how Schwarzschild came by the idea of them. This paper shows that the original idea of action-angle variables was substantially outlined in Charlier's two-volume textbook. Mechanics of Heaven, published in 1902 and 1907. In Volume 2. Charlier extended Jacobi's results and examined a leading function of a canonical transformation. He showed that a complete solution of the Hamilton-Jacobi equation of an intermediate orbit for given canonical equations becomes the leading function. The Hamilton-Jacobi equation for any intermediate orbit was found to be solvable. Charlier was then able to actually perform the canonical transformation, attaining new canonical variables that involved arbitrary constants of the solution to the equation of the intermediate orbits. He related the arbitrary constants to original canonical variables and changed the new canonical variables into new ones (ξ_i, η_i), that depend on an intermediate orbit. In this process, he used Stackers results as demonstrated in Volume 1. Charlier showed that if a Keplerian ellipsis is taken as the intermediate orbit. ξ_1 becomes an element of action integral multiplied by 1/π and η_i=η_it + β_i an argument of angle of a moving point, where n_i is frequency, t is time, and β_i is an arbitrary constant. Schwarzschild noted this fact and thereby attained his formal definition of action-angle variables.

著者

中根 美知代 佐藤 賢一 大谷 卓史 小山 俊士 村上 祐子

出版者

日本科学史学会

雑誌

科学史研究 (ISSN:21887535)

巻号頁・発行日

vol.49, no.256, pp.232-246, 2010 (Released:2021-07-22)

著者

中根 美知代

出版者

日本科学史学会

雑誌

科学史研究 (ISSN:21887535)

巻号頁・発行日

vol.48, no.251, pp.142-151, 2009 (Released:2021-08-03)

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This paper clarifies Cauchy's and Weierstrass's contributions to the construction of differential calculus represented in terms of epsilonics. In the eighteenth century the limit concept had a geometrical image that is typically represented in >indefinitely approaching to a fixed value>. In 1820s Cauchy described this concept in terms of inequalities and defined the limit. Since his new calculus theory was based on this concept, he could transform previous results from calculus to his new theory developed only by algebraic techniques. He also defined his original concept of infinitesimals based on the limit concept. The relations between the infinitesimals and infinitely large numbers or infinitesimally small changes can be represented in term of epsilon-delta inequalities. Although Cauchy occasionally used the term of infinitesimals in the usual sense, he substantially developed his calculus theory in epsilonics using his infinitesimals. Weierstrass noted the differential calculus needs to apply neither Cauchy's limit nor infinitesimals, but the relations that involve them. Neither isolated limits nor infinitesimals can be written in terms of epsilon-delta inequalities, but their relations can. Weierstrass began his 1861 lectures on the differential calculus by defining the fundamental concepts in terms of epsilon-delta inequalities. His original limit concept was also defined in terms of these, without any geometrical image. In contrast to Cauchy, Weierstrass's theory was pure algebraic and had no geometrical background. Although both mathematicians basically developed their differential calculus in epsilonics, the essential difference between their approaches lies in this point.

著者

伊藤 和行 中田 良一 有賀 暢迪 中根 美知代

出版者

日本科学史学会

雑誌

科学史研究 (ISSN:21887535)

巻号頁・発行日

vol.46, no.243, pp.177-189, 2007 (Released:2021-08-09)

著者

中根 美知代

出版者

日本科学史学会

雑誌

科学史研究 (ISSN:21887535)

巻号頁・発行日

vol.42, no.226, pp.112-113, 2003 (Released:2021-08-13)

著者

中根 美知代

出版者

日本科学史学会

雑誌

科学史研究 (ISSN:21887535)

巻号頁・発行日

vol.29, no.173, pp.30-36, 1990 (Released:2021-09-01)

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W. R. Hamilton proposed, in his articles written between 1824 and 1833, the characteristic function to describe the properties of ray systems. It has been usually presumed that he derived the function from the principle of least action. The present paper aims at revealing the historical connection between the above mentioned function and principle. Firstly discussed are the origin and implication of the principle which he called that of least action. He derived it from the law of reflection and refraction ; what he called ACTION in this process did not imply mechanical action but optical path length. Accordingly, the principle should originally be called that of the shortest optical path length. Secondly dealt with are the basis of his characteristic function and the way of its extension. It was introduced, not on the basis of the above-mentioned principle, but through analysis of rays in focal mirrors and focal refractors; before long, he noticed that the function was compatible with the principle ; then, for the purpose of expressing the ray systems in inhomogeneous medium, he utilized a refined form of the function ; besides, he formulated the function on the basis of the principle; finally the principle was accepted as an axiom.

著者

中根 美知代

出版者

日本科学史学会

雑誌

科学史研究 (ISSN:21887535)

巻号頁・発行日

vol.26, no.164, pp.222-226, 1987 (Released:2021-09-21)

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In his early works, W. R. Hamilton set out to describe optical and mechanical systems by a single characteristic function. It is generally accepted that he based his theory on variational principles. In this note the author shows that Hamilton derived the characteristic function V from the law of refraction and reflection in optics, and the equations of motion in mechanics, but not from variational principles. And she also shows that it was crucial for his description of optical-mechanical systems that differentiations of both characteristic functions have the same mathematical form; i.e. exact differential 1-form.

著者

中根 美知代

出版者

京都大学数理解析研究所

雑誌

数理解析研究所講究録 (ISSN:18802818)

巻号頁・発行日

vol.1130, pp.58-71, 2000-02

著者

中根 美知代

出版者

科学基礎論学会

雑誌

科学基礎論研究 (ISSN:00227668)

巻号頁・発行日

vol.40, no.2, pp.117-121, 2013-03-25 (Released:2017-08-01)

著者

中根 美知代

出版者

科学基礎論学会

雑誌

科学基礎論研究 (ISSN:00227668)

巻号頁・発行日

vol.31, no.1-2, pp.1-7, 2004-03-25 (Released:2009-07-23)

参考文献数

15

著者

中根 美知代

出版者

科学基礎論学会

雑誌

科学基礎論研究 (ISSN:00227668)

巻号頁・発行日

vol.25, no.2, pp.105-110, 1998-03-31 (Released:2009-07-23)

参考文献数

8