学霸学习网 这下你爽了
当前位置:首页 >> 其它课程 >>


BSHM Bulletin, 2014 Did Weierstrass’s differential calculus have a limit-avoiding character? His definition of a limit in ? ? ? style MICHIYO NAKANE Nihon University Research Institute of Science & Technology, Japan In the 1820s, Cauchy founded his calculus on his original limit concept and developed his the-ory by using ? ? ? inequalities, but he did not apply these inequalities consistently to all parts of his theory. In contrast, Weierstrass consistently developed his 1861 lectures on differential calculus in terms of epsilonics. His lectures were not based on Cauchy’s limit and are distin-guished by their limit-avoiding character. Dugac’s partial publication of the 1861 lectures makes these differences clear. But in the unpublished portions of the lectures, Weierstrass actu-ally defined his limit in terms of ? ? ? inequalities. Weierstrass’s limit was a prototype of the modern limit but did not serve as a foundation of his calculus theory. For this reason, he did not provide the basic structure for the modern e d style analysis. Thus it was Dini’s 1878 text-book that introduced the definition of a limit in terms of ? ? ? inequalities. Introduction Augustin Louis Cauchy and Karl Weierstrass were two of the most important mathematicians associated with the formalization of analysis on the basis of the e d doctrine. In the 1820s, Cauchy was the first to give comprehensive statements of mathematical analysis that were based from the outset on a reasonably clear definition of the limit concept (Edwards 1979, 310). He introduced various definitions and theories that involved his limit concept. His expressions were mainly verbal, but they could be understood in terms of inequalities: given an e, find n or d (Grabiner 1981, 7). As we show later, Cauchy actually paraphrased his limit concept in terms of e, d, and n0 inequalities, in his more complicated proofs. But it was Weierstrass’s 1861 lectures which used the technique in all proofs and also in his defi-nition (Lutzen 2003, 185–186).

Weierstrass’s adoption of full epsilonic arguments, however, did not mean that he attained a prototype of the modern theory. Modern analysis theory is founded on limits defined in terms of e d inequalities. His lectures were not founded on Cauchy’s limit or his own original definition of limit (Dugac 1973). Therefore, in order to clarify the formation of the modern theory, it will be necessary to identify where the e d definition of limit was introduced and used as a foundation. We do not find the word ‘limit’ in the published part of the 1861 lectures. Accord-ingly, Grattan-Guinness (1986, 228) characterizes Weierstrass’s analysis as limit-avoid-ing. However, Weierstrass actually defined his limit in terms of epsilonics in the unpublished portion of his lectures. His theory involved his limit concept, although the concept did not function as the foundation of his theory. Based on this discovery, this paper re-examines the formation of e d calculus theory, noting mathematicians’ treat-ments of their limits. We restrict our attention to the process of defining continuity and derivatives. Nonetheless, this focus provides sufficient information for our purposes. First, we confirm that epsilonics arguments cannot represent Cauchy’s limit, though they can describe relationships that involved his limit concept. Next, we examine how Weierstrass constructed a novel analysis theory which was not based

2013 British Society for the History of Mathematics 52 BSHM Bulletin

on Cauchy’s limits but could have involved Cauchy’s results. Then we confirm Weierstrass’s definition of limit. Finally, we note that Dini organized his analysis textbook in 1878 based on analysis performed in the e d style. Cauchy’s limit and epsilonic arguments Cauchy’s series of textbooks on calculus, Cours d’analyse (1821), Resume des lecons? donnees a l’Ecole royale polytechnique sur le calcul infinitesimal tome

premier (1823), and Lecons? sur le calcul differentiel (1829), are often considered as the main referen-ces for modern analysis theory, the rigour of which is rooted more in the nineteenth than the twentieth century. At the beginning of his Cours d’analyse, Cauchy defined the limit concept as follows: ‘When the successively attributed values of the same variable indefinitely approach a fixed value, so that finally they differ from it by as little as desired, the last is called the limit of all the others’ (1821, 19; English translation from Grabiner 1981, 80). Starting from this concept, Cauchy developed a theory of continuous func-tions, infinite series, derivatives, and integrals, constructing an analysis based on lim-its (Grabiner 1981, 77). When discussing the evolution of the limit concept, Grabiner writes: ‘This concept, translated into the algebra of inequalities, was exactly what Cauchy needed for his calculus’ (1981, 80). From the present-day point of view, Cauchy described rather than defined his kinetic concept of limits. According to his ‘definition’— which has the quality of a translation or description—he could develop any aspect of the theory by reducing it to the algebra of inequalities. Next, Cauchy introduced infinitely small quantities into his theory. ‘When the successive absolute values of a variable decrease indefinitely, in such a way as to become less than any given quantity, that variable becomes what is called an infinitesimal. Such a variable has zero for its limit’ (1821, 19; English translation from Birkhoff and Merzbach 1973, 2). That is to say, in Cauchy’s framework ‘the limit of variable x is c’ is intuitively understood as ‘x indefinitely approaches c’, and is represented as ‘jx cj is as little as desired’ or ‘jx cj is infinitesimal’. Cauchy’s idea of defining infinitesimals as variables of a special kind was original, because Leibniz and Euler, for example, had treated them as constants (Boyer 1989, 575; Lutzen 2003, 164). In Cours d’analyse Cauchy at first gave a verbal definition of a continuous function. Then, he rewrote it in terms of infinitesimals:

[In other words,] the function f ?x? will remain continuous relative to x in a given interval if (in this interval) an infinitesimal increment in the variable always produces an infinitesimal increment in the function itself. (1821, 43; English translation from Birkhoff and Merzbach 1973, 2). He introduced the infinitesimal-involving definition and adopted a modified version of it in Resume (1823, 19–20) and Lecons? (1829, 278). Following Cauchy’s definition of infinitesimals, a continuous function can be defined as a function f ?x? in which ‘the variable f ?x ? a? f ?x? is an infinitely small quantity (as previously defined) whenever the variable a is, that is, that f ?x ? a? f ?x? approaches to zero as a does’, as noted by Edwards (1979, 311). Thus, the definition can be translated into the language of e d inequalities from a modern viewpoint. Cauchy’s infinitesimals are variables, and we can also take such an interpretation. Volume 29 (2014) 53

Cauchy himself translated his limit concept in terms of e d inequalities. He changed ‘If the difference f ?x ? 1? f ?x? converges towards a certain limit k, for increasing values of x, (. . .)’ to ‘First suppose that the quantity k has a finite value, and denote by e a number as small as we wish. . . . we can give the number h a value large enough that, when x is equal to or greater than h, the difference in question is always contained between the limits k e; k ? e’ (1821, 54; English translation from Bradley and Sandifer 2009, 35). In Resume , Cauchy gave a definition of a derivative: ‘if f ?x? is continuous, then its derivative is the limit of the difference quotient,

?y f ( x ? i ) ? f ( x ) ? ?x i
as i tends to 0’ (1823, 22–23). He also translated the concept of derivative as follows: ‘Designate by d and e two very small numbers; the first being chosen in such a way that, for numerical values of i less than d, [. . .], the ratio f ?x ? i? f

?x?=i always remains greater than f ’?x ? e and less than f ’?x? ? e’ (1823, 44–45; English transla-tion from Grabiner 1981, 115). These examples show that Cauchy noted that relationships involving limits or infinitesimals could be rewritten in term of inequalities. Cauchy’s arguments about infinite series in Cours d’analyse, which dealt with the relationship between increasing numbers and infinitesimals, had such a character. Laugwitz (1987, 264; 1999, 58) and Lutzen (2003, 167) have noted Cauchy’s strict use of the e N characterization of convergence in several of his proofs. Borovick and Katz (2012) indicate that there is room to question whether or not our representation using e d inequalities conveys messages different from Cauchy’s original intention. But this paper accepts the inter-pretations of Edwards, Laugwitz, and Lutzen. Cauchy’s lectures mainly discussed properties of series and functions in the limit process, which were represented as relationships between his limits or his infinitesimals, or between increasing numbers and infinitesimals. His contemporaries presum-ably recognized the possibility of developing analysis theory in terms of only e, d, and n0 inequalities. With a few notable exceptions, all of Cauchy’s lectures could be rewrit-ten in terms of e d inequalities. Cauchy’s limits and his infinitesimals were not func-tional relationships,1 so they were not representable in terms of e d inequalities. Cauchy’s limit concept was the foundation of his theory. Thus, Weierstrass’s full epsilonic analysis theory has a different foundation from that of Cauchy.

Weierstrass’s 1861 lectures Weierstrass’s consistent use of e d arguments Weierstrass delivered his lectures ‘On the differential calculus’ at the Gewerbe Insti-tut Berlin2 in the summer semester of 1861. Notes of these lectures were taken by

1Edwards (1979, 310), Laugwitz (1987, 260–261, 271–272), and Fisher (1978, 16 –318) point out that Cauchy’s infinitesimals equate to a dependent variable function or a?h? that approaches zero as h ! 0. Cauchy adopted the latter infinitesimals, which can be written in terms of e d arguments, when he introduced a concept of degree of infinitesimals (1823, 250; 1829, 325). Every infinitesimal of Cauchy’s is a vari-able in the parts that the present paper discusses. 2A forerunner of the Technische Universitat Berlin. 54 BSHM Bulletin

Herman Amandus Schwarz, and some of them have been published in the original German by Dugac (1973). Noting the new aspects related to foundational concepts in analysis, full e d definitions of limit and continuous function, a new definition of derivative, and a modern definition of infinitesimals, Dugac considered that the nov-elty of Weierstrass’s lectures was incontestable (1978, 372, 1976, 6–7).3 After beginning his lectures by defining a variable magnitude, Weierstrass gave the definition of a function using the notion of correspondence. This brought him to the following important definition, which did not directly appear in Cauchy’s theory: (D1) If it is now possible to determine for h a bound d such that for all values of h which in their absolute value are smaller than d, f ?x ? h? f ?x? becomes smaller than any magnitude e, however small, then one says that infinitely small changes of the argument correspond to infinitely small changes of the function. (Dugac 1973, 119; English translation from Calinger 1995, 607) That is, Weierstrass defined not infinitely small changes of variables but ‘infinitely small changes of the arguments correspond(ing) to infinitely small changes of function’ that were presented in terms of e d inequalities. He founded his theory on this correspondence. Using this concept, he defined a continuous function as follows:

(D2) If now a function is such that to infinitely small changes of the argument there correspond infinitely small changes of the function, one then says that it is a continuous function of the argument, or that it changes continuously with this argument. (Dugac 1973, 119–120; English translation from Calinger 1995, 607) So we see that in accordance with his definition of correspondence, Weierstrass actually defined a continuous function on an interval in terms of epsilonics. Since (D2) is derived by merely changing Cauchy’s term ‘produce’ to ‘correspond’, it seems that Weierstrass took the idea of this definition from Cauchy. However, Weierstrass’s definition was given in terms of epsilonics, while Cauchy’s definition can only be interpreted in these terms. Furthermore, Weierstrass achieved it without Cauchy’s limit. Luzten (2003, 186) indicates that Weierstrass still used the concept of ‘infinitely small’ in his lectures. Until giving his definition of derivative, Weierstrass actually continued to use the term ‘infinitesimally small’ and often wrote of ‘a function which becomes infinitely small with h’. But several instances of ‘infinitesimally small’ appeared in forms of the relationships involving them. Definition (D1) gives the rela-tionship in terms of e d inequalities. We may therefore assume that Weierstrass’s lectures consistently used e d inequalities, even though his definitions were not directly written in terms of these inequalities. Weierstrass inserted sentences confirming that the relationships involving the term ‘infinitely small’ were defined in terms of e d inequalities as follows: (D3) If h denotes a magnitude which can assume infinitely small values, ’?h? is an arbitrary function of h with the property that for an infinitely small value of h it

3The present paper also quotes Kurt Bing’s translation included in Calinger’s Classics of mathematics. Volume 29 (2014) 55

also becomes infinitely small (that is, that always, as soon as a definite arbitrary small magnitude e is chosen, a magnitude d can be determined such that for all values of h whose absolute value is smaller than d, ’?h? becomes smaller than e). (Dugac 1973, 120; English translation from Calinger, 1995, 607) As Dugac (1973, 65) indicates, some modern textbooks describe ’?h? as infinitely small or infinitesimal. Weierstrass argued that the whole change of function can in general be decomposed as Df ?x? ? f ?x ? h? f ?x? ? p:h ? h?h?; ? 1?

where the factor p is independent of h and ?h ? is a magnitude that becomes infinitely small with h.4 However, he overlooked that such decomposition is not possible for all functions and inserted the term ‘in general’. He rewrote h as dx. One can make the difference between Df ?x? and p:dx smaller than any magnitude with decreasing dx. Hence Weierstrass defined ‘differential’ as the change which a function undergoes when its argument changes by an infinitesimally small magnitude and denoted it as df ?x?. Then, df ?x? ? p:dx. Weierstrass pointed out that the differential coefficient p is a function of x derived from f ?x? and called it a derivative (Dugac 1973, 120–121; English translation from Calinger 1995, 607– 608). In accordance with Weierstrass’s definitions (D1) and (D3), he largely defined a derivative in terms of epsilonics. Weierstrass did not adopt the term ‘infinitely small’ but directly used e d inequalities when he discussed properties of infinite series involving uniform conver-gence (Dugac 1973, 122–124). It may be inferred from the published portion of his notes that Cauchy’s limit has no place in Weierstrass’s lectures. Grattan-Guinness’s (1986, 228) description of the limit-avoiding character of his analysis represents this situation well. However, Weierstrass thought that his theory included most of the content of Cauchy’s theory. Cauchy first gave the definition of limits of variables and

infinitesi-mals. Then, he demonstrated notions and theorems that were written in terms of the relationships involving infinitesimals. From Weierstrass’s viewpoint, they were writ-ten in terms of e d inequalities. Analytical theory mainly examines properties of functions and series, which were described in the relationships involving Cauchy’s limits and infinitesimals. Weierstrass recognized this fact and had the idea of consis-tently developing his theory in terms of inequalities. Hence Weierstrass at first defined the relationships among infinitesimals in terms of e d inequalities. In accor-dance with this definition, Weierstrass rewrote Cauchy’s results and naturally imported them into his own theory. This is a process that may be described as fol-lows: ‘Weierstrass completed the transformation away from the use of terms such as “infinitely small”’ (Katz 1998, 728).

Weierstrass’s definition of limit Dugac (1978, 370–372; 1976, 6–7) read (D1) as the first definition of limit with the help of e d. But (D1) does not involve an endpoint that variables or functions

4Dugac (1973, 65) indicated that ?h? corresponds to the modern notion of o?1?. In addition, h?h? corre-sponds to the function that was introduced as ’?h? in the former quotation from Weierstrass’s sentences.



极限思想外文翻译.pdf_其它课程_高中教育_教育专区 暂无评价|0人阅读|0次下载|举报文档极限思想外文翻译.pdf_其它课程_高中教育_教育专区。极限思想外文翻译 ...


搜试试 3 悬赏文档 全部 DOC PPT TXT PDF XLS 广告 百度文库 专业资料 自然科学 数学极限思想中ε-N语言和ε-δ语言的注释及应用_数学_自然科学_专业资料...


搜试试 3 帮助 全部 DOC PPT TXT PDF XLS ...MIT牛人解说数学体系(增加部分英文翻译和备注)_数学_...在极限思想的支持下,实数理论 在这个时候被建立起来...


极限思想及其应用 - 广东金 融学院 2008-JX16- 本科毕业论文(设计) 极限思想及其应用 学生姓名: 孙金龙 学系专号: 071611140 部: 应用数学系 ...


格式:pdf 关键词:极限思想学习资料 1/2 相关文档推荐 巧用极限思想解题 暂无评价...巧用极限思想解题巧用极限思想解题隐藏>> 分享到: X 分享到: 使用一键分...


巧用极限思想解题 - 中学 生数学 ? 201 0年11 月上 ? 第4 05期


搜试试 3 帮助 全部 DOC PPT TXT PDF XLS 百度文库 专业资料 自然科学 数学...极限思想在解题中的运用_数学_自然科学_专业资料。2一 数 学款 学 2016 年...


极限思想的演变及其应用 - 89 第3 O卷 第 4期 《疆师范大学学报》 自然




极限思想在经济生活中的渗透 - 余继光( 绍兴柯桥中学 . 浙江 3 1 2 0 3 0)... 200 3年第l 2期 数学通讯 2 3 极限思想在经济生活中的渗透 余 继光 ( ...


格式:pdf 关键词:期刊杂志 1/2 相关文档推荐 极限思想在解题中的导向作... ...数学通讯 一200 9年第 7期( 下半月) 1 5 极限思想在解题中的作用 王炳 ...


极限思想的演变及其应用 - 89第3 卷第4期021年1 月012《( 新疆师范


极限思想在经济学中的应用 - 学习数学不仅要学重要的数学概念、方法和结论,更要领会到数学的精神实质和思想方法。极限思想方法,是微积分中一个重要的内容,是应用...


极限思想在代数中的应用 - 本文利用极限思想解决了行列式的计算以及矩阵的证明,并


极限思想在数学中的应用 - 课程教育研究 Course Education Re

发现特殊值 渗透极限思想谈《鸡兔同笼》中尝试列表....pdf

格式:pdf 关键词:暂无1/2 相关文档推荐 发现特殊值渗透极限思想... 1页 ...发现特殊值 渗透极限思想谈《鸡兔同笼》中尝试列表法的优化策略 隐藏>> ...


搜试试 3 帮助 全部 DOC PPT TXT PDF XLS 百度文库 专业资料 自然科学 数学...用极限思想妙解立体几何题_数学_自然科学_专业资料。中学生 数学 年! 月上 第...


搜试试 5 悬赏文档 全部 DOC PPT TXT PDF XLS 广告 百度文库 教育专区 高等教育 理学数学分析中极限思想与极限概念教学_理学_高等教育_教育专区 ...


极限与极限思想在中学数学中的应用 - 学校代码: 分类号: 11658 O12 学密 号: 2012204510405 级: 无 硕士学位论文 极限与极限思 想在 中学...


搜试试 5 悬赏文档 全部 DOC PPT TXT PDF XLS 广告 百度文库 ...辅教导学 ? 巧用极限思想解题齐相国 ( 山东省济南市长清第五中学 , 250309)...

网站首页 | 网站地图
All rights reserved Powered by 学霸学习网
copyright ©right 2010-2021。