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BSHM Bulletin, 2014Did Weierstrasss differential calculus have a limit-avoiding character? His definition of a limit in styleMICHIYO NAKANENihon University Research Institute of Science & Technology, JapanIn 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 Cauchys limit and are distin-guished by their limit-avoiding character. Dugacs 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 But in the unpublished portions of the lectures, Weierstrass actu-ally defined his limit in terms ofinequalities. Weierstrasss 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 Dinis 1878 text-book that introduced the definition of a limit in terms of inequalities.IntroductionAugustin 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 Weierstrasss 1861 lectures which used the technique in all proofs and also in his defi-nition (Lutzen 2003, 185186).Weierstrasss 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 Cauchys 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 Weierstrasss 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 Cauchys limit, though they can describe relationships that involved his limit concept. Next, we examine how Weierstrass constructed a novel analysis theory which was not based2013 British Society for the History of Mathematics52BSHM Bulletinon Cauchys limits but could have involved Cauchys results. Then we confirm Weierstrasss definition of limit. Finally, we note that Dini organized his analysis textbook in 1878 based on analysis performed in the e d style.Cauchys limit and epsilonic argumentsCauchys series of textbooks on calculus, Cours danalyse (1821), Resume des lecons donnees a lEcole 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 danalyse, Cauchy defined the limit concept as fol-lows: 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 con-cept, 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 definitionwhich has the quality of a translation or descriptionhe 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 suc-cessive 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 Cauchys 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. Cauchys 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 danalyse Cauchy at first gave a verbal definition of a continuous func-tion. 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 pro-duces an infinitesimal increment in the function itself. (1821, 43; English transla-tion from Birkhoff and Merzbach 1973, 2).He introduced the infinitesimal-involving definition and adopted a modified version of it in Resume (1823, 1920) and Lecons (1829, 278).Following Cauchys 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. Cauchys infinitesimals are variables, and we can also take such an interpretation.Volume 29 (2014)53Cauchy 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,as i tends to 0 (1823, 2223). 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, 4445; 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. Cauchys arguments about infinite series in Cours danalyse, 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 Cauchys 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 Cauchys original intention. But this paper accepts the inter-pretations of Edwards, Laugwitz, and Lutzen.Cauchys lectures mainly discussed properties of series and functions in the limit process, which were represented as relationships between his limits or his infinitesi-mals, 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 Cauchys lectures could be rewrit-ten in terms of e d inequalities. Cauchys limits and his infinitesimals were not func-tional relationships,1 so they were not representable in terms of e d inequalities.Cauchys limit concept was the foundation of his theory. Thus, Weierstrasss full epsilonic analysis theory has a different foundation from that of Cauchy.Weierstrasss 1861 lecturesWeierstrasss consistent use of e d argumentsWeierstrass 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 by1Edwards (1979, 310), Laugwitz (1987, 260261, 271272), and Fisher (1978, 16318) point out that Cauchys infinitesimals equate to a dependent variable function or ah that approaches zero as h ! 0. Cauchy adopted the latter infinitesimals, which can be written in terms of e d arguments, when he intro-duced a concept of degree of infinitesimals (1823, 250; 1829, 325). Every infinitesimal of Cauchys is a vari-able in the parts that the present paper discusses.2A forerunner of the Technische Universitat Berlin.54BSHM BulletinHerman 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 Weierstrasss lectures was incontestable (1978, 372, 1976, 67).3After 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 Cauchys 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, 119120; 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 Cauchys term produce to correspond, it seems that Weierstrass took the idea of this definition from Cauchy. However, Weierstrasss definition was given in terms of epsilonics, while Cauchys definition can only be interpreted in these terms. Furthermore, Weierstrass achieved it without Cauchys 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 Weierstrasss 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 it3The present paper also quotes Kurt Bings translation included in Calingers Classics of mathematics.Volume 29 (2014)55also 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 decom-posed asDf x f x h f x p:h hh;1where 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, 120121; English translation from Calinger 1995, 607608). In accordance with Weierstrasss 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, 122124).