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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 But in the unpublished portions of the lectures, Weierstrass actu-ally defined his limit in terms of ?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¨C186).

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¡¯¡ª which has the quality of a translation or description¡ªhe 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¨C20) 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¨C23). 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¨C45; 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¨C261, 271¨C272), and Fisher (1978, 16 ¨C318) 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 Universit€at 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¨C7).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¨C120; 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¨C121; English translation from Calinger 1995, 607¨C 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¨C124). 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¨C372; 1976, 6¨C7) 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.

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