Continuity between Cauchy and Bolzano: Issues of antecedents and priority
Jacques Bair, Piotr Blaszczyk, Elias Fuentes Guillen, Peter Heinig, Vladimir Kanovei, Mikhail G. Katz
CCONTINUITY BETWEEN CAUCHY AND BOLZANO:ISSUES OF ANTECEDENTS AND PRIORITY
JACQUES BAIR, PIOTR B(cid:32)LASZCZYK, EL´IAS FUENTES GUILL´EN,PETER HEINIG, VLADIMIR KANOVEI, AND MIKHAIL G. KATZ
Abstract.
In a paper published in 1970, Grattan-Guinness ar-gued that Cauchy, in his 1821 book
Cours d’Analyse , may haveplagiarized Bolzano’s book
Rein analytischer Beweis (RB), firstpublished in 1817. That paper was subsequently discredited inseveral works, but some of its assumptions still prevail today. Inparticular, it is usually considered that Cauchy did not develop hisnotion of the continuity of a function before Bolzano developed hisin RB, and that both notions are essentially the same. We arguethat both assumptions are incorrect, and that it is implausible thatCauchy’s initial insight into that notion, which eventually evolvedto an approach using infinitesimals, could have been borrowed fromBolzano’s work. Furthermore, we account for Bolzano’s interest inthat notion and focus on his discussion of a definition by K¨astner(in Section 183 of his 1766 book), which the former seems to havemisrepresented at least partially.Cauchy’s treatment of continuity goes back at least to his 1817course summaries, refuting a key component of Grattan-Guinness’plagiarism hypothesis (that Cauchy may have lifted continuityfrom RB after reading it in a Paris library in 1818). We exploreantecedents of Cauchy and Bolzano continuity in the writings ofK¨astner and earlier authors.Keywords: Bolzano; Cauchy; K¨astner; continuity; infinitesi-mals; variables 01A55; 26A15
Contents
1. Introduction 22. Evolution of Cauchy’s ideas documented by Guitard 42.1. Continuity in 1817 42.2. Continuity in
Cours d’Analyse
Rein analytischer Beweis
Rein analytischer Beweis a r X i v : . [ m a t h . HO ] M a y BAIR, B(cid:32)LASZCZYK, FUENTES GUILL´EN, HEINIG, KANOVEI, AND KATZ
4. Antecedents in K¨astner, Karsten, and others 124.1. K¨astner’s 1760 definition 124.2. K¨astner’s influence on Bolzano 144.3. Bolzano misattributes a definition to K¨astner 144.4. Continuity in Leibniz 175. Conclusion 18Acknowledgments 18References 181.
Introduction
The issue of priority for the definition of the continuity of a functionwas raised in [Grattan-Guinness 1970] in a way that provoked contro-versy. With regard to this issue, Grabiner seeks to shift the focus ofattention away from the Bolzano/Cauchy priority debate, and broadenthe discussion to include an analysis of their common predecessors,particularly Lagrange. She detects an “immediate source of the in-dependent Bolzano–Cauchy definitions” both in Lagrange’s 1798 book
Trait´e de la r´esolution des ´equations num´eriques de tous les degr´es andin his
Th´eorie des fonctions analytiques (see [Grabiner 1984, p. 113]).Grabiner concludes that “these two books are the most likely sourcesfor both Cauchy’s and Bolzano’s definitions of continuous function”(op. cit., p. 114). Grabiner’s analysis challenges Grattan-Guinness’claim that “[Bolzano’s and Cauchy’s] new foundations, based on limitavoidance, certainly swept away the old foundations, founded largelyon faith in the formal techniques” [Grattan-Guinness 1970, p. 382].For sources of Bolzano’s notion of continuity in Lagrange see also[Rusnock 1999, p. 422].Schubring similarly rules out Grattan-Guinness’ hypothesis, and fur-thermore challenges a common assumption that Bolzano’s work wasvirtually unknown in the mathematical community during the first halfof the 19th century [Schubring 1993]. He reports on a (formerly) un-known review of Bolzano’s three important papers from 1816 and 1817,written by a mathematician named J. Hoffmann in 1821 and publishedin 1823.As for the Bolzano–Cauchy continuity, Grattan-Guinness investi-gated the possibility of its antecedents, focusing on the following threesources: (1) Cauchy’s work prior to 1821, (2) Legendre, and (3) Fourier;see [Grattan-Guinness 1970, p. 286]. His search reportedly did notturn up any reasonable antecedents: “of the new ideas that were to
ONTINUITY BETWEEN CAUCHY AND BOLZANO 3 achieve that aim – of them, to my great surprise, I could find noth-ing” (ibid.). His investigation led him to his well-known controversialconclusions. What he missed were the following sources: (1) Cauchy’searlier course summaries that were only discovered over a decade afterGrattan-Guinness’ article (see Section 2); (2) Lagrange (as argued byGrabiner); and (3) other 18th century authors, such as K¨astner andKarsten (see Section 4).Some mathematicians and historians of mathematics assume thatBolzano’s definition of the continuity of a function in his 1817
Reinanalytischer Beweis preceded Cauchy’s, and that the latter first gaveone in his 1821 textbook
Cours d’Analyse . Both assumptions turn outto be incorrect. Scholars commonly assume the following claims to betrue:(Cl 1) Bolzano and Cauchy gave essentially the same definition of con-tinuity, and(Cl 2) Bolzano gave it earlier.We give some examples below. • Jarn´ık: “Bolzano defines continuity essentially in the same wayas Cauchy does a little later” [Jarn´ık 1981, p. 36]. • Segre: “This led [Bolzano], in his
Rein analytischer Beweis (written in 1817, four years before Cauchy published his
Coursd’analyse ), to give a definition of continuity and derivative verysimilar to Cauchy’s, etc.” [Segre 1994, p. 236]. • Ewald: “[Bolzano’s] definition is essentially the same as thatgiven by Cauchy in his
Cours d’analyse in 1821; whether Cauchyknew of Bolzano’s work is uncertain” [Ewald 1996, p. 226]. • Heuser: “Cauchy defines continuity substantially in the sameway as Bolzano: . . . ” Now claim (Cl 1) is problematic since, as noted by L¨utzen,Bolzano did not use infinitesimals in his definition ofcontinuity. Cauchy did. [L¨utzen 2003, p. 175]L¨utzen’s claim that Cauchy used infinitesimals in his definition of con-tinuity is not entirely uncontroversial. While Cauchy indisputably usedthe term infiniment petit , the meaning of Cauchy’s term is subject todebate. Judith Grabiner [Grabiner 1981], Jeremy Gray [Gray 2015, In the original German: “Stetigkeit definiert Cauchy inhaltlich so wie Bolzano”[Heuser 2002, p. 691]. Heuser goes on to present Cauchy’s first 1821 definition interms of f ( x + α ) − f ( x ) (see Section 2.2), but fails to mention the fact that Cauchydescribes α as an infinitely small increment . Note, however, that Bolzano did exploit infinitesimals in his later writings; seee.g., [Grattan-Guinness 1970, note 29, p. 379], [Trlifajov´a 2018], and [Fila 2020].
BAIR, B(cid:32)LASZCZYK, FUENTES GUILL´EN, HEINIG, KANOVEI, AND KATZ
Figure 1.
Cauchy’s treatment of continuity datingfrom 4 march 1817 in the gregorian calendar (which wasa tuesday). The “Mar.” in the figure stands for mardi ,tuesday. The glyph resembling ∂ to the right of the dateseems to be shorthand for ditto , referring to the monthof march mentioned on earlier lines in this Registre del’Instruction for 1817.p. 36], and some other historians feel that a Cauchyan infinitesimalis a sequence tending to zero. Others argue that there is a differ-ence between null sequences and infinitesimals in Cauchy (see e.g.,[Bair et al. 2019]).In sum, Cauchy’s 1821 definitions exploited infinitesimals (and/orsequences), whereas Bolzano’s definition in
Rein analytischer Beweis exploited the clause “provided ω can be taken as small as we please” ina way that can be interpreted as an incipient form of an (cid:15), δ definitionrelying on implied alternations of quantifiers. Such manifest differ-ences make it difficult to claim that the definitions were “essentiallythe same.”To determine the status of claim (Cl 2), we will examine the primarysources in Bolzano and Cauchy and compare their dates.2. Evolution of Cauchy’s ideas documented by Guitard
Primary sources published in the 1980s suggest that an evolutiontook place in Cauchy’s ideas concerning continuity. On 4 march 1817,Cauchy presented an infinitesimal-free treatment of continuity in termsof variables which is procedurally identical with the modern definitionof continuous functions via commutation of taking limit and evaluatingthe function, as we discuss in Section 2.1.2.1.
Continuity in 1817.
In modern mathematics, a real function f is continuous at c ∈ R if and only if for each sequence ( x n ) convergingto c , one has f (lim n →∞ x n ) = lim n →∞ f ( x n ), or briefly f ◦ lim = lim ◦ f at c . In 1817, Cauchy wrote (see Figure 1): The equivalence of such a definition with the (cid:15), δ one requires the axiom of choice.
ONTINUITY BETWEEN CAUCHY AND BOLZANO 5
Figure 2.
Cauchy’s first 1821 definition of continuity
La limite d’une fonction continue de plusieurs variablesest la mˆeme fonction de leur limite . Cons´equence dece Th´eor`eme relativement `a la continuit´e des fonctionscompos´ees qui ne d´ependent que d’une seule variable. (Cauchy as quoted in [Guitard 1986, p. 34]; emphasisadded; cf. [Belhoste 1991, p. 255, note 6 and p. 309])The Intermediate Value Theorem is proved in the same lecture. Cauchy’streatment of continuity in 1817 contrasts with his definitions based oninfinitesimals given four years later in Cours d’Analyse (CdA).2.2.
Continuity in
Cours d’Analyse . In CdA, Cauchy defines con-tinuity as follows (see Figure 2):Among the objects related to the study of infinitelysmall quantities, we ought to include ideas about the Translation: “The limit of a continuous function of several variables is [equal to]the same function of their limit. Consequences of this Theorem with regard to thecontinuity of composite functions dependent on a single variable.” The referencefor this particular lesson in the Archives of the Ecole Polytechnique is as follows:Le 4 Mars 1817, la le¸con 20. Archives E. P., X II C7, Registre d’instruction 1816–1817. Belhoste places it even earlier, in 1816: “according to the
Registres , Cauchy knewthe modern concept of continuity as far back as March 1817, but the ‘invention’ wasanterior, as shown by the instructional program of December 1816” [Belhoste 1991,p. 255, note 6].
BAIR, B(cid:32)LASZCZYK, FUENTES GUILL´EN, HEINIG, KANOVEI, AND KATZ
Figure 3.
Cauchy’s second 1821 definition of continuitycontinuity and the discontinuity of functions. In viewof this, let us first consider functions of a single vari-able. Let f ( x ) be a function of the variable x , and sup-pose that for each value of x between two given limits,the function always takes a unique finite value. If, be-ginning with a value of x contained between these lim-its, we add to the variable x an infinitely small incre-ment α , the function itself is incremented by the differ-ence f ( x + α ) − f ( x ), which depends both on the newvariable α and on the value of x . Given this, the func-tion f ( x ) is a continuous function of x between the as-signed limits if, for each value of x between these limits,the numerical value of the difference f ( x + α ) − f ( x )decreases indefinitely with the numerical value of α .(Cauchy as translated in [Bradley–Sandifer 2009, p. 26]; emphasis on “continuous” in the original; emphasis on“infinitely small increment” added)This definition can be thought of as an intermediary one between themarch 1817 definition purely in terms of variables and containing nomention of the infinitely small, and his second 1821 definition statedpurely in terms of the infinitely small (see Section 2.3).2.3. Second definition of continuity in CdA.
Cauchy goes on tosummarize the definition given above as follows (see Figure 3):In other words, the function f ( x ) is continuous with re-spect to x between the given limits if, between these lim-its, an infinitely small increment in the variable alwaysproduces an infinitely small increment in the functionitself . (ibid.; emphasis in the original). Reinhard Siegmund-Schultze writes: “By and large, with few exceptions to benoted below, the translation is fine” [Siegmund-Schultze 2009]. In the original: “En d’autres termes, la fonction f ( x ) restera continue par rapport`a x entre les limites donn´ees, si, entre ces limites, un accroissement infiniment petit ONTINUITY BETWEEN CAUCHY AND BOLZANO 7
Since Cauchy prefaced his second definition with the words en d’autrestermes (“in other words”), he appears to have viewed the pair of 1821definitions as being equivalent. Cauchy sums up his discussion of con-tinuity in CdA as follows:We also say that the function f ( x ) is a continuous func-tion of the variable x in a neighborhood of a particularvalue of the variable x whenever it is continuous betweentwo limits of x that enclose that particular value, even ifthey are very close together. Finally, whenever the func-tion f ( x ) ceases to be continuous in the neighborhoodof a particular value of x , we say that it becomes dis-continuous, and that there is solution of continuity forthis particular value. (ibid.; emphasis in the original)Note that none of the 1821 definitions exploited the notion of limit.We therefore find it puzzling to discover the contrary claim in a recenthistorical collection:Cauchy gave a faultless definition of continuous func-tion, using the notion of ‘limit’ for the first time. Fol-lowing Cauchy’s idea, Weierstrass popularized the (cid:15) - δ argument in the 1870s, etc. [Dani–Papadopoulos 2019,p. 283]In a related vein, V¨ath opines that “formulat[ing] properties whichhold for infinitesimals (which have been used by Leibniz) in an (cid:15) - δ -type manner . . . was first propagated by Cauchy” [V¨ath 2007, p. 74].Similarly, Goldbring and Walsh claim the following:[T]he mathematical status of [infinitesimals] was viewedas suspect and the entirety of calculus was put on firmfoundations in the nineteenth century by the likes ofCauchy and Weierstrass, to name a few of the moresignificant figures in this well-studied part of the historyof mathematics. The innovations of their “ (cid:15) - δ method”. . . allowed one to give rigor to the na¨ıve arguments oftheir predecessors. [Goldbring–Walsh, p. 843]Presentist views of this type are, alas, not the exception, and muchwork is required to counter them. Recent work on Cauchy’s stance onthe infinitely small and their applications includes [Bair et al. 2017a],[B(cid:32)laszczyk et al. 2017], [Bascelli et al. 2018], and [Bair et al. 2020]. de la variable produit toujours un accroissement infiniment petit de la fonction elle-mˆeme ” [Cauchy 1821, pp. 34–35]. meaning dissolution , i.e., absence (of continuity). BAIR, B(cid:32)LASZCZYK, FUENTES GUILL´EN, HEINIG, KANOVEI, AND KATZ
To summarize, in 1817 Cauchy gave a characterisation of continu-ity in terms of variables, whereas the second 1821 definition involvedonly infinitesimals. Meanwhile, the first 1821 definition exploited bothvariables and infinitesimals.3.
Bolzano’s
Rein analytischer Beweis
Could Bolzano’s
Rein analytischer Beweis (RB) [Bolzano 1817/18]have influenced Cauchy’s definition of continuity? Grattan-Guinnesswrote: Bolzano had given his paper [RB] two opportunitiesfor publication, for not only did he issue it as a pam-phlet in 1817, but – with the same printing – inserted itinto the 1818 volume of the Prague Academy
Abhand-lungen . That journal was available in Paris: indeed,the
Biblioth`eque Imp´eriale (now the
Biblioth`eque Na-tionale ) began to take it with precisely the volume con-taining Bolzano’s pamphlet . [Grattan-Guinness 1970,p. 396] (emphasis in the original)Of particular interest to us is Grattan-Guinness’ reliance on the avail-ability of RB in the Paris
Imperial Library in 1818; see Section 3.1.The papers [Freudenthal 1971] and [Sinaceur 1973] provided evidenceagainst Grattan-Guinness’ hypothesis. However, as noted by Jan Sebes-tik, their work does not rule out the possibility that “Cauchy could haveread Bolzano’s
Rein analytischer Beweis (or heard about it) and couldhave been inspired by it” [Sebestik 1992, pp. 109, 111]. Thirty yearsafter the Benis-Sinaceur paper, Russ wrote:There has been discussion in the literature on the possi-bility that Cauchy might have plagiarized from Bolzano.See Grattan-Guinness (1970), Freudenthal (1971) andSinaceur (1973). ([Russ 2004, p. 149]; emphasis added)It is our understanding that referring to the issue as a “discussion”tends to imply that the hypothesis of plagiarism has not been defini-tively refuted. Arguably, therefore, the issue continues to have rele-vance.3.1.
Grattan-Guinness’ hypothesis.
Having summarized the his-torical background, Grattan-Guinness proceeds to state his hypothesis: Similarly, in a recently published book, Rusnock and ˇSebest´ık mention that “therehas been speculation that Cauchy may have learned a thing or two from Bolzano”[Rusnock–ˇSebest´ık 2019, p. 49]; see also note 3 there.
ONTINUITY BETWEEN CAUCHY AND BOLZANO 9
So here is at least one plausible possibility for Cauchy tohave found a copy of Bolzano’s paper, quite apart fromthe book-trade: he could have noticed a new journal inthe library’s stock and examined it as a possible course of interesting research. [Grattan-Guinness 1970, p. 396]Grattan-Guinness specifically includes the concept of continuity in hishypothesis (op. cit., p. 374).It is our understanding that, while the evidence provided in the arti-cles [Freudenthal 1971] and [Sinaceur 1973] shows clear and profounddifferences between Cauchy and Bolzano’s stance, it does not entirelyrefute the aforementioned hypothesis. We will provide a refutation ofa key component of Grattan-Guinness’ hypothesis concerning the con-cept of continuity. Our refutation is based on the facts of the chronologyof the relevant works. Namely, we will show that Cauchy possessed aconcept of continuity(1) earlier than the date of the acquisition of a journal version ofRB by the Imperial Library in Paris, and(2) even earlier than, or at least contemporaneously with, the dateof the Leipzig fair where RB was first marketed.Note that, according to Grattan-Guinness, the
Biblioth`eque Imp´eriale started to take the journal where RB appeared in the year 1818. Read-ing the 1818 journal version of RB could not therefore have influencedCauchy’s treatment of continuity in 1817 (see Section 2). This re-futes a key component of the plagiarism hypothesis as proposed in[Grattan-Guinness 1970] with regard to the concept of continuity. Thecomparison of dates establishes that Cauchy’s initial insight into conti-nuity could not have been borrowed from Bolzano’s RB, though it doesnot rule out the possibility that Cauchy may have been acquainted withBolzano’s work before formulating the later, 1821 definitions in CdA.Grattan-Guinness also brought broader plagiarism charges againstCauchy, which are not refuted by our comparison of dates. Notice,however, that it is implausible that Cauchy may have seen Bolzano’s1816 text Der binomische Lehrsatz [Bolzano 1816], where the latteralso gave a definition of continuity, since there is no evidence thatthis text was available in France. It seems that this is why Grattan-Guinness found it necessary to speculate specifically concerning theversion of Bolzano’s RB available in a Paris library in 1818, so as to Grattan-Guinness apparently means “source.” Cauchy had discussed continuity even earlier, in an 1814 article on complex func-tions (see [Freudenthal 1971, p. 380]). However, that discussion stayed at the intu-itive level and cannot be described as reasonably precise. bolster the plausibility of the plagiarism claim. Grattan-Guinness mayhave had more of a point with regard to E. G. Bj¨orling. Apparently inthe 1850s, Cauchy may not have been transparent about possible influ-ence of Bj¨orling’s ideas related to uniform convergence. The issue wasstudied in [Br˚ating 2007]. For an analysis of Cauchy’s 1853 approachto uniform convergence see [Bascelli et al. 2018].3.2.
Bolzano’s definition in
Rein analytischer Beweis . In hisRB, Bolzano criticized some proofs of IVT for polynomials that fromhis stance were “based on an incorrect concept of continuity ,” givenfor example their use of “a truth borrowed from geometry ” or “theintroduction of the concepts of time and motion [Bolzano 1817/18,pp. 6, 8–9, 11]. Instead, he defined continuity as follows:According to a correct definition, the expression that afunction f x varies according to the law of continuity forall values of x inside or outside certain limits means onlythat, if x is any such value the difference f ( x + ω ) − f x can be made smaller than any given quantity, provided ω can be taken as small as we please or (in the notationwe introduced in §
14 of
Der binomische Lehrsatz etc.,Prague, 1816) f ( x + ω ) = f x +Ω. (Bolzano as translatedin [Russ 2004, p. 149, 256])The dating of RB will be analyzed in Section 3.3 below. Bolzano’sdefinition is reasonably precise, as is Cauchy’s approach. Here “rea-sonably precise” means “easily transcribable as a modern definition”(rather than merely an intuitive notion of continuity). A modernformalisation of Bolzano’s 1817 definition would involve alternatingquantifiers, whereas a modern formalisation of Cauchy’s 1817 defini-tion would retain almost verbatim the commutation of (a) evaluat-ing f and (b) taking lim (see Section 2.1). Apparently neither Jarn´ıknor Ewald (see Section 1) were aware of Cauchy’s treatment of bothcontinuity and the IVT dating from 4 march 1817.3.3. The dating of Bolzano’s RB.
The earliest known written recordof Bolzano’s RB is in a catalog of the Easter book fair at Leipzig.According to [Evenhuis 2014, p. 4], both the catalog [Olms 1817,p. 30] and the fair itself date from 27 april 1817, over a month laterthan the earliest written record of Cauchy’s treatment of continuity. Note that we take no position with regard to which definition was closer to amodern one, Bolzano’s or Cauchy’s (Bolzano’s was arguably closer to the modern
Epsilontik standard). The point we are arguing is that both were reasonably precisein the sense specified.
ONTINUITY BETWEEN CAUCHY AND BOLZANO 11
Figure 4.
Bolzano’s definition of continuityIt should be noted, however, that Bolzano also gave a definition ofcontinuity in an 1816 publication [Bolzano 1816] (see Figure 4):For a function is called continuous if the change whichoccurs for a certain change in its argument, can be-come smaller than any given quantity, provided thatthe change in the argument is taken small enough. (Bolzano as translated in [Russ 2004, p. 184])This definition is immediately followed by an attempted proof ofan erroneous assertion. Namely, Bolzano claims to prove that if afunction F is differentiable then its derivative, f , is continuous. Thisindicates that Bolzano’s definition of continuity was still sufficientlyambiguous to accomodate errors, as was his ω/ Ω notation. Recently[Fuentes Guill´en–Mart´ınez Adame 2020, Abstract] have argued in
His-toria Mathematica that “those quantities [i.e., Bolzano’s ω ] are notclearly ‘proto-Weierstrassian’.”It is worth noting that an even earlier mention of ideas in the di-rection of Bolzano’s definition of continuity occurs in Bolzano’s math-ematical diaries of early 1815: “if therefore ξ is taken smaller than anygiven quantity, i.e. = ω , the value of f ( x + ω ) − f x must be able to be-come as small as desired” (see op. cit., note 86). Insofar as Cauchy hadno access either to Bolzano’s diaries or the latter’s 1816 work, and theformer would have formulated his first definition of continuity shortlybefore or in any case at about the same time as the 1817 Easter bookfair at Leipzig, it is implausible that Cauchy’s 1817 definition couldhave been borrowed from Bolzano’s work. In the original: “Stetig heißt n¨ahmlich eine Function, wenn die Ver¨anderung, diesie bey einer gewissen Ver¨anderung ihrer Wurzel erf¨ahrt, kleiner als jede gegebeneGr¨oße zu werden vermag, wenn man nur jene klein genug nimmt” [Bolzano 1816,p. 34]. Note that Bolzano repeatedly uses
Wurzel in the sense of “input to afunction”; see e.g., footnote on page 11 of [Bolzano 1817/18]. The issue is discussedin [Russ 2004, p. 256, note f]. Antecedents in K¨astner, Karsten, and others
There exists a historiographic controversy with regard to the issue ofcontinuity in the historical development of mathematics. Unguru andhis disciples adopt a radical posture against such continuity. Otherscholars endorse continuity at various levels and to varying extent. Weadopt the latter view, to the extent that we detect continuity between,for example, the work of K¨astner, on the one hand, and that of Bolzanoand Cauchy, on the other. For more details see [Katz 2020].The mathematical diaries of Bolzano written during 1814–1815 alsocontain criticism of, e.g., [Carnot 1797] and [Crelle 1813] because oftheir assumption of the law of continuity: in the first case he statedthat in such a law “[lay] the key for the resolution of the whole riddleof infinitesimal calculus” [Bolzano 1995, p. 152]; in the latter case, hepointed out that K¨astner had “already drawn attention to the surrep-titious acceptance of this law” [Bolzano 1997, p. 144]. As we alreadymentioned, the first published record of a definition of continuity givenby Bolzano dates from the following year, after which he published hisreasonably precise definition included in RB.As his later works and mathematical diaries show, Bolzano contin-ued to be interested in that issue. Thus, in his
Theory of Functions ,written in the 1830s, he would have “sharpened” his 1817 definition[Rusnock–Kerr-Lawson 2005, p. 306]. Rusnock and Kerr-Lawson arguethat, as early as the 1830s, Bolzano not only grasped the distinction be-tween pointwise continuity and uniform continuity but also presenteda pair of key theorems concerning the latter (ibid.). Moreover, in thatwork Bolzano acknowledged that “[t]he concept of continuity has al-ready been defined essentially as I do here by [other contemporaryauthors]” such as Cauchy and Ohm [Russ 2004, p. 449]. However, atthe same time, in that work he criticized certain specific definitions,including one by A. G. K¨astner in 1766. On the one hand, Bolzano’sdefinition surely constitutes an improvement upon the definition of lo-cal continuity by K¨astner in 1760 (see Figure 5). On the other hand,Bolzano seems to have misrepresented, at least partially, the relevantpassage from K¨astner’s work of 1766.4.1.
K¨astner’s 1760 definition.
K¨astner’s definition included in hisvolume on the analysis of finite quantities (
Analysis endlicher Gr¨ossen ),or letter-algebra, and which can be found in a section entitled “Oncurved lines,” runs as follows:
ONTINUITY BETWEEN CAUCHY AND BOLZANO 13
Figure 5.
K¨astner’s 1760 definition of continuityIn a sequence of magnitudes, their increase or decreasetakes place in accordance with the law of continuity (legecontinui), if after each term of the sequence, another onefollows or precedes the given term that differs from it[i.e., from the given term] by as little as one wishes;as a consequence, the difference of two consecutiveterms can amount to less than any given magnitude. [K¨astner 1760, paragraph 322, p. 180] We translated
Reihe as ‘sequence’, even though it is often translated as ‘series’,since ‘series’ nowadays is a standard technical term which is not appropriate here,and moreover the German term
Reihe can mean either ‘sequence’ or ‘series’. The German conjunction so dass , especially in K¨astner’s (now obsolete) spellingas two separate words, resembles the English ‘such that’; in the present case, how-ever, this is a false friend. In fact ‘as a consequence’ is one of several standardtranslations of the German conjunction sodass . K¨astner’s phrasing nach einander folgender could possibly be interpreted as thestatement that the terms mentioned here are immediate successor elements, inparticular since the standard technical translation for ‘immediate sucessor element’is
Nachfolger . This, however, could not be what K¨astner meant to say. K¨astner’sphrasing (note that he does not say
Nachfolger outright) is sufficiently vague toallow for an interpretation where he means to speak of two terms which follow shortly one after another, though there are other terms in between. In the original: “In einer Reihe von Gr¨ossen, erfolgt das Wachsthum oder dasAbnehmen derselben, nach dem Gesetze der Stetigkeit (lege continui) wenn nachjedem Gliede der Reihe eines folget, oder vor ihm vorhergehen kann, das so wenigals man nur will von dem angenommenen Gliede unterschieden ist, so daß der Un-terschied zweyer nach einander folgender Glieder, weniger als jede gegebene Gr¨ossebetragen kann.” This was quoted in [Spalt 2015, p. 283]. In our translation, wetried to strike a balance between literalness and readability in line with an approachtaken in [Bl˚asj¨o–Hogendijk 2018].
K¨astner’s influence on Bolzano.
Russ notes K¨astner’s influ-ence on Bolzano in the following terms:[T]here were two authors, Wolff and K¨astner, whosework, between them, dominated the century in the Ger-man-speaking regions. . . . [T]hey were both commit-ted to education and wrote highly systematic and com-prehensive multivolume textbooks on mathematics thatwent through many editions and were very influential.Not surprisingly, they were both authors to whom Bol-zano makes frequent reference in his early works.[Russ 2004, p. 14]Indeed, in Bolzano’s mathematical diaries there is a note from theearly 1820s, entitled “On the law of continuity.” Bolzano’s note in-cludes a reference to paragraph 183 of K¨astner’s work on mechanics[K¨astner 1766] and to paragraph 235 of W. J. G. Karsten’s work onmechanics [Karsten 1769]; see [Bolzano 2005, p. 63]. The formulationof both authors ultimately relied on the notion of continuity accordingto which “[a] continuous quantity (continuum) is that [quantity] whoseparts are all connected together in such a way that where one ceases,another immediately begins, and between the end of one and the begin-ning of another there is nothing that does not belong to this quantity”[Russ 2004, p. 17]; see [Karsten 1767, p. 209]; but only that of Karstenwould be equivalent to IVT [Karsten 1769, p. 223]. Interestingly, aswe already mentioned, in a later work Bolzano went back to discussthe notion of continuity in that paragraph of K¨astner’s work. We willanalyze such a reception of the latter’s ideas in Section 4.3.4.3.
Bolzano misattributes a definition to K¨astner.
We reviewedK¨astner’s 1760 definition in Section 4.1. In his
Theory of Functions ,Bolzano seems to have mistakenly attributed a different definition toK¨astner in 1766, which he (Bolzano) considered to be “too broad”:Some very respected mathematicians like
K¨astner ( h¨ohereMechanik , Auflage 2, §§
183 ff.) and
Fries ( Naturphiloso-phie , §
50) define the continuity of a function
F x as thatproperty of it by virtue of which it does not go froma certain value
F a , to another value
F b , without first having taken all the values lying in between . However,it will be seen subsequently that this definition is toowide if in fact the concept intended is to be equivalentto the one above. (Bolzano as translated in [Russ 2004, Perhaps a better translation is “too broad”.
ONTINUITY BETWEEN CAUCHY AND BOLZANO 15 p. 449]; emphasis on K¨astner and Fries in the original;emphasis on “having taken all the values lying in be-tween” and “too wide” added)As we already noted, K¨astner’s formulation to which Bolzano refershere ultimately relied on the former’s geometric notion of continuity.So, while K¨astner’s paragraph 183 is part of a section “On the lawof continuity” (which in turn is part of a chapter “On the movementof solid bodies with determined magnitude and shape”), he explicitlyrefers to the note in his definition 6 (straight and curved lines) of hisbook on geometry. In that note K¨astner points out that before thecurved line that goes from A to B reaches B , “all the minor changesin between must occur” [K¨astner 1758, p. 161].Bolzano would seem to attribute a different definition (via the satis-faction of the Intermediate Value Theorem) to K¨astner (as well as toFries) in the particular case of that paragraph. Nonetheless, Bolzano’sattribution appears to be incorrect.In fact, K¨astner’s discussion of the law of continuity in his section 183resembles, to some extent, Cauchy’s definition of continuity based oninfinitesimals given in Section 2.3 above (though of course K¨astner’sviewpoint is geometric rather than analytic):On the Law of Continuity. 183. In the investigationwhich we now present, it is assumed that the speed ofa body does not change instantaneously, but rather byinfinitely small gradations. Just the same can be saidof the direction. If one views the matter from that per-spective, then a body which is being reflected does notchange its direction instantaneously to the opposite di-rection: its speed becomes smaller and smaller in theprevious direction, finally vanishes, and then transformsinto a velocity having the opposite direction. This is the Law of Continuity (applied to these matters). To wit,by the latter law one claims that generally, no changehappens suddenly, but that every change always movesthrough infinitely small gradations (of which already themovement of a point along a curve is an example; [cf.K¨astner’s] Geom. 6. Erkl. Anm.). ([K¨astner 1766, p. 350, § emphasis on “law of continuity” on the original;emphasis on “every change, etc.” added) According to [Kr¨oger 2014, Abbildung 10], there were two edititions of this trea-tise. These are [K¨astner 1766] and [K¨astner 1793]. In the 1793 edition of K¨astner’streatise referred to by Bolzano as
Auflage 2 , Section 183 appears on page 543.
What may have led Bolzano to claim that K¨astner defined continuitybased on the satisfaction of IVT? Note that K¨astner’s text contains thefollowing three sentences:(K1) If one views the matter from that perspective, then a bodywhich is being reflected does not change its direction instan-taneously to the opposite direction: its speed becomes smallerand smaller in the previous direction, finally vanishes, and thentransforms into a velocity having the opposite direction.(K2) This is the Law of Continuity (applied to these matters).(K3) To wit, by the latter law one claims that generally, no changehappens suddenly, but that every change always moves throughinfinitely small gradations.Possibly, Bolzano interpreted sentence (K1) as the definition of the lawof continuity mentioned in sentence (K2). Now sentence (K1) doessound like (a physical interpretation of) the IVT.However, reading the three sentences together, it is clear that K¨astnermeant sentence (K3) to be the detailed formulation of the law of conti-nuity. Meanwhile, in sentence (K2), K¨astner specifically uses the verb applied . This indicates that K¨astner thinks of sentence (K1) as an ap-plication of the law of continuity, rather than the formulation thereof.Now in modern mathematics it is certainly true that continuity impliesIVT, though the converse is incorrect, as Bolzano himself argued (see[Russ 2004, §
84, pp. 471–472]). In his
Theory of Functions , Bolzanooutlines an idea for a function that takes every intermediate value with-out being continuous, as follows.Bolzano starts with an everywhere discontinuous function W ( x ) de-scribed in §
37, defined only on a collection of rational points, and builtout of a pair of linear functions of different slope. In §
39, Bolzano as-serts that the remaining infinitely many points can be used to assignthe values of the function so as to “fill in” whatever values are miss-ing. Bolzano’s argument is mentioned in [Sebestik 1992, p. 395] and[Smory´nski 2017] (see p. 52 and note 49 there). For a study of coun-terexamples to the implication “if f satisfies IVT then f is continuous”see [Oman 2014], [Radcliffe 2016], and [De Marco 2018].In conclusion, Bolzano may have interpreted sentence (K1) as theformulation of continuity (rather than an application thereof). Un-like Cauchy, Bolzano seems never to have formulated a definition of Sebestik also points out that Bolzano and Cauchy’s definitions of continuity couldhave been “the result of a critical reflection on the texts by Euler and Lagrange”[Sebestik 1992, pp. 110, 81–83].
ONTINUITY BETWEEN CAUCHY AND BOLZANO 17 continuity in terms of infinitesimals. It is possible that K¨astner’s sen-tence (K3) made no sense to Bolzano, who was therefore led to takesentence (K1) to be the formulation of continuity. Thus, while Friesmay perhaps have given a different definition of continuity via the sat-isfaction of IVT (as Bolzano claimed), K¨astner apparently did not.4.4.
Continuity in Leibniz.
An even earlier source for local continu-ity may have influenced K¨astner and other 18th century authors. Sucha source is in Leibniz’s 1687 formulation of the principle of continuity:When the difference between two instances in a givenseries or that which is presupposed can be diminisheduntil it becomes smaller than any given quantity what-ever, the corresponding difference in what is sought orin their results must of necessity also be diminished orbecome less than any given quantity whatever. (Leib-niz as translated by Loemker in [Leibniz 1989, p. 351];emphasis added)In modern terminology, Leibnizian “what is sought” is the dependentvariable while “that which is presupposed” is the independent vari-able. What Leibniz refers to as the principle of continuity involves,in modern terminology, the condition that a convergent sequence in thedomain should get mapped to a convergent sequence in the range. Cauchy’s approach dating from 4 march 1817 is not the final word oncontinuity, but it can be described as reasonably precise (in the senseexplained in Section 3.2). This is unlike many intuitive definitions givenearlier that cannot be so formalized.Notice that Bolzano’s definition is similarly reasonably precise butalso not without its problems. Thus, the Ω appearing there seems to bedefined as the difference f ( x + ω ) − f ( x ), whereas the corresponding (cid:15) in the modern definition is a ∀ -quantified variable entirely unrelatedto f . It is possible that this was also Bolzano’s intention, but it mustbe admitted that such an intention was only imperfectly expressed byBolzano’s formula f ( x + ω ) = f x + Ω and accompanying comments;see [Fuentes Guill´en–Mart´ınez Adame 2020] for a fuller discussion. Not to be confused with his law of continuity . For a detailed discussion see[Katz–Sherry 2013], [Sherry–Katz 2014], [Bascelli et al. 2016], [Bair et al. 2017b],[Bair et al. 2018]. In modern analysis, the sequence-condition is equivalent to continuity for first-countable spaces. Including Cauchy’s own definition in 1814, in an article on complex functionsquoted by Freudenthal; cf. note 11. Conclusion
We have re-examined the priority issue with regard to the concept ofcontinuity. Course notes available at the Ecole Polytechnique indicatethat Cauchy had a reasonably precise concept of continuity of a func-tion earlier than is generally thought. In particular Cauchy’s conceptwas earlier than, or at least contemporaneous with, the first writtenrecord of Bolzano’s 1817 work
Rein analytischer Beweis .In 1970, Grattan-Guinness speculated that Cauchy may have read aversion of Bolzano’s
Rein analytischer Beweis found in a Paris libraryin 1818, and subsequently plagiarized some of Bolzano’s insights, in-cluding continuity, when writing the 1821
Cours d’Analyse . Such ahypothesis is refuted by a written record of a reasonably precise treat-ment of continuity by Cauchy dating from march 1817, and hence an-terior to the Paris library acquisition, on which, among other things,Grattan-Guinness based his hypothesis.The proximity of the dates indicates an independence of Cauchy’sand Bolzano’s scientific insight, and should contribute not only to endspeculations as to possible plagiarism (with regard to the notion ofcontinuity) on either side, but also to improve our understanding oftheir respective developments of such a notion.The prototypes of both Bolzano’s and Cauchy’s definitions of con-tinuity in formulations found in 18th century and early 19th centuryworks, such as those of K¨astner, are yet to be explored fully.
Acknowledgments
We thank the anonymous referees for numerous suggestions thathelped improve the article. We are grateful to Olivier Azzola, archivistat the Ecole Polytechnique, for granting access to Cauchy’s handwrit-ten course summaries reproduced in Figure 1. E. Fuentes Guill´en wassupported by the Postdoctoral Scholarship Program of the Direcci´onGeneral de Asuntos del Personal Acad´emico (DGAPA-UNAM). V.Kanovei was supported by RFBR grant no. 18-29-13037.
References [Bair et al. 2018] Bair, J.; B(cid:32)laszczyk, P.; Ely, R.; Heinig, P.; Katz, M. “Leib-niz’s well-founded fictions and their interpretations.”
Mat. Stud. (2018), no. 2, 186–224. See http://dx.doi.org/10.15330/ms.49.2.186-224 andhttps://arxiv.org/abs/1812.00226[Bair et al. 2020] Bair, J.; B(cid:32)laszczyk, P.; Heinig, P.; Kanovei, V.; Katz, M.“Cauchy’s work on integral geometry, centers of curvature, and other appli-cations of infinitesimals.” Real Analysis Exchange (2020), no. 1, 1–23. ONTINUITY BETWEEN CAUCHY AND BOLZANO 19 [Bair et al. 2017a] Bair, J.; B(cid:32)laszczyk, P.; Ely, R.; Henry, V.; Kanovei, V.; Katz, K.;Katz, M.; Kudryk, T.; Kutateladze, S.; McGaffey, T.; Mormann, T.; Schaps,D.; Sherry, D. “Cauchy, infinitesimals and ghosts of departed quantifiers.”
Mat. Stud. (2017), no. 2, 115–144. See http://dx.doi.org/10.15330/ms.47.2.115-144 and https://arxiv.org/abs/1712.00226 [Bair et al. 2017b] Bair, J.; B(cid:32)laszczyk, P.; Ely, R.; Henry, V.; Kanovei, V.; Katz,K.; Katz, M.; Kutateladze, S.; McGaffey, T.; Reeder, P.; Schaps, D.; Sherry,D.; Shnider, S. “Interpreting the infinitesimal mathematics of Leibniz andEuler.” Journal for General Philosophy of Science (2017), no. 2, 195–238. See http://dx.doi.org/10.1007/s10838-016-9334-z and https://arxiv.org/abs/1605.00455 [Bair et al. 2019] Bair, J.; B(cid:32)laszczyk, P.; Heinig, P.; Kanovei, V.; Katz, M. “19thcentury real analysis, forward and backward.” Antiquitates Mathematicae (2019), 19–49. See http://dx.doi.org/10.14708/am.v13i1.6440 and https://arxiv.org/abs/1907.07451 [Bascelli et al. 2018] Bascelli, T.; B(cid:32)laszczyk, P.; Borovik, A.; Kanovei, V.; Katz, K.;Katz, M.; Kutateladze, S.; McGaffey, T.; Schaps, D.; Sherry, D. “Cauchy’sinfinitesimals, his sum theorem, and foundational paradigms.” Foundationsof Science (2018), no. 2, 267–296. See http://dx.doi.org/10.1007/s10699-017-9534-y and https://arxiv.org/abs/1704.07723 [Bascelli et al. 2016] Bascelli, T.; B(cid:32)laszczyk, P.; Kanovei, V.; Katz, K.; Katz, M.;Schaps, D.; Sherry, D. “Leibniz versus Ishiguro: Closing a Quarter Centuryof Syncategoremania.” HOPOS: The Journal of the International Society forthe History of Philosophy of Science (2016), no. 1, 117–147. See http://dx.doi.org/10.1086/685645 and https://arxiv.org/abs/1603.07209 [Belhoste 1991] Belhoste, B. Augustin-Louis Cauchy. A biography . Translated fromthe French and with a foreword by Frank Ragland. Springer-Verlag, NewYork, 1991.[Bl˚asj¨o–Hogendijk 2018] Bl˚asj¨o, V.; Hogendijk, J. “On translating mathematics.”
Isis; An International Review Devoted to the History of Science and itsCultural Influences (2018), no. 4, 774–781. See https://doi.org/10.1086/701258 [B(cid:32)laszczyk et al. 2017] B(cid:32)laszczyk, P.; Kanovei, V.; Katz, K.; Katz, M.; Kutate-ladze, S.; Sherry, D. “Toward a history of mathematics focused on pro-cedures.”
Foundations of Science (2017), no. 4, 763–783. See http://dx.doi.org/10.1007/s10699-016-9498-3 and https://arxiv.org/abs/1609.04531 [Bolzano 1816] Bolzano, B. Der binomische Lehrsatz, und als Folgerung aus ihmder polynomische, und die Reihen, die zur Berechnung der Logarithmen undExponentialgr¨oßen dienen, genauer als bisher erwiesen . Prague, 1816.[Bolzano 1817/18] Bolzano, B.
Rein analytischer Beweis des Lehrsatzes, dass zwis-chen je zwey Werthen, die ein entgegengesetztes Resultat gew¨ahren, wenig-stens eine reelle Wurzel der Gleichung liege (1817), Prague = Abh. K¨onighB¨ohm. Gesell. Wiss. (3) 5 (1814–1817; publ. 1818), 60 p. Translations inEwald [Ewald 1996] and Russ [Russ 2004].[Bolzano 1995] Bolzano, B. Miscellanea Mathematica 10. In van Rootselaar, Bob,van der Lugt, Anna (Eds.), Bernard Bolzano Gesamtausgabe (BGA).Band 2B6/2. Frommann-Holzboog, Stuttgart-Bad Cannstatt, 1995. [Bolzano 1997] Bolzano, B. Miscellanea Mathematica 12. In van Rootselaar, Bob,van der Lugt, Anna (Eds.), Bernard Bolzano Gesamtausgabe (BGA).Band 2B7/2. Frommann-Holzboog, Stuttgart-Bad Cannstatt, 1997.[Bolzano 2005] Bolzano, B. Miscellanea Mathematica 20. In van Rootselaar, Bob,van der Lugt, Anna (Eds.), Bernard Bolzano Gesamtausgabe (BGA).Band 2B11/2. Frommann-Holzboog, Stuttgart-Bad Cannstatt, 2005.[Bradley–Sandifer 2009] Bradley, R.; Sandifer, C. (Eds.)
Cauchy’s Cours d’analyse.An annotated translation . Sources and Studies in the History of Mathematicsand Physical Sciences. Springer, New York, 2009.[Br˚ating 2007] Br˚ating, K. “A new look at E. G. Bj¨orling and the Cauchy sumtheorem.”
Archive for History of Exact Sciences (2007), no. 5, 519–535.[Carnot 1797] Carnot, L. R´eflexions sur la M´etaphysique du Calcul Infinit´esimal.Duprat, Paris, 1797.[Cauchy 1821] A. L. Cauchy, Cours d’Analyse de L’ ´Ecole Royale Polytechnique.Premi`ere Partie. Analyse alg´ebrique . Imprim´erie Royale, Paris, 1821. Trans-lation in Bradley–Sandifer ([Bradley–Sandifer 2009], 2009).[Crelle 1813] Crelle, A. Versuch einer rein algebraischen und dem gegenw¨artigenzustande der mathematik, etc. Vandenhoek und Ruprecht, 1813. See https://books.google.com.mx/books?id=hDo7AQAAIAAJ [Dani–Papadopoulos 2019] Dani, S.; Papadopoulos, A. (Eds.).
Geometry inHistory . Springer, Cham, 2019. See http://dx.doi.org/10.1007/978-3-030-13609-3 [De Marco 2018] De Marco, G. An almost everywhere constant function surjectiveon every interval.
Amer. Math. Monthly (2018), no. 1, 75–76.[Evenhuis 2014] Evenhuis, N. “Dates of the Leipzig Book Fairs (1758–1860), withNotes on the Book Catalogs.”
Sherbornia (2014), no. 1, 1–4.[Ewald 1996] Ewald, W. (Ed.) From Kant to Hilbert: a source book in the founda-tions of mathematics . Vol. I, II. Compiled, edited and with introductions byWilliam Ewald. Oxford Science Publications. The Clarendon Press, OxfordUniversity Press, New York, 1996.[Fila 2020] Fila, M. “Why and how did Bernhard Bolzano develop calculus of theinfinite quantities in
Paradoxien des Unendlichen ?” In preparation (2020).[Freudenthal 1971] Freudenthal, H. “Did Cauchy plagiarize Bolzano?”
Archive forHistory of Exact Sciences (1971), 375–392.[Fuentes Guill´en–Mart´ınez Adame 2020] Fuentes Guill´en, E.; Mart´ınez Adame, C.“The notion of variable quantities ω in Bolzano’s early works.” Histo-ria Mathematica (2020), 25–49. See https://doi.org/10.1016/j.hm.2019.07.002 [Goldbring–Walsh] Goldbring, I.; Walsh, S. “An invitation to nonstandard analysisand its recent applications.” Notices Amer. Math. Soc. (2019), no. 6,842–851.[Grabiner 1981] Grabiner, J. The origins of Cauchy’s rigorous calculus.
MIT Press,Cambridge, Mass.–London, 1981.[Grabiner 1984] Grabiner, J. “Cauchy and Bolzano. Tradition and transformationin the history of mathematics.” Chapter 5 in
Transformation and Traditionin the Sciences: Essays in Honour of I Bernard Cohen , 105–124. Ed. byEverett Mendelsohn. Cambridge University Press, Cambridge, 1984. See https://books.google.co.il/books?id=mJ_u4t1wrLsC
ONTINUITY BETWEEN CAUCHY AND BOLZANO 21 [Grattan-Guinness 1970] Grattan-Guinness, I. “Bolzano, Cauchy and the ‘newanalysis’ of the early nineteenth century.”
Arch. History Exact Sci. (1970),no. 5, 372–400.[Gray 2015] Gray, J. The real and the complex: a history of analysis in the 19th cen-tury . Springer Undergraduate Mathematics Series, Springer, Cham, 2015.[Guitard 1986] Guitard, T. “La querelle des infiniment petits `a l’´Ecole Polytech-nique au XIXe si`ecle.”
Historia Sci. (1986), 1–61.[Heuser 2002] Heuser, H. Lehrbuch der Analysis. Teil 2.
Lehrbegrif der gesamten Mathematik: DieRechenkunst und Geometrie. 1.
R¨ose, Greifswald, 1767.[Karsten 1769] Karsten, W. J. G.
Lehrbegrif der gesamten Mathematik: DieMechanik der festen K¨orper. 4.
R¨ose, Greifswald, 1769. See https://books.google.com.mx/books?id=CNxEAAAAcAAJ [K¨astner 1758] K¨astner, A. G.
Anfangsgr¨unde der Arithmetik, Geometrie, ebe-nen und sph¨arischen Trigonometrie und Perspectiv . Witwe Vandenhoeck,G¨ottingen, 1758.[K¨astner 1760] K¨astner, A. G.
Anfangsgr¨unde der Analysis endlicher Gr¨oßen .Wittwe Vandenhoeck, G¨ottingen, 1760. See http://gdz.sub.uni-goettingen.de/no_cache/dms/load/img/?IDDOC=116776 [K¨astner 1766] K¨astner, A. G.
Anfangsgr¨unde der h¨ohern Mechanik.
Verlag derWittwe Vandenhoek, G¨ottingen, 1766.[K¨astner 1793] K¨astner, A. G.
Anfangsgr¨unde der h¨ohern Mechanik, Auflage 2.
Verlag der Wittwe Vandenhoek, G¨ottingen, 1793.[Katz 2020] Katz, M. “Mathematical conquerors, Unguru polarity, and the task ofhistory.”
Journal of Humanistic Mathematics (2020), no. 1, 475–515. See http://dx.doi.org/10.5642/jhummath.202001.27 and https://arxiv.org/abs/2002.00249 [Katz–Sherry 2013] Katz, M.; Sherry, D. “Leibniz’s infinitesimals: Their fictional-ity, their modern implementations, and their foes from Berkeley to Rus-sell and beyond.” Erkenntnis (2013), no. 3, 571–625. See http://dx.doi.org/10.1007/s10670-012-9370-y and https://arxiv.org/abs/1205.0174 [Kr¨oger 2014] Kr¨oger, D. Abraham Gotthelf K¨astner als Lehrbuchautor. Disserta-tion, Bergische Universit¨at Wuppertal, 2014. urn:nbn:de:hbz:468-20150311-103303-7[Leibniz 1989] Leibniz, G. Philosophical papers and letters . Second Edition. Syn-these Historical Library, Vol. 2. Leroy E. Loemker, Editor and Translator.Kluwer Academic Publishers, Dordrecht–Boston–London, 1989.[L¨utzen 2003] L¨utzen, J. “The foundation of analysis in the 19th century.” Chap-ter 6 in
A history of analysis , 155–195, H. Jahnke (Ed.), History of Mathe-matics, 24, Amer. Math. Soc., Providence, RI, 2003.[Olms 1817] Olms (publisher). Allgemeines Verzeichniß der B¨ucher, welche inder Frankfurter und Leipziger Ostermesse des Jahres entweder ganz neugedruckt, oder sonst verbessert, wieder aufgelegt worden sind . . . (1817)
Accessible to registered libraries at and accessible also at https://books.google.de/books?id=DlpeAAAAcAAJ&pg=PA1
See also Evenhuis [Evenhuis 2014].[Oman 2014] Oman, G. “The converse of the intermediate value theorem: fromConway to Cantor to cosets and beyond.”
Missouri J. Math. Sci . (2014),no. 2, 134–150.[Radcliffe 2016] Radcliffe, D. “A function that is surjective on every interval.” Amer. Math. Monthly (2016), no. 1, 88–89.[Rusnock 1999] Rusnock, P. “Philosophy of mathematics: Bolzano’s responses toKant and Lagrange. Math´ematique et logique chez Bolzano.”
Rev. HistoireSci. (1999), no. 3–4, 399–427.[Rusnock–Kerr-Lawson 2005] Rusnock, P.; Kerr-Lawson, A. “Bolzano and uniformcontinuity.” Historia Mathematica , (2005), no. 3, 303–311.[Rusnock–ˇSebest´ık 2019] Rusnock, P.; ˇSebest´ık, J. Bernard Bolzano: His life andwork.
Oxford University Press, Oxford, 2019.[Russ 2004] Russ, S.
The mathematical works of Bernard Bolzano . Oxford Univer-sity Press, Oxford, 2004.[Schubring 1993] Schubring, G. “Bernard Bolzano–not as unknown to his contem-poraries as is commonly believed?”
Historia Mathematica (1993), no. 1,45–53.[Sebestik 1992] Sebestik, J. Logique et math´ematique chez Bernard Bolzano.
L’Histoire des Sciences: Textes et ´Etudes. Librairie Philosophique J. Vrin,Paris, 1992.[Segre 1994] Segre, M. “Peano’s axioms in their historical context.”
Archive forHistory of Exact Sciences (1994), no. 3–4, 201–342.[Sherry–Katz 2014] Sherry, D.; Katz, M. “Infinitesimals, imaginaries, ideals,and fictions.” Studia Leibnitiana (2012), no. 2, 166–192. See and https://arxiv.org/abs/1304.2137 (Article was published in 2014 even though the journal issue lists theyear as 2012)[Siegmund-Schultze 2009] Siegmund-Schultze, R. Review of Bradley–Sandifer[Bradley–Sandifer 2009] for Zentralblatt (2009). See https://zbmath.org/?q=an%3A1189.26001 [Sinaceur 1973] Sinaceur, H. “Cauchy et Bolzano.” Rev. Histoire Sci. Appl. (1973), no. 2, 97–112.[Smory´nski 2017] Smory´nski, C. MVT: a most valuable theorem.
Springer, Cham,2017.[Spalt 2015] Spalt, D.
Die Analysis im Wandel und Widerstreit . Verlag Karl Auber2015.[Trlifajov´a 2018] Trlifajov´a, K. “Bolzano’s infinite quantities.”
Foundations of Sci-ence (2018), no. 4, 681–704.[V¨ath 2007] V¨ath, M. Nonstandard analysis . Birkh¨auser Verlag, Basel, 2007.
ONTINUITY BETWEEN CAUCHY AND BOLZANO 23
J. Bair, HEC-ULG, University of Liege, 4000 Belgium
E-mail address : [email protected] P. B(cid:32)laszczyk, Institute of Mathematics, Pedagogical University ofCracow, Poland
E-mail address : [email protected] E. Fuentes Guill´en, Department of Mathematics, Faculty of Sci-ences, UNAM, Mexico
E-mail address : [email protected] P. Heinig
E-mail address : [email protected] V. Kanovei, IPPI RAS, Moscow, Russia
E-mail address : [email protected] M. Katz, Department of Mathematics, Bar Ilan University, RamatGan 5290002 Israel
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