Cauchy's Cours d'analyse. An annotated translation
Augustin-Louis Cauchy's (1789--1857) famous ``Cours d'Analyse de l'École Royale Polytechnique:\ 1.re Partie.\ Analyse Algébrique'' (Paris 1821) has now been translated into English for the first time (the English translation will henceforth be quoted as ET). The Cours d'Analyse is based on lectures which Cauchy held at the École Polytechnique in Paris from 1815. It is a well-known fact that Cauchy was not allowed to teach from this text-book when it appeared in 1821, because the authorities found it superfluously rigorous for the use of students. A second part of the Cours d'Analyse with a similar foundational approach never appeared. It is, however, clear that the Cours d'Analyse was enormously influential in the history of mathematics. This was due to Cauchy's -- in his prolific research -- and other mathematicians' references to and use of the fundamental definitions and notions introduced in Cauchy's book. It is sufficient here to mention Cauchy's rigorous introduction of limits, his use of the \(\varepsilon\)-\(\delta\)-formalism in the more complicated proofs, the intermediate value theorem for continuous functions, his insistence that divergent series have no sums, the Cauchy criterion for convergence of series, etc. As an example of Cauchy's style of presentation I quote here his definition of continuity and discontinuity from the translation ET which follows -- except for not reproducing italics for the word ``discontinuous'' which are in the original -- exactly Cauchy's wording and symbolism: ``Let \(f(x)\) be a function of the variable~\(x\), and suppose that for each value of \(x\) between two given limits the function always takes a unique finite value. If, beginning with a value of \(x\) contained between these limits, we add to the variable an infinitely small increment~\(\alpha\), the function itself is incremented by the difference \[ f(x+\alpha) - f(x), \] which depends both on the new variable \(\alpha\) and on the value of~\(x\). Given this, the function \(f(x)\) is a continuous function of \(x\) between the assigned limits if, for each value of \(x\) between these limits, the numerical value of the difference \[ f(x+\alpha) - f(x) \] decreases indefinitely with the numerical value of~\(\alpha\). \dots Finally, whenever the function \(f(x)\) ceases to be continuous in the neighborhood of a particular value of~\(x\), we say that it becomes discontinuous, and there is a \textit{solution of continuity} for this particular value.'' (ET, p.~26) There has been, however, an intense historical discussion in the last four decades or so how to interpret certain apparent remnants of the past or -- as compared to J. L. Lagrange's (1736--1813) rigorous ``Algebraic Analysis'' -- even steps backwards in Cauchy's book, particularly his use of infinitesimals and his apparently flawed proof that a point-wise convergent series of continuous functions has a continuous function as limit. In 1970 the historian Ivor Grattan-Guinness put forward a hypothesis according to which Cauchy received crucial inspiration for the Cours from a nowadays famous paper [``Rein analytischer Beweis des Lehrsatzes, daß zwischen je zwey Werthen, die ein entgegengesetztes Resultat gewähren, wenigstens eine reelle Wurzel der Gleichung liege'' (Zbl 1191.01036)] of 1817 written by the Prague theologian and philosopher Bernhard Bolzano (1781--1848) on the intermediate value theorem for continuous functions, without acknowledging his debt to Bolzano. As to that hypothesis the majority of historians has remained unconvinced, pointing e.g.\ to the deeper understanding of Bolzano of the completeness of the real numbers, to his example of a nowhere differentiable continuous function, and to the fact that Bolzano had to be rediscovered by H.~Hankel and H.~A.~Schwarz around 1870, such that his real impact remained restricted [\textit{J.~Lützen}, ``The foundation of analysis in the 19th century.'' In: H.~N.~Jahnke (ed.), A history of analysis. Providence, RI:\ American Mathematical Society, pp.~155--196 (2003; Zbl 1088.01001); p.~169/70]. Moreover, it has been remarked that enough similarities exist between certain points in the works of Euler, Lagrange, Lacroix, Poisson and the young Cauchy to suggest natural roots of Cauchy's concepts, theorems and proofs (Lützen 2003, p.~161). As to Cauchy's flawed proof mentioned above which was apparently lacking the notion of uniform convergence, certain still existing ambiguities in Cauchy's definition of continuity (see above), in particular not defining continuity point-wise, have invited some historians to interpret his proof as being nevertheless correct. The main argument against this interpretation is Cauchy's own admission in a later publication (1853) that his theorem ``cannot be accepted without restriction'' (Lützen 2003, p.~168). Other historians and mathematicians [\textit{D.~Laugwitz}, Hist. Math. 14, 258--274 (1987; Zbl 0638.01012), \textit{I.~Lakatos}, ``Proofs and refutations. The logic of mathematical discovery.'' Cambridge University Press (1976; Zbl 0334.00022)] have tried to understand Cauchy's use of infinitesimals in the sense of modern non-standard analysis and were thus able to save several theorems and proofs in Cauchy's works, which otherwise look problematic with modern eyes. Lützen, however, argues convincingly against this ``reading post-Weierstrassian ideas into Cauchy's work'' (Lützen 2003, p.~164), not least because a non-Archimedian interpretation of the continuum would clash with the Euclidean theory, which was still the basis of Cauchy's book. Indeed, Cauchy writes in the ``introduction'' to the Cours d'Analyse: ``As for methods, I have sought to give them all the rigor that one demands in geometry, in such a way as never to revert to reasoning drawn from the generality of algebra. Reasoning of this kind, although commonly admitted, particularly in the passage from convergent to divergent series and from real quantities to imaginary expressions, can, it seems to me, only occasionally be considered as inductions suitable for presenting the truth, since they accord so little with the precision so esteemed in the mathematical sciences.'' (quoted from the translation on p.~102 in [\textit{U.~Bottazzini}, The Higher Calculus, Springer (1986; Zbl 0597.01011)]) Unfortunately, the English version ET of the Cours d'Analyse under review translates parts of this passage from Cauchy's introduction differently: ``Arguments of this kind \dots may be considered \dots only as examples serving to introduce the truth some of the time \dots '' (ET, p.~1/2) Obviously the French ``inductions propres'' (suitable inductions) has been replaced by ``introductions propres'' (``serving to introduce''), which clearly has a different meaning. The correct quotation with its methodological intent (as given by Bottazzini) has to be seen in the context of Cauchy's effort to reduce his notions to ``numbers'' (nombres), i.e., basically to points of the Euclidean continuum, which he found mathematically unproblematic. The editors/translators of the English translation remark: ``Cauchy takes pain to define his terms, carefully distinguishing, for example, between \textit{number} and \textit{quantity}. To Cauchy, numbers had to be positive and real, but a quantity could be positive, negative or zero, real or imaginary [on this specific claim see my remarks below; R.S.], finite, infinite or infinitesimal.'' (ET, p.~ix) As we will see, Cauchy's distinctions are even more careful than the editors/translators want us to believe. Anyway, Cauchy's renunciation of ``algebra'' seems to collide, at first sight, with the subtitle of the book: ``1.re Partie.\ Analyse Algébrique''. But the latter has to be read in a sense opposed to ``Analyse infinitésimale''. In fact, one does not find even the full notion of a derivative in the Cours d'Analyse, let alone parts of Cauchy's theory of complex functions, for which he is probably most famous. Cauchy's Cours d'Analyse in its original 1821 edition comprises xvi~+ 576~pages. It does not have a single diagram, possibly in order to rely on verbal and symbolic rigor, although in some places, for instance at the verbal explanation of a coordinate system by means of a ``vertical plane'' (ET, p.~7, for ``le plan est supposé vertical''), a diagram would undoubtedly have helped. (For a general overview of the book see also [\textit{I.~Grattan-Guinness}, ``A.-L.~Cauchy, Cours d' Analyse (1821) and Résumé of the Calculus (1823).'' In:\ I.~Grattan-Guinness (ed.), Landmark Writings in Western Mathematics, 1640--1940; Elsevier, Amsterdam etc., pp.~341--353 (2005; Zbl 1090.01002)], which contains, however, several mistakes in the page numbers.) The Cours in its original edition of 1821 consists of an 8-page introduction, 18~pages ``preliminaries'', 12~chapters (chapitres) and 9~notes (notes). The latter comprise pages 403 to 576 and present ``the derivations which may be useful both to professors and students of the Royal Colleges, as well as to those, who wish to make a special study of analysis'' (ET, pp.~ix/x). Among the notes one finds for instance a special case of the intermediate value theorem for continuous functions (note~III), an extension of Lagrange's interpolation formula to rational functions (note~V) and one (note~IX) on ``products composed of an infinite number of factors''. The content of the main chapters is described by the translators in the following manner: ``The first six chapters deal with real functions of one and several variables, continuity, and the convergence and divergence of series. In \dots Chapters~7 to 10 Cauchy turns to complex variables, what he calls \textit{imaginary quantities}. Much of this parallels what he did with real numbers, but it also includes a very detailed study of roots of imaginary equations. We find here the first use of the words \textit{modulus} and \textit{conjugate} in their modern mathematical senses. Chapter~10 gives Cauchy's proof of the fundamental theorem of algebra, that a polynomial of degree~\(n\) has \(n\) real or complex roots.'' (ET, p.~ix) One should add to this description of the Cours's contents (which is, as we shall see, not totally correct) that Cauchy introduces infinitely small and infinitely large quantities (``quantités infiniment petites ou infiniment grandes'') in the second chapter, before he bases his definition of the continuity of a function on them (see above). Now what is the purpose and the quality of the English translation under review? One should first make it perfectly clear that it is a great achievement to have this influential text for the first time in English, given that many students and scholars today do not read French. Translation has priority here. After all, the French original is today easily accessible online in non-searchable pdf-format in both French editions of 1821 and 1897 (through Google-books and \url{http://gallica.bnf.fr/} respectively). The notion ``Annotated Translation'' which the editors/translators use for their publication, has to be understood in the rather restricted sense of trying to be literally ``correct''. I sympathize with the dedication contained in the edition where the editors/translators thank three historians of mathematics for teaching them ``the importance and satisfaction of reading original sources'' (ET, p.~v). The translators' intent ``to let Cauchy speak for himself'' (ET, p.~xv) is, however, a bit difficult to realize anyhow, because the 1897 edition, on which the translation is based, was published by the French Academy of Sciences in Cauchy's Collected Works after his death. To be sure, the second French edition is very close to the original of 1821, although the different print leads to about 20\%~less pages. In their footnotes, the editors spare no pains finding small typographical errors, introduced in the second French edition. One may ask though whether this is of much relevance for a reader of the English edition. It is helpful, however, to have a page concordance of the two French editions in the appendix of the translation (ET, pp.~397--401). Generally, the reader would have liked to find the French original more often in parentheses in translation, in particular when new notions are introduced. By and large, with few exceptions to be noted below, the translation is fine. It does not try to unduly modernize, except when ``de plus and plus approchées'' is translated as ``better and better approximations'' (ET, p.~6), which seems an unnecessary use of value-laden vocabulary. Nevertheless, given the resources spent on a new edition, and taking into account that providing a translation does not rule out adding a helpful commentary and more thorough introduction as well, the edition leaves some wishes unfulfilled. Bradley and Sandifer, who are known as Euler scholars, make it perfectly clear that they do not aim at a critical historical edition. They state in the ``Translators' Preface'': ``We believe that the primary purpose of a translation such as this is to make the work available in English, and not to provide a platform for our opinions on how this work should be interpreted. Towards this end, we have generally limited our commentary to expository remarks rather than interpretative ones. For those passages that are controversial and subject to a variety of interpretations, we try to refer the interested reader to appropriate entry-point sources and do not try to be comprehensive.'' (ET, p.~xiii/xiv) Even when referring to the literature, in particular to \textit{B.~Belhoste} [``Augustin-Louis Cauchy: A Biography.'' Springer, New York (1991; Zbl 0726.01015)], \textit{H.~Freudenthal} [``Cauchy.'' Dictionary of Scientific Biography, vol.~3, pp.~131--148; Scribners, New York (1971)], \textit{J.V.~Grabiner} [``The Origins of Cauchy's Rigorous Calculus.'' MIT Press, Cambridge (1981; Zbl 0517.01002)], \textit{I.~Grattan-Guinness} [``Bolzano, Cauchy, and the `new analysis' of the early nineteenth century.'' Arch.\ Hist.\ Exact Sci.\ 6, 372--400 (1970; Zbl 0198.00601)], \textit{I.~Grattan-Guinness} [loc. cit. (2005)] and \textit{J.~Lützen} [loc.\ cit.\ (2003)], the editors/translators rarely summarize the state of research, as it was partly described in the introduction of this review. Two of the most extensive contributions, those by \textit{U.~Bottazzini} [``Geometrical rigour and `modern' analysis. An introduction to Cauchy's Cours d'analyse.'' In: U.~Bottazzini (ed.), A.L.~Cauchy, Cours d'Analyse \dots 1821, Clueb, Bologna, pp.~xi--clxvii (1990)] and \textit{C.~Gilain} [``Cauchy et le Cours d'Analyse de l'École Polytechnique.'' Bull.\ Soc.\ Amis Bibl.\ Éc.\ Polytech.\ 5, 3--46 (1989; Zbl 0684.01006)], appear in the bibliography, but are not used. When mentioning Cauchy's proofs of the intermediate value theorem and of the fundamental theorem of algebra the editors/translators do not make an effort to compare them to the proofs by contemporaries such as Bolzano and Gauß, in particular with regard to the standard of rigor. Basically only in one instance do they deviate from their position of objective reserve, and that is when attacking Hans Freudenthal's biography (1971) of Cauchy in the Dictionary of Scientific Biography for allegedly ``many inaccurate citations of Cauchy's work,'' and accusing Freudenthal of ``hero worship'' (ET, p.~xiii). The referee is at a loss about these undocumented accusations. Reading in Freudenthal's biography that ``Cauchy had no clear overall view of his own work'' (Freudenthal 1971, p.~140) and (which Freudenthal quotes from another source) that ``Cauchy was a child who was as naïve as he looked'' (Freudenthal 1971, p.~134) does not sound like ``hero worshipping.'' While the reproduction of the formulas generally seems to be impeccable one finds inaccuracies in the text at several places in the translation. Into the description of the book's content quoted above, into the index and occasionally also into the translation itself (ET, p.~132) the editors/translators smuggle the notion of an ``imaginary quantity'' (even highlighting it in italics) while Cauchy always talks about ``imaginary expressions'' for what we today would call ``complex numbers.'' Apparently Cauchy wanted to reserve ``quantity'' for entities which can be ordered according to their size. Also the notion ``imaginary equation'' is misleadingly quoted, because Cauchy defines the latter just as equality between imaginary expressions (ET, p.~119) and not, as expected from the translators' quote, as a polynomial with complex coefficients. Indeed it is a discussion of the roots of imaginary expressions (i.e., of complex numbers) which Cauchy undertakes in \S 4 of chapter~VII. Cauchy expresses in this context nicely -- by shaded root-symbols -- the many-valuedness of the roots. Cauchy's discussion of polynomials in chapter~X, however, is restricted to polynomials with rational coefficients. Cauchy never uses the word ``infinitésimale'' for ``infiniment petite'', although the editors use the English equivalent ``infinitesimal'' in their translation of Cauchy's ``Preliminaries'' (ET, p.~7). Unfortunately it is exactly this early chapter of Cauchy's Cours d'Analyse, on which much of the work is based, where the English translation contains quite a few slips. ``Quantité'' is occasionally translated as ``number'' (ET, p.~6), despite the warning uttered by the editors themselves in their preface. Now it is fair to say that any translation contains mistakes; and maybe mistakes matter less today because of the existence of the online publications of the original versions for possible checks. However, there is clearly a strong connection between translating and commenting, and the more translators are engaged in interpreting the text the more likely are they to find discrepancies and to avoid mistakes in the translation. By going into the discussion about Cauchy and non-standard analysis (mentioned above) the mistake related to ``imaginary quantities'' would most likely have been avoided. By discussing in more detail Cauchy's notion of singular values (valeurs singulières) the translators would have discovered that the original contains the word ``often'' (souvent), when Cauchy says that points of discontinuity ``often'' give rise to singular values (ET, p.~33). By referring, as Lützen (2003, p.~166) does, to the difference between the definitions of point-wise discontinuity and interval-related continuity the translators would have noticed that Cauchy emphasizes the notion of discontinuity by writing ``discontinue'' in italics. While the translators add correctly in a footnote (to the quotation given above) that ``solution of continuity'' has to be understood as ``that continuity dissolves or disappears'' (ET, p.~26), their later remark about ``what we would call a point of discontinuity'' (ET, p.~33) seems misleading. The examples given may have indicated that the present English translation of Cauchy's Cours d'Analyse would have further benefited by historical and mathematical commentary, which is largely lacking. It seems that the editor of the series in which the translation appeared, Sources and Studies in the History of Mathematics and Physical Sciences, could have mustered one member of his editorial board, Jesper Lützen (to whom this review owes much), to add a historical article to the edition. Maybe this could be a proposal for a second, improved English edition of this historically still valuable text?
scientific article; zbMATH DE number 5557088