Cours d'analyse infinitésimale. Tome I. (Q1505345)

Editorial remark: The original JFM entry refers here to a review by \textit{Jules Tannery} [Darboux Bull. 27, 121--126 (1903)]. With the kind permission of the editors of the Jahresberichte of the DMV, we add here a reprint the 2014 review of \textit{J. Mawhin} [Jahresber. Dtsch. Math.-Ver. 116, No. 4, 243--259 (2014; Zbl 1316.01010)] (with slight adaptations of the references): The \textit{Cours d'analyse infinitésimale} of Charles-Jean de La Vallée Poussin: from innovation to tradition \textbf{1 Introduction} The French mathematical literature has a long tradition of extensive textbooks in mathematical analysis, which seems to have started with the publication of more or less extended versions of the lectures given at the {École polytechnique}, and later in Faculties of Science. Let us just mention the most famous ones published in the XIX\(^{th}\) century by Lagrange, Lacroix, Cauchy, Sturm, Bertrand, Hermite, Jordan at the \textit{École polytechnique}, and by Picard and Goursat at the Faculty of Science of Paris. The corresponding Belgian production is less impressive, started later and is dominated by the \textit{Cours d'analyse infinitésimale} of Charles-Jean de La Vallée Poussin (1866--1962) (in short VP), a two-volumes set, which remains famous (like its author!) for an exceptionally long life and for its international influence, both consequences of its innovative, elegant and rigorous character. This book remained until 1970 the basis for the course on differential and integral calculus of the first two years of studies leading, at the \textit{Université Catholique de Louvain}, either to the degree of \textit{docteur en sciences physiques et mathématiques} (later \textit{licencié en sciences mathématiques} and \textit{licencié en sciences physiques}) or to the degree of \textit{ingénieur civil}. The engineer candidates formed the majority of the class population, and the content and level of the course was a compromise between the wishes of the pure and of the applied users of the material. VP, who remains famous for his proof (independently of Jacques Hadamard (1865--1963)) of the prime number theorem, and for his original contributions to integration, Fourier series, approximation theory, conformal maps and potential theory, graduated in engineering at the \textit{Université Catholique de Louvain} in 1890, and in mathematics and physics in 1891. He was called by the same university in October 1891 to replace Louis-Philippe Gilbert (1832--1892), who was ill, for the course of differential and integral calculus. Gilbert died in 1892, and, from this moment, VP was in charge of those lectures until 1935 (see {[8]}, Vol. 1 for more details). Gilbert had already traced the way by publishing several editions of a \textit{Cours d'analyse infinitésimale. Partie élémentaire} {[26]}, covering his lectures on differential and integral calculus. If the \(1^{st}\) edition (1872) was still defining continuity through the intermediate value property, and claimed its equivalence with Cauchy's definition as well as the differentiability of any continuous function, all this was amended in the \(2^{nd}\) edition (1878), where Darboux's example of a nowhere differentiable continuous function was presented. The \(3^{rd}\) edition (1887) incorporated the recent advances made in Germany on the foundations of analysis and Gilbert, in the \textit{Préface}, defended the opinion that rigor is not the enemy of simplicity. The posthumous \(4^{th}\) edition (1892), used by VP during several years as a support of his lectures, was replaced in 1898--1899 by a mimeographed \textit{Cours d'analyse infinitésimale} in two volumes, that can be seen as the ``zero\(^{th}\) edition'' of VP's famous book. The order and presentation of the material remained very close to the last edition of Gilbert's treatise. \textbf{2 The first edition (1903--1906)} Volume I of the \(1^{st}\) edition of VP's \textit{Cours d'Analyse infinitésimale}, was published in 1903, in a format motivated in the \textit{Preface} by its author who explains that ``{this book [...] must serve together the future engineers and the students preparing the doctorate. To reach this double aim, we have adopted two different types. The text in large ones for the beginners, the one in small types providing to the preceding one all the necessary complements for advanced studies. In the text in large types, the questions are presented in their most elementary form, without never renouncing to rigor. They are reconsidered in the text in small types [...] under the most general viewpoint. [...] The text in large types of this first volume contains the material of the lectures of the first year.}'' The end of the \textit{Preface} reveals that the sources of the author's inspiration are ``{the \textit{Cours d'analyse} of Mr. C. Jordan, the one of our former master Ph. Gilbert, so remarkable by its qualities of exposition, and also the \textit{Leçons sur les applications géométriques de l'analyse} by M. Raffy.}'' Several of the first papers of VP had been motivated by correcting some mistakes in the \(1^{st}\) edition of the \textit{Cours d'analyse} of Camille Jordan (1838--1922), who acknowledged him in the \textit{Preface} of Volume 2 of the substantially modified \(2^{nd}\) edition of his \textit{Cours}. ``{We had given for the conditions under which the differentiation under the sign \(\int\) is legitimate an inexact assertion. Trying to correct this error, that M. de la Vallée Poussin had mentioned to us, we have been led to discuss in detail the essential propositions concerning the definite integrals.}'' A first noticeable difference with the mimeographed edition is the larger space devoted, in the introductory part, to the theory of real numbers, based upon Dedekind's cuts, followed by a much more precise and detailed study of the continuous functions of one or several variables and their properties. There is little novelty in the chapters on differentiation. One should notice that, the author ``proved'' the chain rule for \(f[x(t),y(t)]\) under the too weak assumptions that \(f(x,y)\) has partial derivatives, and \(x(t)\) and \(y(t)\) are differentiable, leading to some imprecision in stating the validity conditions for Taylor's formula. An important addition is a precise study of the existence and uniqueness of implicit functions, using the intermediate value property for one unknown function followed by induction on the dimension. This is applied, in small types, to a rigorous presentation of Lagrange multipliers for constrained extrema. In the presentation of definite simple integrals, Darboux's approach for Riemann integral is used, with, in small types, a general version of the rule of change of variables, and a thorough treatment of Darboux lower and upper integrals. Further conditions for Riemann integrability require considerations on the topology of one-dimensional sets and their Jordan's measurability. At this occasion, the \textit{characteristic function} \(e\) of a bounded set \(E\) is introduced, and Jordan's outer and inner lengths of \(E\) are respectively defined as the upper and the lower integral of \(e\) over some \([a,b] \supset E\). As noticed by Hawkins {[29]}, ``{this connection between measure and integration could not have been stated more clearly.}'' In the applications of calculus to the geometry of curves and surfaces, an orientation is given to the binormal of a curve in space, which provides a sign to the torsion. After a classical presentation of the length of a curve, Jordan's viewpoint based upon functions with bounded variation is adopted and a new proof of Jordan's theorem for simple closed curves is given, with applications to curvilinear integrals. The last chapter, devoted to series, contains Bertrand's logarithmic criteria and du Bois-Reymond's theorem on the impossibility of constructing a complete scale of convergence and divergence. Weierstrass' example of nowhere differentiable continuous function is presented, and completed in VP's paper {[11]}. The \(1^{st}\) edition of Volume 2 of the \textit{Cours d'analyse infinitésimale} came out in 1906. In the \textit{Avertissement}, it is said that one begins ``{by presenting the theory of double integrals, of improper integrals, and in particular of Eulerian integrals in the simplest possible way, [...] rather different from the classical one, but equally natural for students.}'' This is essentially an approach, for continuous functions of two variables, by two successive simple integrations, followed by showing its equivalence to the limit of some double sums. Triple integrals are defined in a similar way. Special attention is paid to Jacobians and change of variable in multiple integrals. Three definitions are given for the area of a surface, and the reduction formulas of volume integrals into surface integrals and of surface integrals into line integrals are proved with care, but in a rather literary way. A more general study of multiple integrals using again Darboux's approach is printed in small types, following the required properties of the sets of integration and their Jordan's measurability. For improper simple integrals, the distinction between convergence and absolute convergence is introduced. In the case of improper double integrals, special attention is paid to their reduction to simple ones, a topic to which VP contributed in 1899 {[10]}. His results {[9]} from 1892 on the uniform convergence of parametric improper integrals are also presented. The substantial part of the volume devoted to ordinary differential equations starts with an original way of solving Cauchy's problem \(y' = f(x,y),\) \(y(x_0) = y_0\), when \(f\) and \(f'_y\) are continuous near \((x_0,y_0)\). The idea consists in constructing step by step, for every \(\alpha > 0\), an approximate solution by the formula \[ \varphi(x_0) = y_0, \;\; \varphi(x) = y_0 + \int_{x_0}^x f(s,\varphi(s - \alpha))\,ds, \] and showing that its limit for \(\alpha \to 0\) is a solution of the Cauchy problem. Rediscovered in 1925 by Leonida Tonelli (1885--1946) {[44]}, this approach now bears his name. In the explicit integration of special differential equations, the discussion of Riccati's equation is more developed than in most textbooks. The treatment of linear differential equations and systems is rather standard, except for an unusual attention paid to Bessel's equation, and the same is true for the short introduction to first order partial and total differential equations. A completely new chapter with respect to the mimeographed version is entitled: ``Special questions: circular and Eulerian functions. Fourier series''. One finds there the expression of circular and hyperbolic functions as infinite products and series of fractions, Bernoulli's numbers and polynomials, and a very detailed treatment of Euler's Beta and Gamma functions. Their use in analytic number theory may have motivated this choice, and most sections are printed in small types. Trigonometric and Fourier series are presented following Dirichlet's approach, and Cantor's uniqueness theorem of the trigonometric expansion is proved. To respect Belgian official programs, the volume ends with chapters on the calculus of variations and the calculus of differences (including Euler's summation formula). A study of the singular points of planar curves and of the curves defined on surfaces concludes the volume. The \textit{Avertissement} of Volume 2 ends by observing that ``{this volume having taken considerable proportions, we have renounced to include the principles of the theory of functions of a complex variable. We hope to be able to publish this theory later with other questions.}'' This should take place in a Volume 3 announced several times but never published. The restriction to functions of real variables remained a characteristic distinction of VP's treatise, with respect to the other more or less contemporary French ones by Jordan, Émile Picard (1856--1941), Édouard Goursat (1858--1936), Georges Humbert (1859--1921), René Baire (1874--1932), and others. \textbf{3 The second edition (1909--1912)} Presenting the second edition of the \textit{Cours d'analyse infinitésimale} as ``considerably reworked'' is not an understatement. In the \textit{Preface} of Volume I, VP, after recalling that the structure and the use of different types was conserved, ``{will not insist on the many modifications made to our first redaction and will only mention here the main one. The theory of definite integrals has been completely renewed, since our first edition, by the beautiful writings of M. Lebesgue. We have thought necessary to introduce in this course the fundamental results obtained in this new way; but we have rather considerably modified the proofs of the author, in order to eliminate the notion of transfinite, which has not yet entered our teaching methods.}'' This introduction in a textbook of the new integral introduced in 1902 by Henri Lebesgue (1875--1941) {[33]} requires a deeper study of set theory. The introductory part is completed, in small types, by a description of the cardinality of sets and of perfect sets. In the chapter on differentiation, the Dini derivatives are considered, including Scheeffer's theorem. For definite integrals, Darboux's presentation of Riemann integral is now followed by measure theory in the real line according to Émile Borel (1871--1956) and Lebesgue, starting with Borel-Lebesgue's lemma and its consequences. The exterior and interior measures are introduced, as well as Borel measurable sets. A necessary and sufficient condition for measurability is proved, followed by the properties of measure with respect to the operations on sets. Measurable real functions of a real variable are defined through the measurability of the counter-image of intervals. The integral for bounded measurable functions is introduced following Lebesgue's division of the range of the function. Two definitions for the extension to unbounded functions are proposed, including the cut-off procedure introduced by VP in {[9]} for improper absolute integrals. The existence of primitives for not necessarily continuous functions follows a new way, whose interest was underlined by Lebesgue himself in his 1910 memoir on integration of functions of several variables {[34]}, recalling that ``{in my \textit{Leçons sur l'intégration}, where I treated the case of one variable, to compare the indefinite integral of \(f(x)\) to a function \(F(x)\) assumed to exist and having \(f\) for derivative, I tried to evaluate \(F(b) - F(a) - \int_a^b f(x)\,dx\), \(a\) and \(b\) arbitrary and fixed, by replacing the curve \(F(x) = y\) by a polygonal line circumscribed for which, consequently, one can define the sides using \(f(x)\). [...] M. de la Vallée-Poussin proceeds differently; he compares the function \(F(x)\), \(x\) variable, to the indefinite integral \(\int f(x)\,dx\) using theorems which generalize the fundamental theorem: \textit{two functions having everywhere the same derivative only differ by a constant}, or Ludwig Scheeffer's theorem. To do this, M. de la Vallée-Poussin uses functions close to \(\int f(x)\,dx\) chosen in such a way that one knows their derivative almost everywhere. This a method imitated from the one of M. de la Vallée-Poussin that I use here. I consider as an advantage of this method to require only a minor study of the functions of several variables.}'' The remaining of Volume I is essentially unchanged with respect to the \(1^{st}\) edition. The section on rectifiable curves is enriched by Lebesgue's result about the differentiability almost everywhere of continuous functions having bounded variation, and the formula for the length as a Lebesgue integral. The chapter on series contains Arzelá's quasi-uniform continuity and Lebesgue's theorem on the integration of sums of series of integrable functions. In the \textit{Preface} to the \(2^{nd}\) edition of Volume II, VP explained that ``{the whole redaction of Volume II has seen more or less deep modifications, but the most important one follows from the introduction of the multiple integrals of M. Lebesgue. We have presented this theory following the fundamental memoirs of the author and we have been led to treat a new question which provides interesting applications of it, namely the development of functions in series of polynomials. Furthermore, the theory of trigonometric series, which owes also to M. Lebesgue its most important advances, has been completely rewritten and adapted to the level of the present knowledge. However, because of lack of space, we have sacrified the theory of Eulerian integrals contained in the first edition.}'' The revision starts with Chapter III, now entitled ``Multiple integrals of Riemann and Lebesgue''. Measure theory and Lebesgue integral follow the lines of the one-dimensional case, with the additional property that a bounded measurable function can be arbitrary closely approximated by a continuous one outside of a subset of arbitrary small measure. This result is generally attributed to Nikolai N. Lusin (1883--1950), but Lusin's note {[35]} is dated June 17, 1912 and VP's Volume II, May 15, 1912, so that the contributions are independent. The study of indefinite multiple integral follows the two years old memoir of Lebesgue {[34]}. The \textit{indefinite integral} of an integrable function \(f\) on \(E\) is defined on measurable subsets \(e \subset E\), with measure \(m(e)\), by \(F(e) = \int_e f(x)\,dx\), and provides an example of countably additive and absolutely continuous set function (\(F(e) \to 0\) when \(m(e) \to 0\)). The study of its derivative requires the introduction of Vitali's covering theorem. The derivative of \(F\) in restricted sense at \(x\) is the limit of \(F(\gamma)/m(\gamma)\), where \(\gamma\) is a ball centered at \(x\) whose radius tends to zero. The general derivative of \(F\) at \(x\) is the limit of \(F(\omega)/m(\omega)\) when \(\omega\) is a measurable set containing \(x\) whose measure tends to \(0\) in such a way that the ratio of \(m(\omega)\) with the measure of the smallest ball \(\gamma \supset \omega\) remains bounded away from zero (regularity condition). The main result is Lebesgue's theorem stating that a countably additive and absolutely continuous set function has a unique finite derivative almost everywhere and is the indefinite integral of this derivative. Lebesgue-Fubini's reduction theorem for multiple integrals, which takes its most elegant and general form within Lebesgue integration theory, is followed by the Leibniz rule for differentiation under the integral sign and the Green formula in the same setting. The approximation of functions by polynomials is treated in Chapter IV. For a continuous function \(f\) on \([a,b]\) with, without loss of generality, \(0 < a < b < 1\), the \(n^{th}\) approximating polynomial \(P_n\) is defined by the integral formula \[ P_n(x) = \frac{3 \cdot 5 \cdot \ldots \cdot (2n+1)}{2(2 \cdot 4 \cdot \ldots \cdot 2n)} \int_a^b f(u)[1 - (u-x)^2]^n \,du. \] This integral, that VP thought to be new when he introduced it in 1908 {[14]}, had been considered a few months earlier by Edmund Landau (1877--1938) for the same purpose {[32]}. For a Lebesgue integrable function \(f\), the theorem of Frederic Riesz (1880--1956) [40] that \(P_n(x)\) converges to \(f(x)\) at any point where \(f(x)\) is the derivative of its indefinite integral (and hence almost everywhere) is proved. The approximation of functions of several variables is also considered with, for Lebesgue integrable functions, a variant of a recent theorem of Leonida Tonelli (1885--1946) {[43]}. Chapter IV ends with the study of trigonometric and Fourier series, completing the \(1^{st}\) edition, in small types, by the important advances made by Lebesgue and others using the new integral. To Dini's and Jordan's convergence criteria of the Fourier series of \(f\), VP added the one he published in 1911 {[17]}, on the convergence to the limit for \(\alpha = 0\) of \((1/2\pi) \int_0^\alpha [f(x+\alpha) + f(x-\alpha)]\,d\alpha\), at any point \(x\) such that this function has bounded variation in \(\alpha\) in a small interval \([0,\varepsilon]\). The summation of divergent Fourier series uses essentially Fejér's method, both for continuous and Lebesgue integrable functions. The new summation method introduced by VP in {[14]} is only mentioned. Follows Parseval formula for \(L^2\)-functions, du Bois-Reymond and Lebesgue singularities for continuous ones. Cantor's uniqueness result is completed by recent results of Lebesgue. The space given to the integration of total differential equations is tripled with respect to the \(1^{st}\) edition and includes VP's results in {[13]}. The remaining of Volume II is essentially unchanged, except that Bernoulli's numbers and polynomials appear in the chapter on difference equations. When discussing the envelopes of planar curves, VP's complete treatment given in {[15]} is quoted. This \(2^{nd}\) edition of VP's \textit{Cours d'analyse} is, in 1912, the only textbook on analysis containing both Lebesgue integral and its application to Fourier series, and a general theory of approximation of functions by polynomials. \textbf{4 The third edition (1914) and its ``ghost'' Volume II} Once more, the \textit{Avertissement} to the \(3^{rd}\) edition of Volume I, published in the Spring of 1914, summarizes the modifications introduced there. The first one concerns differentiability where one has ``{abandoned the old definition of the total differential and adopted Stolz' one [41]. Its superiority has been emphasized by the works of MM. S. Pierpont [sic] {[37]}, Fréchet {[25]} and mostly W. H. Young [46].}'' In 1893, in Volume I of a remarkably modern book on differential and integral calculus [41], Otto Stolz (1842--1905), a former student of Weierstrass, had defined for the first time the modern concept of (total) differential of a real function \(f\) of \(n\) variables at a point \(x\), equivalent to classical differentiability when \(n = 1\), and lying between the existence of the partial derivatives at \(x\) and their continuity at \(x\) when \(n > 1\). In VP's own terminology, the function \(u(x,y)\) is differentiable at \((x,y)\) if it is defined in a neighborhood of this point and if its variation \(\Delta u = u(x + \Delta x,y + \Delta y) - u(x,y)\) can be decomposed in two parts \((A \Delta x + B \Delta y) + (|\Delta x| + |\Delta y|)\varepsilon,\) with \(A\) and \(B\) independent of \(\Delta x\) and \(\Delta y\), and \(\lim_{|\Delta x| + |\Delta y| \to 0} \varepsilon = 0.\) As observed by VP in his \textit{Avertissement}, its ``{superiority [...] is unquestionable: the theorems follow more directly from the principles, the theory of differentiation of explicit and implicit functions becomes sharper, and, consequently, more satisfactory.}'' The (justified) enthusiasm of VP led him to the following version of the implicit function theorem, attributed to William H. Young (1863--1942): given \(n\) Functions \(F_1,\ldots,F_n\) of the \(m+n\) variables \((x,y,\ldots,u,v,w,\ldots)\) which vanish, are totally differentiable, and have a non-zero Jacobian with respect to the \(u,v,w,\ldots\) at \((a,b,\ldots,u_0,v_0,w_0,\ldots)\), there exists at least a system of functions \(u,v,w,\ldots\) of \((x,y,\ldots)\) equal to \(u_0,v_0,w_0,\ldots\) at \((a,b,\dots)\), and satisfying identically the equations \(F_1 = 0,\) \(\ldots,\) \(F_n = 0\) in a neighborhood of this point. This result is proved for \(n = 1\) using Bolzano's intermediate value theorem, which is correct, but VP's induction argument to go from this result to arbitrary \(n\) is not. A correct proof requires more sophisticated tools, like Brouwer's fixed point theorem. Notice that, despite of VP's generous attribution, this generalized implicit function theorem can hardly be found in [46]. The other substantial modification in the \(3^{rd}\) edition of Volume I of the \textit{Cours d'analyse} concerns advanced measure and integration. The main lines are again summarized in the \textit{Avertissement}. ``{We have moved to the introductory part, and simplified, measure theory, previously located in the chapter on definite integrals. We have completely reworked the theory of Lebesgue integral, but conserved the processus we had previously introduced to go from the derivative to the primitive. [...] Its use occurs in two new sections, one devoted to the problem of change of variables in a definite integral, which seems to receive here its definitive solution, the other one to the search of the primitive of a generalized second order derivative, fundamental in the theory of Fourier series.}'' For Lebesgue integration, much more emphasis is put on the limit process under the integral sign, with the dominated convergence and the monotone convergence theorems. For the search of primitive functions, VP's original approach of the \(2^{nd}\) edition is developed by expliciting the concepts of \textit{major (minor) functions} of a Lebesgue integrable function \(f\) on \([a,b]\). They are continuous functions, infinitely close from above (below) to \(\int_a^x f(t)\,dt\), having Dini derivatives greater (smaller) than \(f(x)\) at any point where \(f\) is finite. They are used to express, for a function \(F\) having bounded variation on \([a,b]\), \(F(b) - F(a)\) in terms of \(\int_a^b F'(x)\,dx\) and the total variation of \(F\) in the set of points where \(F'\) is not determined or infinite \textit{(VP's decomposition theorem)}, and to show that the absolute continuity of \(F\) is necessary and sufficient for being the indefinite integral of its derivative. Generalizations of the major and minor functions have been the starting point of Oskar Perron (1880--1975), for introducing in 1914 his generalization of Lebesgue integral {[36]}. The \textit{Encyclopédie des sciences mathématiques} (t. II, vol. 1, p. 100) having reproached to VP's original proof of Jordan's theorem for a closed simple curve ``to be only indicated'', details are added, ``to which one can give a pure arithmetical sense''. The volume ends with an ``Addition to Volume II (\(2{nd}\) edition)'', describing VP's recent results {[19]} on the uniqueness of the expansion of a function in trigonometrical series. The \(3^{rd}\) edition of Volume II of the \textit{Cours d'analyse infinitésimale} never appeared. A brief explanation is given in the Introduction of VP's paper of 1915 on Lebesgue integral {[21]} which recalls that ``{many of the obtained results [...] were already printed in August 1914 and were supposed to appear at the end of this same year in the \(3^{rd}\) edition of Volume II of my \textit{Cours d'analyse}. All that has been burnt in Louvain with many other more precious things.}'' During World War I, the German troops invading Belgium reached Louvain ion August 19, 1914. Alleging the activity of franc-tireurs whose existence was never proved, they reacted in a very brutal way. In particular, they set fire to the old buildings of the University of Louvain during the night of August 25, destroying completely the library and its precious collections. The Uystpruyst printing house publishing VP's \textit{Cours d'analyse}, next to the library, burnt as well, including the material related to the \(3^{rd}\) edition of Volume II. VP left Belgium until the end of the war, as, successively, guest professor at Harvard University, the Faculty of Science and the \textit{Collège de France} in Paris, and the University of Geneva. In Paris, he published his famous monograph on Lebesgue integral {[22]}, repeating in its Introduction the story of the \(3^{rd}\) edition of Volume II of the \textit{Cours d'analyse}. For a long time, the last three sections of {[21]} on multiple Lebesgue integrals remained the unique source for guessing the evolution of the corresponding chapter in this lost \(3^{rd}\) edition. The main novelty was the concept of derivative on a dyadic net, for completely additive set functions having bounded variation, which, as mentioned by VP, developed a remark made in passing by Lebesgue in {[34]}. This notion allowed to prove generalizations of the decomposition theorem of Volume I, and to apply it to continuous functions of two variables having bounded variation. The last section of {[21]} treated the change of variables in double integrals, where, again, Stolz' differentiation was used, ``{showing its superiority, once more, in the present question.}'' Some years ago, when I was working with Paul Butzer and Pasquale Vetro on the publication of the \textit{Collected Works} of VP {[8]}, a great-grandson of the Belgian mathematician showed me a collection of galley proofs he was keeping in memory of his great-grandfather. He kindly allowed me to study them and copy what could be interesting. Most corresponded to published material, but a careful analysis showed me that some where galley proofs of parts of the \(3^{rd}\) edition of Volume II of the \textit{Cours d'analyse}. They covered the end of chapter II on improper integrals and exact differentials, the whole Chapter III on multiple integrals and the beginning of Chapter IV on the analytical representation of functions and Fourier series. Other ones were galley proofs of the \(2^{nd}\) edition of the remaining part of Chapter IV, annotated by VP for the \(3^{rd}\) edition. With this material, that I hope to publish in the future, it is possible to reconstruct the most original part of the ``ghost'' edition. With respect to the \(2^{nd}\) edition, Chapter II is enriched by VP's extension {[20]} of Goursat's technique {[27]} for proving Cauchy's integral theorem, to the obtention of necessary and sufficient conditions for the line integral of \(P\,dx + Q\,dy\), with \(P\) and \(Q\) totally differentiable, to depend only upon its extremities. In Chapter III, the main addition is the derivative on a dyadic net of additive functions with bounded variation, similar to the one in {[21]}. The section on change of variables in integrals is the one given in the \(2^{nd}\) edition, except mentioning that the proof comes from VP's paper {[16]}. For Chapter IV, the modifications are essentially stylistic or consist in adding a few computational details. The reference to VP's memoir {[14]} is added to the theorem of the approximation of the derivatives of the function by the derivatives of polynomials. The addition to the \(2^{nd}\) edition of Volume II given in the \(3^{rd}\) edition of Volume I, mentioned above, becomes the last section of Chapter IV. A beautiful analysis of the contributions of VP to Fourier series, including the evolution of their treatment in the \textit{Cours d'analyse}, is given by Jean-Pierre Kahane on pp. 573--586 of {[8]}, Vol. 3. Another collateral damage of World War I to VP's \textit{Cours d'analyse} is that its German translation, planned just before the war, never materialized. \textbf{5 The fourth (1921--1922) and subsequent editions} One could think \textit{a priori} that the loss of the \(3^{rd}\) edition of Volume II would have been compensated by the publication of the \(4^{th}\) edition. That it is not the case is explained in the \textit{Avertissement} to this edition of Volume I. ``{The printing of Volume I, started in 1919, has met, at the beginning, serious material difficulties. To save time, I have suppressed the questions printed in small types in the old edition and, in particular, the theories related to Lebesgue integral, that I hope to include in a third volume.}'' The \textit{Avertissement} to \(4^{th}\) edition of Volume II is more precise. ``{The \(3^{rd}\) edition of Volume II of this \textit{Cours d'Analyse} was burnt in Louvain in 1914 before its completion and never came to light. It contained a rather extended contribution to set theory and Lebesgue integral. Since this time, I returned to those questions and I have published [...] my book \textit{Intégrales de Lebesgue, fonctions d'ensemble, classes de Baire (Paris, Gauthier-Villars, 1916)}. For the reasons I have given in the \textit{Avertissement} of Volume I, those questions have been kept apart from the present volume, but maybe will find place with other ones in a Volume III.}'' This planned Volume III never appeared. With the \(4^{th}\) edition, VP's \textit{Cours d'analyse} more or less returned to the content of the \(1^{st}\) edition and remained essentially unchanged in the many subsequent editions, except for the applications of analysis to geometry, as described later, and for a more classical approach to Cauchy's existence theorem for differential equations. The elegance and rigor of the style had not been lost, but most of the XX\(^{th}\) century material had disappeared. In Volume II, the chapter on Eulerian integrals of the \(1^{st}\) edition, suppressed in the \(2^{nd}\) one, was reintroduced, and, for Bessel equation, VP's paper of 1905 {[12]} on its integration in finite terms was mentioned. Because of the absence of the \(3^{rd}\) edition of Volume II, the best available complete set for the \textit{Cours d'analyse infinitésimale} is the \(3^{rd}\) edition of Volume I (1914) joined to the \(2^{nd}\) edition of Volume II (1912). They have been translated in Russian, and have been reprinted in 2003 by Gabay in Paris. At the time of their publication, their unique competitor was the first edition of \textit{The Theory of Functions of a Real Variable and the Theory of Fourier Series} {[31]} published in 1907 by Ernest W. Hobson (1856--1933), giving, in a much less elegant style and a less original way, the first presentation of Lebesgue integral in book form and English language. The first treatise in German on real functions including Lebesgue's measure and integral, published in 1918, was the \textit{Vorlesungen über reelle Funktionen} [2] of Constantin Carathéodory (1873--1950), who wrote that ``{äusserst originell sind ausserdem die Darstellungen bei de la Vallée Poussin.}'' Carathéodory published in 1939 the first volume of another book on real functions [3]. Irony of history, the second volume never came to light, the whole edition being destroyed in the publishing house Teubner during the bombing of Leipzig by the Royal Air Force of December 4, 1943! Volume I of the \textit{Cours d'analyse} has seen a total of twelve editions, the last one in 1959, and Volume II nine, the last one in 1957, an exceptionally long life for a mathematical book. The difference in the number of editions for the two volumes is explained by the fact that Volume I covered the material for the first year students at the \textit{Université Catholique de Louvain}, and Volume II the one for the second year students, whose population was substantially smaller. In the \textit{Preface} of the \(6^{th}\) edition of Volume II, VP announced that he ``{revised with the greatest care the part of volume devoted to the geometrical applications. The principles of the theory of envelopes, which could look somewhat wavering, have got all the wanted precision. I have thought that the time had come to introduce in the theory of space curves the consideration of the moving frame whose use has much spread. [...] I have developed somewhat more than before the theory of the curvature of surfaces: I have made more precise the interpretation of signs, I have exposed O. Bonnet's theorem establishing the relation between the total curvature of a surface and the geodetic curvature of its contour, and finally I have extended the determination of a surface through its six parameters.}'' The volume gained some fifty pages, and some of the improvements came from VP's papers {[23]} and {[24]}. In 1935, VP abandoned the course on differential and integral calculus to his former student Fernand Simonart (1888--1966), a differential geometer, who contributed to the revisions of the last editions of the \textit{Cours}. The transition was courteously anounced in the \textit{Preface} of the \(8^{th}\) edition. ``{After having had the honor to teach myself the material of this course at the University of Louvain during forty-five years, I have been particularly happy to see this work assigned to one of my most distinguished former students, M. Fernand Simonart, to-day my colleague since already many years [...]. He has spontaneously offered me his precious help for the revision of material and the publication of this eighth edition. The book has remained essentially the same, but M. Simonart has introduced many improvements of details and judicious additions.}'' Those additions essentially deal with the geometrical applications of differential and integral calculus. Simonart also introduced the language of vector analysis in the theorems of Green, Stokes and Ostrogradsky. \textbf{6 Reception, influence and modernity} The preceding lines have, I hope, convinced the reader of the exceptional quality, both in content and in style, of the \textit{Cours d'analyse infinitésimale}. The first editions, with their important changes, received very positive reviews. Jules Tannery (1848--1910) [42] observed, for the \(1^{st}\) edition of Volume I, that ``{although the good books on this topic are numerous, nobody will regret the publication of this new course of analysis; the effort made by the author to found exclusively the teaching of analysis on perfectly rigorous notions is worthy of attention, especially because it is really an elementary book that he wanted to write, and that he has written.}'' The same reviewer noticed, concerning the same edition of Volume II, ``{the very simple proof of the existence theorem for ordinary differential equations of the first order, ingeniously based upon the consideration of a function satisfying such an equation with an error smaller than any given quantity.}'' Robert d'Adhémar (1874--1941) {[4]} wrote, for the \(2^{nd}\) edition of Volume I, that ``{the first edition of this book was excellent; the present one is a true jewel. [...] This chapter [on Lebesgue integral] where M. de la Vallée has put originality and a great force of synthesis, will be remarked by the scientists,}'' and, for the \(2^{nd}\) edition of Volume II, that ``{written in this way by a very rigorous and penetrating mind, the book has the beauty of strong and classical things, and I am sure that more than one professor of analysis, before giving his lecture, [...] will read the corresponding chapter of the book of M. de la Vallée Poussin.}'' The same author noticed that ``{Volume I, \(3^{rd}\) edition, and Volume II, \(2^{nd}\) edition, will be translated in German and we impatiently wait for a Volume III, by which the book would do a precious service to French students. [...] This sole book where one can find a didactic presentation of Lebesgue integral [...] can be compared to the one of M. Camille Jordan. There is nothing more to say.}'' The analysis was confirmed by Paul Mansion (1844--1919) in JFM 45.1281.03. ``{Dieses Buch, dessen deutsche Übersetzung sich beim Ausbruch des Krieges in Vorbereitung befand, ist wohl -- besonders wegen der Strenge der Darstellung -- das beste unter allen französischen Lehrbüchern.}'' Reviewing the same volumes for the AMS {[38]}, M.B. Porter observes that ``{the handling throughout is clear, elegant, and concise; the various topics are illustrated by numerous carefully chosen examples selected with rare pedagogic skill to develop a real understanding of the text. [...] It is impossible to point out all the merits of these volumes, so rich in varied topics, so lucid in exposition and elegant in presentation. A unique feature of the book is that it does for Lebesgue's integral what Jordan did for Riemann's theory.}'' For Rolin Wavre {[45]}, analyzing the \(4^{th}\) edition, ``{for the real domain, the course of M. de la Vallée-Poussin is, in many points, more detailed than those of MM. Jordan, Picard or Goursat, to which one will often be tempted to compare it. A deeper analysis would be necessary to show the personal contribution of the author to the treated matters. This contribution is undoubtly very important.}'' For the \(5^{th}\) edition of Volume I, Porter {[39]} observes that ``{even in following the conventional order of the French treatises, de la Vallée Poussin displays his usual elegance and simplicity of presentation so that the most hackneyed matters acquire a new interest. [...] The treatment of indeterminate forms is the best the reviewer knows of. [...] In conclusion, it may be said that this is one of the most valuable handbooks on modern analysis in any language and an English translation of it would be a welcome addition to our literature of the subject.}'' This wish was not realized, but the \(8^{th}\) edition was reprinted by Dover in 1946. The influence of VP's \textit{Cours d'analyse infinitésimale} has been deep and long lasting, as revealed by many testimonies. In the Introduction to the \(8^{th}\) edition of his \textit{Course of pure Mathematics} {[28]}, Godefrey Harold Hardy (1877--1947) expressed his indebtedness. ``{I have rewritten the parts of Chs. VI and VII which deal with the elementary properties of differential coefficients. Here I have found de la Vallée-Poussin's \textit{Cours d'analyse} the best guide, and I am sure that this part of the book is much improved.}'' When publishing Volume I of his monograph \textit{Analysis}, Einar Hille (1894--1980) recalled in the Introduction that ``{fifty years ago, in preparing for a comprehensive examination, I read Ch.J. de la Vallée Poussin \textit{Cours d'analyse infinitésimale}, vol. 1 (1909). This treatise left a lasting impression and, when my book was planned, that of the Belgian master served as the model although the final product differs from it in many respect.}'' In his obituary of VP [1], John Charles Burkill (1900--1993) noticed that ``{the contribution to mathematical literature for which he is most widely known is his \textit{Cours d'Analyse} [...]. If Jordan's is the most noble of the \textit{Cours d'Analyse} and perhaps Goursat's [...] the most widely read, it can hardly be doubted that VP's is the most elegant and lucid. After half a century it is still put before the more able undergraduates as a model of style, and there are parts of it which no other writer has presented with anything like the same economy and clarity.}'' Almost one century after its publication, VP's \textit{Cours d'analyse infinitésimale} in its \(4^{th}\) or later edition, would not require substantial modifications to be used to-day as a reference text in an advanced calculus course. The XIX\(^{th}\) century concept of limit of a variable should be replaced by that of limit of a sequence, with a sequence defined as a mapping from the natural integers to the real numbers. The given (correct) proofs of the fundamental properties of a continuous function on a closed interval could be simplified by replacing the rather cumbersome result about the oscillation of the function by Borel-Lebesgue lemma. Similarly for functions of several variables, where the concept of domain bounded by a curve in the plane should be replaced by the more general one of compact. All proofs about differentiation and differentiable functions are still up-to-date. The presentation of the implicit function theorem could be saved by replacing the generalized assumptions mentioned above by the standard ones. The approach to indefinite integrals needs no modification and definite (Riemann) integration in dimension one is well described using Darboux's method. VP's approach of multiple integrals is somewhat discursive and should be made more rigorous. The presentation of numerical series and series of functions is better and more complete than in many contemporary textbooks, and the same is true for improper, curvilinear and Eulerian integrals. The theory of Fourier series, in the absence of Lebesgue integral, can hardly be improved and the chapters on differential equations do not differ essentially from the corresponding ones in a modern book of advanced calculus. Under the influence of Bourbaki's \textit{Éléments de mathématique}, the mathematical style has been formalized in the second half of the XX\(^{th}\) century, and one could be tempted, to translate VP's \textit{Cours d'analyse infinitésimale} in the language of Jean Dieudonné's \textit{Éléments d'analyse}. The price paid for a greatest apparent rigor would be a substantial loss in style and elegance. Since the pioneering period where VP was including Lebesgue integration in two early editions of his \textit{Cours d'analyse}, many other approaches have been devised to introduce Lebesgue measure and integral. The question of presenting measure before integral, or integral before measure has been warmly discussed, and is a matter of taste. However, VP's presentation remains, after one century, a very readable and valuable reference for learning Lebesgue's theory in \(\mathbb R^n\). \textbf{7 Conclusion} The \textit{Cours d'analyse infinitésimale} of VP, with his many editions, has been most influential all around the world in providing to beginners a clear and rigorous introduction to the foundations and applications of differential and integral calculus. When travelling abroad and telling that I was professor at the \textit{Université Catholique de Louvain}, I was surprised to hear comments from so many interlocutors, about the fundamental role the \textit{Cours d'analyse} had played in their mathematical training. They all praised the elegance and clarity of the style, the choice of the topics and their presentation. On the other hand, the 2nd and 3rd editions have been the royal way used by many mathematicians of the first quarter of the XXth century to get acquainted with Lebesgue integral and its application to Fourier series. In the French literature, one had to wait for World War II to see a timid and short introduction to Lebesgue integral in the ultimate edition of Picard's \textit{Traité d'analyse} and in Valiron's \textit{Cours d'analyse}. No one could be compared in elegance and in comprehensiveness to the two volumes of the Belgian mathematician. One could hardly find an introduction to analysis containing so many original contributions of his author. The \textit{Cours d'analyse infinitésimale} is a beautiful example of the indispensable fruitful relation between research and teaching. VP's teaching influenced his research, and VP's research influenced his teaching. This is the secret of great mathematical books. In his already quoted paper in [8], Vol. 3, Jean-Pierre Kahane insisted that ``{there is something in common to the productions of youth and of maturity of de La Vallée Poussin, about Fourier analysis as well as the other domains of his activity. He is simple, elegant and precise. To read him to-day is both a good lecture of mathematics and a beautiful lecture on language and on style.}'' I hope this paper will contribute to motivate mathematicians to pay a first or another visit to VP's \textit{Cours d'analyse infinitésimale}. \textbf{References} {\parindent=10mm \begin{itemize}\item[[1]] \textit{J. C. Burkill}, Charles-Joseph [sic] de la Vallée Poussin (1866--1962), J. Lond. Math. Soc. 39, 165--175 (1964; Zbl 0118.01205) \item[[2]] \textit{C. Carathéodory}, Vorlesungen über reelle Funktionen. Leipzig und Berlin: B. G. Teubner (1918; JFM 46.0376.12) \item[[3]] \textit{C. Carathéodory}, Reelle Funktionen. Bd. 1: Zahlen-Punktmengen-Funktionen. Leipzig, Berlin: B. G. Teubner (1939; Zbl 0022.11904; JFM 65.0192.03) \item[[4]] \textit{R. d'Adhémar}, Cours d'analyse infinitésimale par Ch. J. de la Vallée Poussin. Tome I, 2e éd., Revue Questions Sci. 16, 247--248 (1909); tome II, 2\(^e\) éd., ibid. 23, 615--616 (1913); tome I, 3e éd., ibid. 25, 251 (1914) \item[[5]] \textit{Ch. J. de La Vallée Poussin}, Cours d'analyse infinitésimale. Mimeographed. in-\(4^\circ\). Uystpruyst, Louvain. 1\(^{re}\) partie, 212 pp. (1898); 2\(^{e}\) partie, 222 pp. (1899) \item[[6]] \textit{Ch. J. de La Vallée Poussin}, Cours d'analyse infinitésimale. Tome I. Uystpruyst, Louvain and Gauthier-Villars, Paris. \(1^{st}\) ed. 1903, xiv+372 pp. \(2^{nd}\) ed. 1909, xii+423 pp. \(3^{rd}\) ed. 1914, ix+452 pp. (Reprint: Gabay, Paris, 2003. Russian translation: Petrograd, 1921). \(4^{th}\) ed. 1921, x+434 pp. \(5^{th}\) ed. 1923, vii+436 pp. \(6^{th}\) ed. 1926, vii+436 pp. \(7^{th}\) ed. 1930, viii+448 pp. \(8^{th}\) ed. 1938, x+460 pp. (with collaboration of F. Simonart). (Reprint: Dover, New York, 1946). \(9^{th}\) ed. 1943, vii+472 pp. (with collaboration of F. Simonart). \(10^{th}\) ed. 1947, x+480 pp. (with collaboration of F. Simonart). \(11^{th}\) ed. 1954, viii+480 pp. (with collaboration of F. Simonart). \(12^{th}\) ed. 1959, viii+484 pp. (with collaboration of F. Simonart) \item[[7]] \textit{Ch. J. de La Vallée Poussin}, Cours d'analyse infinitésimale. Tome II. Uystpruyst, Louvain and Gauthier-Villars, Paris. \(1^{st}\) ed., 1906, xvi + 440 pp. \(2^{nd}\) ed., 1912, ix + 464 pp. (Reprint: Gabay, Paris, 2003. Russian translation: Leningrad, 1933). \(3^{rd}\) ed., 1914 (destroyed in World War I; unpublished). \(4^{th}\) ed., 1922, xiv + 478 pp. \(5^{th}\) ed., 1925, xi + 478 pp. \(6^{th}\) ed., 1928, viii + 525 pp. \(7^{th}\) ed., 1937, x + 525 pp. (Reprint: Dover, New York, 1946). \(8^{th}\) ed., 1949, viii + 548 pp. (with collaboration of F. Simonart). \(9^{th}\) ed., 1957, viii + 552 pp. (with collaboration of F. Simonart) \item[[8]] \textit{Ch. J. de La Vallée Poussin}, Collected Works. Œuvres scientifiques, P. Butzer, J. Mawhin, P. Vetro ed. Académie royale de Belgique, Bruxelles and Circolo Matematico di Palermo, Palermo. Vol. I. Biography and number theory (2000); Vol. II. Integration and measure, probability, ordinary differential equations, mechanics, geometry (2001); Vol. III. Approximation theory, Fourier analysis, quasi-analytic functions (2004); Vol. IV. Complex functions, conformal representation, potential theory (to appear) \item[[9]] \textit{Ch. J. de La Vallée Poussin}, Étude des intégrales à limites infinies pour lesquelles la fonction sous le signe est continue, Ann. Soc. Sci. Bruxelles II, 16, 150--180 (1892; JFM 24.0262.01) (CW II, 33--63) \item[[10]] \textit{Ch. J. de La Vallée Poussin}, Réduction des intégrales multiples généralisées, Journ. de Math. (5) 5, 191--204 (1899; JFM 30.0278.01) (CW II, 115--128) \item[[11]] \textit{Ch. J. de La Vallée Poussin}, Sur la fonction sans dérivée de Weierstrass, Ann. Soc. Sci. Bruxelles I, 27, 92--95 (1903; JFM 34.0410.04) (CW III, 483--486) \item[[12]] \textit{Ch. J. de La Vallée Poussin}, Intégration de l'équation de Bessel sous forme finie, Ann. Soc. Sci. Bruxelles II, 29, 140--143 (1905; JFM 36.0397.03) (CW II, 499--502) \item[[13]] \textit{Ch. J. de La Vallée Poussin}, Sur les équations différentielles totales, Ann. Soc. Sci. Bruxelles II, 30, 295--298 (1906; JFM 37.0329.01) (CW II, 511--514) \item[[14]] \textit{Ch. J. de La Vallée Poussin}, Sur l'approximation des fonctions d'une variable réelle et de leurs dérivées par des polynomes et des suites limitées de Fourier, Bull. Cl. Sci. Acad. Roy. Belgique (4) 10, 193--254 (1908; JFM 39.0329.02) (CW III, 3--64) \item[[15]] \textit{Ch. J. de La Vallée Poussin}, Sur les enveloppes de courbes planes qui ont un contact d'ordre supérieur avec leurs enveloppées, Mem. Pontif. Accad. Romana Nuovi Lincei 28, 43--50 (1910; JFM 41.0644.01) (CW II, 641--648) \item[[16]] \textit{Ch. J. de La Vallée Poussin}, Réduction des intégrales doubles de Lebesgue. Application à la définition des fonctions analytiques, Bull. Cl. Sci. Acad. Roy. Belgique (4) 12, 768--798 (1910; JFM 41.0329.03) (CW II, 131--798) \item[[17]] \textit{Ch. J. de La Vallée Poussin}, Un nouveau cas de convergence des séries de Fourier, Rend. Circ. Mat. Palermo 31, 296--299 (1911; JFM 42.0283.01) (CW III, 487--490) \item[[18]] \textit{Ch. J. de La Vallée Poussin}, Sur l'unicité du développement trigonométrique, Bull. Cl. Sci. Acad. Roy. Belgique (5) 2, 702--718 (1912; JFM 43.0320.03) (CW III, 495--511) [anounced in C. R. Acad. Sci., Paris 155, 951--953 (1912; JFM 43.0320.02), same title (CW III, 491--493)] \item[[19]] \textit{Ch. J. de La Vallée Poussin}, Sur l'unicité du développement trigonométrique. Note additionnelle, Bull. Cl. Sci. Acad. Roy. Belgique (5) 3, 9--14 (1913; JFM 44.0486.01) (CW III, 513--518) \item[[20]] \textit{Ch. J. de La Vallée Poussin}, Sur la définition de la différentielle totale et sur les intégrales curvilignes qui ne dépendent que de leurs limites, Ann. Soc. Sci. Bruxelles I, 38, 67--72 (1914; JFM 45.0452.03) (CW IV, to appear) \item[[21]] \textit{Ch. J. de La Vallée Poussin}, Sur l'intégrale de Lebesgue, Trans. Am. Math. Soc. 16, 435--501 (1915; JFM 45.0441.06) (CW II, 165--501) \item[[22]] \textit{Ch. J. de La Vallée Poussin}, Intégrales de Lebesgue. Fonctions d'ensemble. Classes de Baire. Paris: Gauthier-Villars. 1st ed. (1916; JFM 46.1519.01); 2nd ed. (1934; JFM 60.0193.06) \item[[23]] \textit{Ch. J. de La Vallée Poussin}, Sur les théorèmes d'existence de la théorie du plan osculateur, Ann. Soc. Sci. Bruxelles I, 46, 524--542 (1926; JFM 52.0691.11) (CW II, 649--667) \item[[24]] \textit{Ch. J. de La Vallée Poussin}, Sur les enveloppes de courbes planes, Ann. Soc. Sci. Bruxelles I, 48, 5--9 (1928; JFM 54.0723.02) (CW II, 669--673) \item[[25]] \textit{M. Fréchet}, Sur la notion de différentielle totale, \textit{Nouv. Ann. Math.} (4) 12, 385--403 (1912; JFM 43.0481.03), ibid, 433--449 (1912; JFM 43.0482.01) \item[[26]] \textit{L. P. Gilbert}, Cours d'analyse infinitésimale. Partie élémentaire. Paris: Gauthier-Villars. 1st ed. (1872; JFM 04.0118.02); 2nd ed. (1878; JFM 10.0197.02); 3rd ed. (1887; JFM 19.0247.01); 4th ed. (1892; JFM 24.0247.04) \item[[27]] \textit{E. Goursat}, Sur la définition générale des fonctions analytiques d'après Cauchy, Trans. Am. Math. Soc. 1, 14--16 (1900; JFM 31.0398.01) \item[[28]] \textit{G. H. Hardy}, A course of pure Mathematics. Cambridge: Cambridge Univ. Press. 7th ed. (1938; Zbl 0019.20303; JFM 64.1001.04) \item[[29]] \textit{Th. Hawkins}, Lebesgue's theory of integration. Its origin and development. London etc.: The University of Wisconsin Press (1970; Zbl 0195.33901) \item[[30]] \textit{E. Hille}, Analysis. Vol. I. New York: Blaisdell (1964; Zbl 0119.28303) \item[[31]] \textit{E. W. Hobson}, The theory of functions of a real variable and the theory of Fourier's series. Cambridge: Cambridge Univ. Press (1907; JFM 38.0414.01) \item[[32]] \textit{E. Landau}, Über die Approximation einer stetigen Funktion durch eine ganze rationale Funktion, Rend. Circ. Mat. Palermo 25, 337--346 (1908; JFM 39.0472.02) \item[[33]] \textit{H. Lebesgue}, Intégrale, longueur, aire, Ann. Mat. Pura Appl. (3) 7, 231--359 (1902; JFM 33.0307.02) \item[[34]] \textit{H. Lebesgue}, Sur l'intégration des fonctions discontinues, Ann. Sci. Éc. Norm. Supér. (3) 27, 361--450 (1910; JFM 41.0457.01) \item[[35]] \textit{N. Lusin}, Sur les propriétés des fonctions mesurables, C. R. Acad. Sci., Paris 154, 1688--1690 (1912; JFM 43.0484.04) \item[[36]] \textit{O. Perron}, Über den Integralbegriff, Sitz.-Ber. Heidelberger Akad. Wiss. 14, 3--16 (1914; JFM 45.0445.01) \item[[37]] \textit{J. P. Pierpont}, Lectures on the theory of functions of real variables. Volume I. Chicago, London: Ginn (1905; JFM 36.0346.01) \item[[38]] \textit{M. B. Porter}, Vallée Poussin's Cours d'Analyse, Bull. Am. Math. Soc. 22, 77--85 (1915) \item[[39]] \textit{M. B. Porter}, Shorter Notices. Cours d'Analyse Infinitésimale. By Ch. J. de la Vallée Poussin. Vol. I, 5th ed., Bull. Am. Math. Soc. 31, 83 (1925) \item[[40]] \textit{F. Riesz}, Über die Approximation einer Funktion durch Polynome, Jahresber. Dtsch. Math.-Ver. 17, 196--211 (1908; JFM 39.0471.04) \item[[41]] \textit{O. Stolz}, Grundzüge der Differential- und Integralrechnung. Erster Teil: Reelle Veränderliche und Functionen. Leipzig. B. G. Teubner (1893; JFM 25.0447.01) \item[[42]] \textit{J. Tannery}, Vallée-Poussin (Ch.J. de la). Cours d'analyse infinitésimale. Tome I, Bull. Sci. Math. 27, 121--126 (1903); tome II, ibid. 30, 100--102 (1906) \item[[43]] \textit{L. Tonelli}, Sulla rappresentazione analitica delle funzioni di più variabili reali, Rend. Circ. Mat. Palermo 29, 1--36 (1910; JFM 41.0490.02) \item[[44]] \textit{L. Tonelli}, Sull'equazioni funzionali del tipo di Volterra, Bull. Calcutta Math. Soc. 20, 31--48 (1930; JFM 56.0346.04) \item[[45]] \textit{R. Wavre}, Ch.J. de la Vallée-Poussin. Cours d'analyse infinitésimale. Quatrième édition. 2 vol. 323--324 (1921-1922) \item[[46]] \textit{W. H. Young}, The fundamental theorems of the differential calculus. Cambridge: University Press (1910; JFM 41.0306.01) \end{itemize}}