Following the steps of spanish mathematical analysis: from Cauchy to Weierstrass between 1880 and 1914

Rey. Acad. Canar. Cienc., XX (Núms. 1-2), 119-134 (2008) (publicado en septiembre de 2009) FOLLOWING THE STEPS OF SPANISH MATHEMATICAL ANALYSIS: FROM CAUCHY TO WEIERSTRASS BETWEEN 1880 AND 1914 José M. Pacheco'Z, Francisco J. Pérez-Fernández**3 & Carlos O. Suárez**4 •DepartmenlO de Matemáticas. Universidad de Las Palmas de Gran Canaria ** Departamento de Matemáticas. Universidad de Cádiz Abstraet Rigorous Mathematical Analysis in the Cauchy style was not accepted in astraightforward manner by the European mathematical community of the central years of the 19 th Century. In average, only around forty years after the 1821 Cours d'Analyse did Cauchy's treatment become astandard in the more mathematically advanced countries, as aparadigm that remained in use until the arithmetisation of Analysis by Weierstrass replaced it befare the end of lhe century. ln this paper the aUlhors show how rigorous Malhematical Analysis a la Cauchy was adopted in Spain quite late -around 1880and how in sorne more forty years, the Weierstrassian formulation became the usual presentation in Spanish texts. 2000 MSC Numbers: 01 A55, OlA72 1Introduction It is known that the definitive introduction of rigorous Mathematical Analysis in Spain was achieved by Julio Rey Pastor (1888-1962) when he gave to print his two basic books Elementos de Análisis Algebraico (Rey Pastor, 1917) and Teoría de las Funciones Reales (Rey Pastor, 1918) after two stages in Gerrnany: with Frobenius, Schottky and Schwartz in Berlin, 1911-12, and with Caratheodory, Courant, Holder and Koebe in Gottingen, 19131914. Inspired on the original theories by Cauchy, Weierstrass, Cantor, and Dedekind, these two books incorporated to Spanish Mathematics the most rigorous standards of the Gerrnan mathematical schools. Nevertheless, several attempts had been made in the presentation of Mathematical Analysis in Spain to introduce rigour before Rey's books: The aim of this paper is to provide a critical description of these mathematical activities. According to Belhoste (Belhoste, 1991), Cauchy's viewpoints on Mathematical Analysis were not accepted in astraightforward manner, neither in France nor elsewhere. When Cauchy fled into exile the year 1830, his followers Navier, Sturrn, Liouville and Duhamel maintained his ideas and ltsed them in teaching and in mathematical writing, as arule in a less accurate way and even mixing them with other mathematical traditions (Grattan1 The first author wishes lo lhank lhe Max·Planck-lnstilul lür Wissenschaftsgeschichte in Berlin lor ils hospitality during lhe six·monlh stay (March-Augusl 2008) when lhis paper was wrilten. A firsl version is available in the Preprint Series 01 lhe MPIWG (N° 353, published in AugusI2008). 2 Corresponding aulhor. Email: [email protected] 3 Email: [email protected] 4 Email: [email protected] 119 © Del documento, de los autores. Digitalización realizada por ULPGC. Biblioteca universitaria, 2010 Guinness, 2000, p.67). It was Duhamel who became one of the forerunners of the introduction of the Cauchy styJe in Spain around 1880 through his Cours d'Analyse, a very popular book in this country. The second step towards rigour, the Weierstrassian revolution, was accepted in France when Camille Jardan (1838-1922) pubJished his own Cours d'Analyse de l'École Polytechnique (Jordan, 1893-96) and adopted the E-Jstyle. In spite of it, the older Cauchy presentation was still in use for several years: Jesper Lützen points out that Sturm's text was used in Copenhagen as late as 1915 (Lützen, 2003). Around 1840, the Spanish educational system was gradually recovering from the long reign of Fernando VII, where most universities and higher education centres were either closed or dismantled. Soon the basic works of the mathematical scene were translated andJor adapted into Spanish and the newer ideas of Gauss, Cauchy, and Abel on Mathematical Analysis, as well as the birth of non-euclidean geometries, the development of projective geometry, and the modern Funktionentheorie according to Riemann became available to the small Spanish mathematicaJ cornmunity, although in those days the books selected for translation or adaptation did not yet include the ideas by Cauchy. In aprevious paper (Pacheco et al., 2007) the authors have shown how notions such as limits, functions, infinitesimals, etc. were introduced in texts by authors Iike Vallejo and Feliu, but those terms and techniques were not used in their proofs or would-be proofs. As Grabiner points out, insistence in proof is acharacteristic feature of the development and foundations of Analysis according to the Cauchy style (Grabiner, 1981), and 1880 was lhe year when the older pre-Cauchy Analysis really disappeared from Spanish higher education and proofs actually entered the Spanish mathematical literature. The mathematician Simón Archilla and the civil engineer Antonio Portuondo wrote the books where Cauchy's vision of AnaJysis was presented in Spanish words for the first time. They elaborated on the new ideas, but through Duhamel's books rather than by studying the original work by Cauchy, even though these last ones were available at sorne leamed libraries, like the one at lhe Real Observatorio de la Armada (Royal Navy Observatory) in San Fernando, close to Cádiz. This paper does not consider Portuondo's contribution (Portuondo, 1880), essentially the small book Tratado sobre el infinito, which is the object of another forthcoming paper by the same authors (Pacheco et al., 2009). More on other mathematical activities of Portuondo can be found in (Pacheco 2008). 2Archilla's Principios del Cálculo Diferencial Simón Archilla (1836-1890) taught Mathematics at the Universities of Barcelona and Madrid. His courses comprised Higher AIgebra, Analytic Geometry, and Differential and Integral Calculus. In order to cater for these last topics, he published in 1880 the fundamental book Principios del Cálculo Diferencial (The Principies of Di fferenti al Calculus, Principios from now on). For availability reasons, in this paper the second edition of 1894 compiled by his son Faustino ArchiJJa will be used as the standard reference (Archilla yEspejo, 1894). The authors al so checked lhe first edition and realised that the onJy difference between them are the page numbers: Not asingle change was introduced in the second edition which strictly speaking was only asecond printing. 120 © Del documento, de los autores. Digitalización realizada por ULPGC. Biblioteca universitaria, 2010 Archilla was elected to the Real Academia de Ciencias Exactas, Físicas yNaturales, (Royal Academy for the Exact, Physical, and Natural Sciences), and he read his inaugural dissertation on June 10 th 1888 with the title Sobre el concepto yprincipios fundamentales del Cálculo Infinitesimal (On the concept and fundamental principies of Infinitesimal Calculus) (Archilla yEspejo and Vicuña yLazcano, 1888) where he made amost interesting historical report on the idea of infinity, ranging from Archimedes to the date the discourse was delivered. In page 61 of this report, Archilla acknowledges the role of Cauchy: "( ...) there was a need to achieve what seemed so difficult from the very beginning: To force the notion of infinity to serve the needs of Analysis. Opposing to this aim were not only the special status of infinite quantities when considered as tools in mathematical applications to number and distance, but also the philosophical criteria upon which the legitimate intervention of infinity in Analysis would be based" (note 1). Moreover, he points out how "( ...) under the light of the new doctrine, it has become common knowledge that the ratio of certain infinitesimally small, although it is always finite quantity, does not tend to any limit whatsoever; it is plain to see that the continuity of a function does not imply that the orders of its increment and that of the variable be the same; and it is possible to conceive and to determine, as Weierstrass has shown, continuous functions without aderivative at any point: Such things, if not unconceivable, were difficult to understand and to explain with the old ideas" (note 2). In the foreword to Principios the author presents the basics of the discourse in his book through aquotation from the preface of Hoüel's Cours de Calcul Infinitésimal, abook translated into Spanish in 1878: "There is only one rigorous method to present Infinitesimal Calculus: It is the method of the infinitely small, or of the limits, the method of Cauchy and Duhamel..." This opinion is clearly detailed in the introduction to Principios: "Our aim in this book is to summarise the most important principIes of Infinitesimal Calculus, trying to explain their natural interrelations, and to study the intimate relationships between the fundamental notions upon which they are based and those that are legitimately obtained from them, according to the doctrine first expounded by Cauchy and then developed by Duhamel..." (note 3). The difference between this text and its predecessors is finally featured in the treatment given to the ideas of continuity and differentiability in the light of infinitesimals (Introduction, page VI): "In the study of functions we have focused on the notion ofcontinuity, by showing the difficulties of directly studying it, and by relating it indirectly with the idea of infinitesimal quantities through the limit notion ... " 121 © Del documento, de los autores. Digitalización realizada por ULPGC. Biblioteca universitaria, 2010 2.1 Archilla 00 iofinitesimals Archilla takes infinitesimals as the fundamental notion in his book, and he introduces the concept of avariable in page 1: Avariable quantity, or simply avariable, is aquantity that can take aseries of succesive values according to any prescribed law. He goes on by explaining how to operate with variables according to the succesive values that determine them, and he defines infinitely small and infinitely large quantities in page 4: "A variable quantily lhat can take values smaller than any given quantity and can indefinitely satisfy this condition is an inflnitely small quantity, or simply an inflnitely small. On the other hand, if a variable quantity can take values larger than any given quantity and can indefinitely satisfy this condition is an inflnicely large quantity, or simply an inflnitely large. Finally, quantities that are constant or are neither infinitely small nor infinitely large are calledflnite quantities". Limits are presentedin a way that directly match es the definition of an infinitesimally small, the limit of avariable quantíty being aconstant quantity such that the varying one approaches it in the following sense: The difference between the constant and the successive values of the variable becomes smaller than any given quantity, but never equals zero. Therefore, from this definitíon it follows that: (1) The difference between avariable and its limit is an infinitesimally small quantity. (2) Infinitesimally small quantities have zero as their limit. (3) Infinitely large quantities have no limit. (4) The limit of avariable is aconstant that can not be found among the successive values of the variable. This last observation is directly inherited from Cauchy and introduces acertain lack of generality, for sequences tending to zero like 1, O, 1, o, '*, o, ~, ... would not comply with such adefinition. Nevertheless, this observation can be eonsidered aminor flaw. More interesting is the idea of an extended real line expounded in this paragraph, obtained from page 6: "Our aim has been to give more generality to this doctrine by encompassing in a common ciassification finite quantities, infinitely small and infinitely large ones, thus establishing a general theory of the orders DI quantity." Then Archilla proeeeds to the definition of infinitely small, or large, or constant quantíties vía the ratios between two quantities: "When the ratio f is an infinitely small a, then x is said to be infinitely small with respect to y. When the ratio ~ is an infinitely large A, then x is said to be infinitely largewith respect to y. Finally, hen the ratio f. is sorne constant a, then x is said to be DI the same order oIy." And he goes on by speeifying aeomparison unit: 122 © Del documento, de los autores. Digitalización realizada por ULPGC. Biblioteca universitaria, 2010 "Once a fundamental infinitesimally small quantity a has been arbitrarily chosen, we shall deem every infinitesimal of the same order asfirst-order infinitesimals". Thus, if f3 is any other first order infinitesimal, then sorne finite quantity a will exist such that ~ =a+úJ, where úJ is an infinitesimally small, from where f3 = a(a + úJ). In the same way the definition of an n-th order infinitesimal with respect to the basic one is obtained as /3" =al/(a+úJ) Moreover, acknowledging that A =;- is an infinitely large quantity, the orders for infinities are likewise defined through the following equalities: B" = A"(a +úJ) = (,b)" (a + úJ) = alI (a+ úJ), and a whole scale of infinitesimals, ranging from the infinitely small to the infinitely large through finite quantities, is obtained by allowing the exponent n to run oyer the real numbers. In the aboye computations there is the implicit assumption that the product of an infinitely small úJand afinite quantity a is, again, an infinitely smal!. The proof of this fact is found in page 15: In order to proye it, it suffices to show that aúJ < t, where t is an arbitrarily small quantity. But this inequality can be al so wrilten in the form aa <.;; and since úJ is an infinitesimally small, its inyerse is larger than any given number. Therefore aúJ is an infinitely small quantity. With the aboye remarks, Archilla presents the usual algebraic rules for infinitesimals and establishes the relationships between limits and infinitesimals by noticing in page 20 that: "If the difference between a variable x and a constant a is infinitely small, then this constant is the limit of the variable. Therefore, the equation x=a+ úJ necessarily implies lim x= a" 2.2 Archilla on continuous functions The notion of continuity is presented by Archilla in two stages. First, ageneral idea of continuity for any variable quantity, is introduced (page 56): "A variable quantity x varies in a continuous manner if it necessarily takes every intermediate value when going from ato b, and, moreover, the same property holds for any subinterval [a"b J] ~ [a,b] ". This definition has ageometric tlavour, and the precision about the restriction to any subinterval is a most interesting one. In asecond stage the notion of continuous function is introduced (page 57): "A function f(x) of a variable x is said to be continuous between the values x =a and x=b when x varies in a continuous way between these values, if the function values can not pass 123 © Del documento, de los autores. Digitalización realizada por ULPGC. Biblioteca universitaria, 2010 from some value m lo any olher one n:f:. m, bolh belween f(a) and f(b), wilhout taking all intermediate values between m and n". This is Medvedev's "continuity in the sense of Dirichlet-Lobachevskii" (Medvedev, 1991, p. 60-62), and it obviously needs one more condition, Le. monotonicity of the function, which is implicitly assumed as shown by the definition of discontinuity explained by the figure in page 58. Nevertheless, when coming to practical questions, Archilla does not forget his programme on infinitesimals: In the same page 58 he recalls that "( ...)a function is continuous on a given interval if, to any infinite1y small increment of the variable in that interval, there exists an associaled infinite1y small increment of lhe funclion". Next let us observe how the continuity of the logarithm is assessed: Let the increment of the logarithm be log(x+ h) -Iog(x) = log ~ =log(l +~). Now it is known -because it had been proved earlier in the bookthat for any fixed x, 10gO +~) and ~ differ in an infinitely small (¡J of order larger than %, namely log(x+ h)-log(x)= ~+ (¡J. Therefore it is plain that Archilla considers that bOLh the Cauchy and the Dirichlet-Lobachevskii viewpoints on continuous functions one are equivalent ones, without taking care of proving it: To him, thaL was simply tme. 2.3 Arehilla on derivable funetions and on differentials Differential calculus deals on how LO establish infinitesimal relationships between the increments of the independent variable(s) and that of the dependent variable or function. Archilla denotes those increments by fu and /1qJ(x) , and he writes (pages 65-66): " Direct consideration of the limit of 6~') when fu ~ O greatly simplifies the research whose aim is to determine the infinitesimal relationships between/1x and /1(¡J(x), and it introduces one of the most important objects in the Mathematical Analysis. The limil is called the derivative, or derived function, of qJ(x) ". To show how Principios is still amixture of old and new ideas, it is enough to notice that Archilla sticks to the old idea that the variable can not take the value of the limit when approaching it and, moreover, he insists on afunction being continuous for it to be differentiable, even acknowledging that this condition is not sufficient. He explains it by considering the infinitesimal orders of the two increments. As aremark, the authors realise that there is no reference in Principios either to Weierstrass or to the existence of eontinuous functions without aderivative at any point. Nevertheless, Archilla might have known about these funetions after the first 1880 edition of Principios, sin ce he spoke about them in his 1888 discourse for the Academia. Ir has been impossible to know whether he thought of including such atopie in apossible second edition, as the 1894 one is simply areprint, most possibly prepared by his son for monetary reasons. 124 © Del documento, de los autores. Digitalización realizada por ULPGC. Biblioteca universitaria, 2010 The treatment of the mean value theorems shows clearly the influence of Cauchy. [n a rather complicated paragraph (see note 4) in page 76, the author explains that if the derivative tp'(x) is acontínuous function for all xbetween Xo and Xo+ h, then it will take finite values, their mean value f1(tp'(x)) being somewhere between the maxímum and the minimum of the derivative on that interva!. Therefore, due to the assumed continuity of the derivative there, exists avalue xdsuch that tp'(xd)=f1(tp'(xd )) and the eguatíon he obtained previously as an "interesting property" of the derivative: tp(x o+ h) - tp(x o)=htp'(x,,), can be rewritten as: tp(x o+ h) - tp(x o)=htp'(x,,) =htp'(x + Oh), from which acompletely modern version ofRolle's Theorem follows. Jt must be noticed that the "interesting property" is simply an integral-free version of the mean value for integrals as applied to the continuous derivative: obtained by the author through arather obscure argument in pages 74 and 75. The largest part of Principios extends from page 115 untí! the end of the treatise, under the title "Book 1II: Differential Calculus", and it starts with the concept ofthe differential ofa function. Thus: "When studying the fonn and general properties of the infinitely small increment of functions of one or several variables, we realised that among the infinite number of quantities differing infinitely little from the increment of the function, there existed one simpler than any other quantity, and it was expressed through the derivative or partial derivatives of the function; moreover, we saw that this unique form, in the case of a fllnction of a single variable y=tp(x), was tp'(x)Lli. This infinitely small quantity, which is unique by his form among all quantities that differ infinitely little from ~y, will be called the differential of y= tp(x) , and will be represented by the special notation dy or dtp(x) ... " This surprisingly modem definition lacks only the words "linear function" to be a fully current one. Nevertheless, Archilla is well aware of the linearity of the differential as a function of the increment of the independent variable: "The vallles of the differential dtp(x) for sorne definite x are proportional to those of Lli = dx and this is a property exclusive to the differential among all quantities infinitely close to the increment ~y." 125 © Del documento, de los autores. Digitalización realizada por ULPGC. Biblioteca universitaria, 2010 Principios ends without any consideration of the Integral Calculus. The authors do not know whether the author intended to publish asequel on this topic, but the fact is that no evidence has be en found to support this view. 3Clariana and his lecture notes on Mathematical Analysis: Complex Analysis Lauro Clariana Ricart (1842-1916) spent his life in Catalonia, mostly in Barcelona and Tarragona, the capital city of the neighbouring province of the same name. He studied in Barcelona the courses needed to beco me Licenciado (1872) in the so called Ciencias Exactas, the name given in Spain to Mathematics for more than acentury. One year later he obtained the Doctorate in the exact Sciences at the same university. Clariana was, for along period beween 1865 and 1916, aprolific writer on many topics ranging from Mathematical Analysis and Rational Mechanics to Music and the construction of several machines. A detailed biographical account and acomplete Iist of Clariana's writings -sorne of them unpublishedcan be found in (Clariana-Clarás, 1993). Clariana taught Integral and Differential Calculus, and Rational Mechanics, and was one of the few Spaniards who participated in several Intemational Congresses and Meetings. He travelled to Paris, Brussels, München, and Freiburg between 1888 and 1900, and in 1888 he was awarded in Paris aprize for his Memoir "On the spirit of Mathematics in the modem times". Clariana was the author of two books on Mathematical Analysis for the use of students at the Escuela Superior de Ingeniería Industrial in Barcelona. They were published in 1892 and 1893 under the titles Resumen de las lecciones de Cálculo Diferencial eIntegral (Clariana Ricart, 1892b) (Resumen from now on), and Complemento a los elementos de los Cálculos (Clariana Ricart, 1892a) (Complemento in what follows). They appeared as lithographed handwritten lecture notes, and ajoint edition under the title Conceptos fundamentales de Análisis Matemático appeared in a more normal printing style the year 1903 (Clariana Ricart, 1903). Asmall favourable review of this last book appeared in the Bulletin of the American Mathematical Society the year 1904 (McFarlane, 1904). Resumen is ageneral introduction te Analysis, and at the very beginning, in page 5, Clariana makes his position clear (note 5): "Because the infinitely small and the infinitely large are the only elements that can beco me the basis of quantity in Mathematics, synthesised in the finite ones, we shall admit three categories of quantity under the following forms: (1) That of the infinitely small. (2) That of the finite. (3) That of lhe infinitely large. And these are the only true concepts of quantily lhat are directly connecled lo the Leibnizian idea of a differentia)" A very long introduction (Prolegómenos) of 45 pages is offered on the various classes of numbers and functions, as well as on the foundations and the history of the infinitesimal method, where the author summons Descartes, lohann Bemouilli and Coumot, and indeed Newton, Leibniz, and D'Alembert. The rest of the book is aclassical treatise on the usual topics on Differential and Integral Calculus presented in astraightforward way. Theorems 126 © Del documento, de los autores. Digitalización realizada por ULPGC. Biblioteca universitaria, 2010 are not highlighted and proofs are not distinctly offered. The emphasis is on the succession of useful and applicable formulas, with a few examples spread over the text. This makes the book very readable and surprisingly modem even to today's standards. Complemento has a different flavour: It is acompilation of loosely knit topics in higher Analysis. Clariana declares in abrief introduction: 'lhe aim of this book is to present what we should call 'modem theories of the infinitesimal and inlegral calculus', not because so me of them are recent ones, bul because they have nol yel been presented in the Spanish education". The first two chapters are devoted to "Jnfinitesimally Small Triangles" and "Orders of Comparison for Curves", where the fundamental Leibnizian triangle is explained and applied in depth, as well as the idea of the order of an infinitesimal is deeply applied to the study of different elements of curves. With this equipment, the book follows with astudy of Classical Differential Geometry. On the remaining chapters, avariety of topics is included. There are the Euler-McLaurin summation formula, special functions and elliptic integrals... To summarise; it is a simplified version of the usual second volume in the c1assical French treatises that c1early inspired the author, and the style is the same of Resumen. In basic questions, Clariana holds the same opinions of Archilla. As an example, the definition of acontinuous function on an interval reads: "The function y= F(x) is continuous belween lhe values a and b attributed lo x if its values pass from one value lO another lhrough values lhat differ between lhem as liule as desired." But the main feature in Clariana's work is that he is the first author to introduce in print Complex Analysis in Spain. He plainly states that "a complex quantity has the form x+ yi, where x and y are real quantities" and goes on by explaining that Gauss was the first to speak of yi as imaginary and that Cauchy denoted as imaginary the whole complex quantity. Of course he points out that when x and y are variable quantities, then x+ yi is a complex variable and that acomplex quantity is infinitesimally small (large) if the real part x and the imaginary part y are infinitesimally small (large) quantities. Continuity of a complex quantity is, of course, assessed form the continuity of its component real variables. Then, astandard theory follows. Jt must be noted that before Clariana no Spanish mathematician had studied complex quantities as the object of Analysis. Only algebraic, geometric or arithmetic considerations had been made in Spain on these numbers, and for nearly forty years the source book was the rather obscure Teoría transcendental de las cantidades imaginarias, the posthumously edited work (1865) of José María Rey Heredia (1818-1861) who inspired several developments, especiaJly in the presentation of Analytic Geometry. The work of Rey Heredia and sorne of his followers has been extensively studied elsewhere by the authors (Pacheco et al., 2006). 127 © Del documento, de los autores. Digitalización realizada por ULPGC. Biblioteca universitaria, 2010 Rey Pastor, J., 1918. Teoría de las funciones reales. (Edición del autor), Madrid. Villafañe yViñals, 1., 1892. Tratado de Análisis Matemático: Curso superior, Imprenta P. de Caridad, Barcelona. 134 © Del documento, de los autores. Digitalización realizada por ULPGC. Biblioteca universitaria, 2010