A Museum for Mathematics
| The Garden of Archimedes
A Museum for Mathematics |
The century following the discovery of calculus is characterised to a great extent by the exploration of physical and mathematical phenomena through the new differential tools. Reflections on its origins appear only occasionally, as, for example, in Eulero. He states that, to be rigorously accurate, one has to say that the infinitesimal are nothing but downright zeros. D'Alembert, in the Encyclopedie maintains that the "true mathematics" of calculus resides in the limits.
At the end of the century, in the same year when the Réflexions sur la métaphysique du calcul infinitésimal by Lazare Carnot came to light, a volume was published by Joseph Luis Lagrange (1736-1813) containing the lectures he gave to the école Polytechnique, entitled Théorie des fonctions analytiques (1797). The programme, to which Lagrange had already dedicated reflections and written records during the previous years, consists in building a theory of functions and founding infinitesimal calculus in an unambiguous manner. In this way it was freed, as indicated in the subtitle, from any consideration of infinitesimal quantities, of limits and of fluxions, returning it to the algebraic analysis of finite quantities. This is done by putting at the basis of calculus the expansion in power-series, and by constructing from them the "derivative" functions. It is here that this term is introduced.
Lagrange's plans did not become popular among his contemporaries even considering his influence. In the year that he published the Théorie, the first of the three volumes of the Traité du calcul différentiel et du calcul intégral by Sylvestre Françoise Lacroix was also published. The treatise by Lacroix, devoted to the various techniques more that to the foundations, is introduced as a downright approach where the vision of Lagrange is placed beside both methods of the differentials and of the limits. The work was remarkably successful, and so were the other manuals by Lacroix for the students of any level of the École Polytechnique and of the École Normale, which represent the body of work that formed generations of mathematical thought. The first edition, completed in 1800, was reprinted many times, and in many languages between that year and 1881, including an Italian translation in 1829.
Interest in the problem of defining the foundations of the analysis more rigorously is first expressed in the manuals written by Augustin Louis Cauchy (1789-1857). The Cours d'analyse de l'École Polytechnique, the first of the three, was published in 1821 and, as is often pointed out, was the starting point of modern analysis. Here, Cauchy overturned Lagrange's point of view, eliminating matters "drawn from the generality of algebra". According d'Alembert's point of view, the concept of limit is put at the basis of all analytic constructions. Cauchy uses it to define the controversial notion of "infinitesimal" and that of infinity, continuity of functions and, in the Résumé des leçons sur le calcul infinitesimal (1823), the "derivative" - keeping the terminology of Lagrange - as the limit of the incremental ratio.
Bernhard Bolzano (1781-1848) moves in a similar direction. In 1817 his pamphlet Rein analytischer Beweis des Lehrsatzes was published. Here, to give a demonstration of the theorem of zeros, some concepts are introduced, such as that of continuity of functions, of convergence of the series,and of upper bounds. Bolzano's contributions, however, remained unknown and were only rediscovered later.