Significado de la contribución matemática de Benito Bails (1731-1797) en la España del siglo XVIII

Résumé

Abstract: The study of mathematics in the Spanish 18th century was mostly taught in academies. In fact, the Bourbon Monarchy, after the War of the Spanish Succession (1700-1714), promoted the modernization of the country by fostering scientific and technological development, creating civilian and military academic institutions where both pure and mixed mathematics were fundamental disciplines. In 1752, Ferdinand VI established the Royal Academy of San Fernando in Madrid, with the aim of conveniently organizing the training of young students of the three Noble Arts —sculpture, painting and architecture— in 18th century Spain. Since the mathematics being taught was not sufficient for the proper instruction of its students, the Academy undertook a series of initiatives to update and modernise its teaching, in order to provide a solid scientific and technical foundation for its disciples. In this context, in the midst of the Enlightenment, we situate the mathematical course Elements of Mathematics (hereinafter Elements) by Benito Bails (1731-1797), composed of ten volumes in eleven issues, which were published between 1779 and 1799, although they had already been printed between 1772 and 1776, with the exception of the two volumes of volume IX. The course presents a singular characteristic in the way it was composed: Bails, with his own criteria, proposed a modernisation of Spanish mathematics based on the selection of the best European treatises available at the time. Our thesis analyses the work Elements, whose relevant mathematical content has not been fully examined so far. The aim is to analyse the mathematical and methodological contributions of the course, situating it in the process of modernisation of mathematical thought, based on the diffusion of modern European mathematics. With the analysis of the mathematical contents of the course, evidence is provided to show that the contribution of Bails and his works to the modernisation of mathematics meant a great advance in scientific progress in Spain, from the last quarter of the 18th century until well into the 19th century. Main goal: To analyze the impact and influence of Bails' work in the modernization of mathematical thought in Spain at the time, based on the diffusion of modern European mathematics and the contribution in the composition and elaboration of mathematical contents. Specific goals: A. To determine to what extent both the historical and academic contexts influenced the commissioning of Bails to prepare a mathematical course designed for the teaching of the disciples of the Academy. B. To study the genesis of the process of creation of the Elements, to describe its structure and contents, and to analyze the prologues of the different volumes, in order to evaluate, among others, the didactic conception of Bails' work. C. To analyze the currents of European scientific production mentioned by Bails, in order to better understand the importance of the treatises of the Elements in the process of mathematical modernization of the time. D. Signifying the figure of Bails as a mathematician. In this sense, the idea of Bails is valued in relation to the concepts of infinity, infinitely large and infinitely small and the limit as a tool of rigour in his version of the algebrization of infinity. E. To illustrate in a notorious way the process of reception, impact and circulation of mathematical knowledge through Bails' works. Methodology: This thesis follows two lines of research in the history of science. One line focuses on the active appropriation and circulation of knowledge in the context of communication as part of the process of scientific production. The second line, more novel, delves into the relationship between mathematics and engineering, and analyzes mathematics as scientific practices for technical development, trying to answer the essential question of how the scientific-technical training imparted through mathematical courses to technical professions impacted society. Conclusions: ‒ Bails proposed a modernization of Spanish mathematics based on the selection of the best European treatises available at the time. Thus, the Elements are part of a tradition of mathematical courses, which emerged in the seventeenth century, where much of the new European knowledge of the time is collected, responding to the demand for scientific and technical knowledge. We have shown that Bails considered mixed mathematics in terms of utility, and pure mathematics as the basis on which to apply mathematics and provide for the progress of mankind. ‒ The Elements marked a turning point in the didactic-cognitive conception of the scientific and technical knowledge taught for the training of the students of the Academy. In the search for the excellence of his work, Bails formed the multitude of contents to be covered in his course based, as we have shown in our research, on three fundamental pillars: the extension of the subjects, the quality of the contents and the order or coordination of the disciplines. An example of a novel updating of knowledge, as an outstanding didactic-cognitive aspect, is Bails' interest in explaining infinity and disseminating the fundamentals of infinitesimal calculus following criteria of rigour and conceptual clarity. ‒ Bails explored the arithmetic-algebraic properties of zero and infinity in an exhaustive and original way. Bails adopted a position, we believe original, by also approaching in a mathematical course the infinite from the theory of the limit. On the other hand, Bails demonstrated the power and rigor of infinitesimal calculus in solving problems that had traditionally been difficult to approach from classical geometry. The approach adopted by Bails denotes a clear conceptual evolution of his mathematical thinking from geometry and arithmetic to the analysis of infinities, functions and series theory. In this sense, it should be noted that Bails' treatise on logarithms is one of the most complete European texts in terms of the treatment of the different doctrines that introduce them, with the inclusion of the most recent infinite algebraic-analytic processes of the time. We conclude that the Elements are on a par with the best European mathematical teaching works of the time and that Bails was one of the leading Spanish mathematicians of the eighteenth century. ‒ Through his work, Bails developed a true process of scientific production that benefited the circulation of theoretical and technical knowledge of mathematics. Bails' up-to-date treatises on pure and mixed mathematics played a decisive role in the processes of scientific and technical production. ‒ Our research shows that the contribution of Bails' works to the updating of Spanish mathematics, with a European character, had a great impact and influence on the scientific-technical progress and advancement through the new mathematicians, engineers, architects, etc., trained with these works. Their influence and diffusion covered both Spain and Latin American countries, from the last quarter of the 18th century until well into the 19th century.