De Lacroix a Cauchy: La fundamentación del cálculo infinitesimal en José Mariano Vallejo (1807-1832)

This paper studies the foundations of infinitesimal calculus in the work of Jose Mariano Vallejo (1779-1846), the most influential mathematician in Spanish during the first half of the 19th century. For this purpose, the analytical foundations of his three main works related to infinitesimal calculus are considered: Memoria sobre la curvatura de las lineas en sus diferentes puntos, sobre el radio de curvatura y sobre las evolutas (1807), and the two editions (1813 and 1832) of the volume of his Tratado Elemental de Matematicas devoted to calculus. The study shows the evolutionary dynamics of Vallejo’s thought, following the European leading references in the development of mathematical analysis in his time, and how Vallejo combines the successive influence of Lagrange, Lacroix, and Cauchy, according to criteria of educational adequacy and mathematical rigor. In particular, the influences and discrepancies on infinitesimal calculus between Vallejo’s Treatise (1813) and Lacroix’ Traite elementaire de calcul differentiel et de calcul integral (1802) are determined, as well as the scope of the introduction of Cauchy’s ideas, that were published for the first time in Spain in the second edition of the volume on infinitesimal calculus of Vallejo’s Treatise (1832).