A limited introduction of calculus in Spain –on a fluxional basis– took place in a few Jesuit and military institution in the second half of the 18th century, until the expulsion of the Jesuits in 1767.

The setting up the colleges and seminars abandoned after the Jesuits’ expulsion eventually facilitated the emergence of teachers and institutions that allowed mathematics to thrive during the following period, up to the Independence War against Napoléon (1808-1814).

This paper presents the network of mathematicians who starred in this process (Jorge Juan, Bails, Juan Justo García, Subirás, Varas, Verdejo, Ibarra, Ciscar, Chaix, and Vallejo, among others) in institutions such as the former Jesuit colleges, military academies, technical institutions, and university. The most relevant works, from Benito Bails’ Elements of Mathematics (10 vols. 1772-83) and Principles of Mathematics (3 vols. 1776) to Vallejo’s Elementary Treatise of Mathematics (5 vols. 1812-1817), are considered.

From these data, conclusions are established as regards foreign influences and main trends in the consolidation of calculus in Spain. The powerful influence of Jorge Juan definitely tipped the scales in favour of continental calculus, partly for pedagogical reasons, but also because differences between fluxional or infinitesimal approaches were not so important for many Spanish authors during this period: by the end of the 18th century, Leibniz’s notation was absolutely dominant, but fluxional concepts were still considered more rigorous. Nevertheless, the concept of limit as defined by D’Alembert and developed by Cousin, gradually emerged from Bails and was adopted by Ciscar, who dismissed differentials despite their usefulness, as a result of his deep understanding of the problem of foundations of calculus.

The theory of limits as the basis for the foundations of calculus, which was a clear precedent for the correct approach to the problem, was also adopted in Chaix’s Instituciones de cálculo diferencial (1801), the first Spanish work entirely devoted to differential calculus. In this work, functions were the central element of calculus, derivatives –not differentials– were the characteristic element, the expression dy/dx was a symbol –not a quotient– and geometry was relegated to applications in favour of algebra.

Finally, the fourth volume of Vallejo’s Elementary Treatise (1813), devoted to differential and integral calculus, also followed D’Alembert in defining calculus through the concept of limit, reduced calculus to algebra in Lagrange’s style, and introduced Lacroix in Spain, albeit avoiding differentials and including finite differences.