The Garden of Archimedes  A Museum for Mathematics

The diffusion of calculus

11. Guillaume François Antoine de l'Hospital, Analyse des infiniment petits pour l'intelligence des lignes courbes, seconde édition, Paris, chez Etienne Papillon, 1716 [first edition 1696].
12. Johann Bernoulli, Opera omnia, tomus primus, Lausannae et Genevae, ex typographia Marci Michaelis Bousquet, 1742.
13. Jacob Bernoulli, De curva celerrimi descensus, in Opera, tomus secundus, Genevae, sumptibus Haeredum Cramer & Fratrum Gilibert, 1744.
14. Leonhard Euler, Introductio in analysin infinitorum, tomus primus, Lausannae, apud Marcum Michaelem Bousquet, 1748.

The methods of Newton and Leibniz follow different developing and diffusion paths. Among the principal followers and supporters of Newton, with the combination of the developments in series and of the method of fluxions, we find James Stirling, with the Methodus differentialis (1707), and Brook Taylor with the Methodus incrementorum directa et inversa (1715), openly inspired by Newton's conception of the De quadratura, work where the development in series appears which still today is called after him.

In the meantime, the validity of the method of fluxions became object of punctual and severe criticism. Weaknesses, mysteries and inconsistencies were highlighted in the pamphlet entitled The Analist or discourse addressed to an infidel mathematician, published by George Berkeley in 1734. In defence of the method of Newton, in 1742 Colin MacLaurin (1698-1746) publishes the Treatise of fluxions, where he attempts a systematic exposition of the theory of fluxions in rigorously geometric terms and avoiding infinites and infinitesimal, prime and ultimate ratios and going by instantaneous velocity.

The diffusion and the influence of the work of Leibniz was a lot wider than Newton's, maybe partly because of the choice of a happier terminology and formalims, partly for sure thanks to the capacities of his followers, first among them the Jacob and Johann Bernoulli brothers, who followed each other in holding the chair of Mathematics in Basilea. They were the first exponents of a family that would give significan contributions in the field of Mathematics and Phisics in the decades to follow. The calculus of Leibniz was used with great ductility in the study of problems connected to curves and differential equations like the properties of the lemniscate, the definition of the evolute of the logarithmical spyral, of the equation of the brachistocrone and of the tautochrone, of the catenary.

In 1691 Johann spent some time in Paris where he met, among the others, the marquis Guillaume Françoise de l'Hospital (1661-1704) to whom he gives lectures on the new method. From the relative handwritten notes takes origin the treatise Analyse des infiniments petits that l'Hospital publishes in Paris in 1696. This first systematic exposition of the differential calculus - which will be completed fifty years later with the Lectiones mathematicae de methodo integralium for the part relative to the calculus that Leibniz called "summatorius" and "integralis" from Jacob onwards - receives great success and becomes the text on which generations of mathematicians were trained.

Strictly connected to the Bernoulli family, since he was a student of Jacob in Basilea and then collaborator and friend of his sons Nicolaus and Daniel, was the very productive mathematician Leonhard Euler (1707-1783). In 1748 he publishes in two volumes the treatise Introductio in analysin infinitorum, followed in 1755 by the Instituitiones calculi differentialisand in1768-1880 by the Institutiones calculi integralis. The Introductio opens with the definition of "functio", term already used by Leibniz and by the Bernoullis, that Euler identifies with an analytical expression. The definition of function will be object of a long and lively discussion connected to the study of phisical-mathematical phenomena, such as the vibrating chord, that involves D'Alembert, among others, and that reaches Lagrange and Fourier.