Mathematics

By the mid-seventeenth century, science had become professionalized enough for some national societies to be instituted, especially the Royal Society of London and the Paris Académie des Sciences. At that time two major mathematicians emerged: Isaac Newton (1642–1727) in Cambridge and Gottfried Wilhelm von Leibniz (1646–1716) in Hanover. Each man invented a version of the differential and integral calculus, Newton first in creation but Leibniz first in print. The use here of Leibniz's adjectives recognizes the superior development of his version. During the early 1700s Newton became so furious (or envious?) that he promoted a charge of plagiarism against Leibniz, complete with impartial committee at the Royal Society. It was a disaster for Britain: Newton's followers stuck with their master's theory of "fluxions" and "fluents," while the Continentals developed "differentials" and "integrals," with greater success. The accusation was also mathematically stupid, for conceptually the two calculi were quite different: Newton's was based upon (abstract) time and unclearly grounded upon the notion of limit, while Leibniz's used infinitesimal increments on variables, explicitly avoiding limits. So even if Leibniz had known of Newton's theory (of which the committee found no impartial evidence), he rethought it entirely.

Leibniz's initial guard was largely Swiss: brothers Jakob (1654–1705) and Johann Bernoulli (1667–1748) from the 1680s, then from the 1720s Johann's son Daniel (1700–1782) and their compatriot Leonhard Euler (1707–1783), who was to be the greatest of the lot. During the eighteenth century they and other mathematicians (especially in Paris) expanded calculus into a vast territory of ordinary and then partial differential equations and studied many related series and functions. The Newtonians kept up quite well until Colin Maclaurin (1698–1746) in the 1740s, but then faded badly.

The main motivation for this vast development came from applications, especially to mechanics. Here Newton and Leibniz differed again. In his Principia mathematica (1687) Newton announced the laws that came to carry his name: (1) a body stays in equilibrium or in uniform motion unless disturbed by a force; (2) the ratio of the magnitude of the force and the mass of the body determines its acceleration; and (3) to any force of action there is one of reaction, equal in measure and opposite in sense. In addition, for both celestial and terrestrial mechanics, which he novelly united, the force between two objects lies along the straight line joining them, and varies as the inverse square of its length.

With these principles Newton could cover a good range of mechanical phenomena. His prediction that the Earth was flattened at the poles, corroborated by an expedition undertaken in the 1740s, was a notable success. He also had a splendid idea about why the planets did not exactly follow the elliptical orbits around the sun that the inverse square law suggested: they were "perturbed" from them by interacting with each other. The study of perturbations became a prime topic in the eighteenth century, with Euler's work being particularly significant. Euler also showed that law 2 could be applied to any direction in a mechanical situation, thus greatly increasing its utility. He and others made important contributions to the mechanics of continuous media, especially fluid mechanics and elasticity theory, where Newton had been somewhat sketchy.