Mathematics

However, Newton's theory was not alone in mechanics. Leibniz and others developed an alternative approach, partly inspired by Descartes, in which the "living forces" (roughly, kinetic energy) of bodies were related to their positions. Gradually this became a theory of living forces converted into "work" (a later term), specified as (force x traversed distance). Engineers became keen on it for its utility in their concerns, especially when impact between bodies was involved; from the 1780s Lazare Carnot (1753–1823) urged it as a general approach for mechanics.

Carnot thereby challenged Newton's theory, but his main target was a recent new tradition partly launched by Jean d'Alembert (1718–1783) in midcentury and developed further by Joseph-Louis Lagrange (1736–1813). Suspicious of the notion of force, d'Alembert had proposed that it be defined by Newton's law 2, which he replaced by one stating how systems of bodies moved when disturbed from equilibrium. At that time Euler and others proposed a "principle of least action," which asserted that the action (a mechanical notion defined by an integral) of a mechanical system took its optimal value when equilibrium was achieved. Lagrange elaborated upon these principles to create Méchanique analitique (1788), in which he challenged the other two traditions; in particular, dynamics was reduced mathematically to statics. For him a large advantage of his principles was that they were formulated exclusively in algebraic terms; as he proclaimed in the preface of his book, there were no diagrams, and no need for them. A main achievement was a superb though inconclusive attempt to prove that the system of planets was stable; predecessors such as Newton and Euler had left that matter to God.

Lagrange formulated mechanics this way in order to make it (more) rigorous. Similarly, he algebraized the calculus by assuming that any mathematical function could be expressed in an infinite power series (the so-called Taylor series), and that the basic notions of derivative (his word) and integral could be determined solely by algebraic manipulations. He also greatly expanded the calculus of variations, a key notion in the principle of least action.

As in mechanics, Lagrange's calculus challenged the earlier ones, Newton's and Leibniz's, and as there, reaction was cautious. A good example for both contexts was Pierre-Simon Laplace (1749–1827), a major figure from 1770. While strongly influenced by Lagrange, he did not confine himself to the constraints of Lagrange's book when writing his own four-volume Traité de mécanique céleste (1799–1805; Treatise on celestial mechanics). His exposition of celestial and planetary mechanics used many differential equations, series, and functions.