Calculus

No mathematical development better demonstrates convergent evolution than calculus. Isaac Newton and Gottfried Wilhelm Leibniz created it independently in the 1670s, working from the same intellectual precursors, facing the same unsolved problems, and arriving at fundamentally equivalent solutions. The conditions had aligned; the discovery was inevitable.

The adjacent possible had been building for decades. Bonaventura Cavalieri's method of indivisibles (1635) showed that areas and volumes could be calculated by summing infinitely thin slices. René Descartes' coordinate system (1637) provided the algebraic framework for expressing curves as equations. Johannes Kepler and Blaise Pascal had explored infinitesimals in various contexts. By the 1660s, mathematicians across Europe were converging on similar techniques for solving problems of tangent lines and areas.

Newton developed his method of fluxions during 1665-1670, treating mathematical quantities as flowing and seeking their instantaneous rates of change. He applied calculus to physics, deriving the laws of motion and gravitation. But Newton, secretive by nature, did not publish. His manuscripts circulated among correspondents but remained formally unpublished until decades later.

Leibniz, working independently around 1674, approached the same problems through a different philosophical lens. Where Newton saw physical motion, Leibniz sought metaphysical explanation of change itself. His notation—the integral sign ∫ and the differential dx—proved more practical than Newton's dots, and he published first, in 1684.

The priority dispute that followed consumed both men's later years and poisoned relations between English and Continental mathematicians for generations. Yet the dispute itself proved the point: both had created equivalent systems because both were working at the frontier of the same accumulated knowledge. The accusation that one copied the other misunderstood how mathematics develops.

Calculus immediately transformed physics, enabling Newton's Principia and later Maxwell's electromagnetic equations, thermodynamics, and general relativity. Every scientific field requiring the analysis of continuous change—which is nearly all of them—builds on calculus. The method of indivisibles suggested it was possible; the Cartesian coordinate system provided the tools; Newton and Leibniz completed the synthesis within years of each other.