How did Descartes turn shapes into equations?

By giving every point numbers. In 1637, René Descartes published an essay called La Géométrie as one of three appendices to his Discourse on the Method. Its method, now called analytic geometry or Cartesian geometry in his honour, studies geometry through a coordinate system: each point in a plane gets a pair of numbers, its coordinates, so that a shape can be defined, and calculated with, as an equation.

The core trick is deceptively simple. Lay down two number lines at right angles, and any point in the plane gets a horizontal number and a vertical number, an x and a y. A circle, a line, a curve: each one becomes a relationship between x and y that you can write down and calculate with, rather than only draw and measure. That single move is why analytic geometry became the foundation for most of the geometry that came after it, and why it is now used from physics and engineering to rocketry, statistics and economics.

Descartes wasn't quite first, and he wasn't alone. Long before him, the Greek mathematician Apollonius of Perga had used reference lines in a way strikingly similar to coordinates, close enough that his work is sometimes thought to have anticipated Descartes by roughly 1800 years, though he never let equations define the curves themselves; curves came first, and the coordinate scaffolding was added afterward. In 11th-century Persia, Omar Khayyam got closer still, solving cubic equations through geometry and narrowing the gap between the two fields. And in Descartes's own era, the French mathematician Pierre de Fermat developed the same basic idea independently, circulating his own manuscript in Paris just before Descartes's book appeared.

The two men approached the idea from opposite directions. Fermat started from an algebraic equation and found the curve it described. Descartes started from a curve and worked out its equation, a path that forced him to grapple with far messier, higher-degree equations along the way. His essay, full of gaps in its arguments and complicated equations, was not well received at first; it won recognition only after a Latin translation with commentary by van Schooten appeared in 1649. Today analytic geometry is the foundation of most modern fields of geometry.