In the 1620s Descartes was a mathematician, military engineer and traveller; in 1628 he settled in the United Provinces (the Dutch Republic) and spent most of the 1630s working there. In 1637 he published the Discours de la méthode in Leiden: at once an autobiographical-philosophical essay and an introduction to three scientific appendices (Dioptrique, Météores, La Géométrie). The work was deliberately written in French rather than Latin — a clear intent to push knowledge beyond the narrow university circle.

The innovation of La Géométrie was a simple step with deep consequences: Descartes assigned every point in the plane a pair of numbers (x, y) and translated geometric curves into algebraic equations between those numbers. Two disciplines kept apart since antiquity — Euclidean geometry and algebra — met in a single symbolic language. A line is an equation; a circle is an equation; the conic-section problems Greek geometers struggled with became polynomial manipulations. In the same essay Descartes also standardised much of modern algebraic notation (x, y, z for unknowns; a, b, c for constants; the use of exponents).

The philosophical side carried its own revolution. To strip knowledge of authority and tradition Descartes tried doubting everything; what remained was the existence of the doubter — cogito, ergo sum. The implication for science was radical: knowledge is reached by observation and logic, by methodical reasoning, not by deference to authority alone. This happened in the same years as Galileo's observational revolution and the Francophone reception of Bacon's empirical method.

The long-term impact of La Géométrie may have been the least visible at first. Once a curve can be written as an equation, rates of change, slopes, areas and volumes on that curve can also be computed by equation — half a century later Newton and Leibniz independently built differential and integral calculus on exactly this ground. Much of the remaining seventeenth-century scientific revolution would be constructed on top of Descartes's coordinate plane.