La Géométrie (The Geometry)
Marry algebra to geometry: a curve becomes an equation, a point a pair of numbers.
Before Descartes, algebra and geometry were two different worlds. He built a bridge between them — and we have been walking across it ever since.
The big idea
Descartes' idea is the one you met the first time you drew a graph. Pick two lines crossing at a corner; now any point on the page can be named by two numbers — how far across, how far up. That pair is the point's coordinates. And here is the magic: a shape is no longer just a picture, it is a rule the numbers obey. A circle becomes 'x² + y² = 9'; a parabola becomes its own short equation. Geometry and algebra turn out to be the same subject seen two ways.
How it happened
In 1637 Descartes published a slim book, the Discourse on Method, about how to think clearly. He tucked three scientific essays behind it as demonstrations; the last, La Géométrie, was the one that changed mathematics. Writing in French rather than scholarly Latin, he showed how to turn a stubborn geometric problem — one the ancient Greeks had wrestled with — into an equation you could simply solve. He was not quite alone: a French lawyer named Pierre de Fermat had hit on the same idea at the same time, working from the other direction. Between them, coordinate geometry was born.
Why it mattered
Once a curve is an equation, you can compute with it. You can ask where two curves cross, or where a curve is steepest, and answer with algebra instead of clever drawing. That made the next leap possible: a generation later Newton and Leibniz used Descartes' grid to invent the calculus — the mathematics of change that underlies all of physics. Nearly every graph you have ever seen, from a fever chart to a stock price to a physics diagram, lives on the plane he set up.
An everyday picture
Think of a city laid out in blocks. Tell a friend 'three blocks east, two blocks north,' and they reach exactly one corner — that is a coordinate. Now trace every corner that is exactly five blocks from the town square: you walk a circle, and 'five blocks from the square' is its equation. Descartes' insight is that the directions (the algebra) and the shape you walk (the geometry) are the same information in two costumes.
Where it sits
Descartes stands between Euclid, whose geometry (also in this Library) was built from constructions with straight-edge and compass, and Newton and Leibniz, whose calculus he made possible. He inherited the new symbolic algebra of the Renaissance and welded it to the ancient geometry of curves. The join held so well that we named it after him: every 'Cartesian' grid, and every coordinate on a map or a screen, descends from this one essay.
Book I — Problems constructible with straight lines and circles
Any problem in geometry can easily be reduced to such terms that a knowledge of the lengths of certain straight lines is sufficient for its construction.
… taking one line which I shall call unity in order to relate it as closely as possible to numbers, and which can in general be chosen arbitrarily, and having given two other lines, to find a fourth line which shall be to one of the given lines as the other is to unity (which is the same as multiplication).
Book II — On the nature of curved lines
I could give here several other ways of tracing and conceiving a series of curved lines, each curve more complex than any preceding one, but I think the best way to group together all such curves and then classify them in order, is by recognizing the fact that all the points of those curves which we may call 'geometric,' that is, those which admit of precise and exact measurement, must bear a definite relation to all points of a straight line, and that this relation must be expressed by a single equation.