The birth of analytic geometry

The chapter we are studying was born in France in the 1630s, out of an idea as simple as it was revolutionary: a curve of the plane can be described by an equation between two coordinates. Before then, ever since Euclid and Apollonius, geometry had consisted of straightedge-and-compass constructions and proportions between segments; algebra, still young, lived apart. Bringing them together meant one could now compute geometry.

Descartes and the Géométrie (1637)

René Descartes published La Géométrie as one of the three appendices to his Discours de la méthode (1637). In that work he shows how, once a reference segment and a direction are fixed, every point of a curve can be located by two variable lengths, which today we call $x$ and $y$; a geometric problem then corresponds to an equation between $x$ and $y$, and vice versa (Boyer). This is not yet our system of two orthogonal axes with negative coordinates — that would come later — but the principle is exactly the one we use to derive the equation of a locus: take a generic point $P(x;y)$ and translate into algebra the property that characterises it.

Fermat and loci

At almost the same time, and quite independently, Pierre de Fermat had arrived at the same idea in his Ad locos planos et solidos isagoge (Introduction to Plane and Solid Loci), which circulated as a manuscript before Descartes’ Géométrie. Fermat starts precisely from the notion of locus — the subject of this chapter — and establishes that “whenever two unknown quantities appear in an equation, there is a locus, and the endpoint of one of them describes a line” (Katz). He recognised that first-degree equations give straight lines and second-degree ones give the conics, thereby classifying curves such as the parabola and the ellipse that we meet again, later, under the unifying definition.

The two works differ in style — Descartes starts from problems to be constructed, Fermat from equations to be interpreted — but together they found what we now call analytic geometry, the bridge between algebra and geometry that makes the whole Cartesian plane possible (Kline).

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Topics: Geometric loci Concepts: Locus · Cartesian plane Skills: Analytical geometry People: René Descartes (Cartesio) · Pierre de Fermat