La Géométrie

La Géométrie was published in 1637 as an appendix to Descartes's Discourse on the Method. Alongside La Dioptrique and Les Météores, it formed a set of essays illustrating, through concrete examples, the results obtained with the method the author set out in the Discourse. Descartes wrote it in French rather than Latin, the language customary for academic publications at the time, which helped it reach a broader readership.

The work is divided into three books. The first, titled Problems That Can Be Constructed with Circles and Straight Lines Only, introduces an algebraic notation very close to the one still in use: the final letters of the alphabet stand for unknowns and the initial ones for constants. Descartes also adopts modern exponential notation for powers, though he keeps the traditional form for squares, and he breaks with the Greek habit of linking powers to areas or volumes, treating them all as lengths of line segments. This allows him to associate numbers with segments constructible by straightedge and compass. Much of the book is devoted to solving the locus problems of Pappus, where the germ of what would later be called the Cartesian coordinate system appears.

The second book, On the Nature of Curved Lines, distinguishes between geometrical curves, which can be expressed by algebraic equations in two variables, and mechanical curves, such as the quadratrix or the spiral, which Descartes regarded as unsuitable for mathematical study. He also proposes an algebraic method for finding the normal at any point of a known curve, from which the construction of tangents follows.

The third book, On the Construction of Solid and Supersolid Problems, deals with the nature of equations and how to solve them. Descartes recommends gathering all terms on one side and setting the equation equal to zero, states an intuitive version of the factor theorem, and systematically examines negative and imaginary roots, using what is now known as Descartes's rule of signs.

The style is deliberately spare: methods are indicated rather than fully proved, and many details are left to the reader, which made the work difficult to follow. Frans van Schooten, professor at Leiden, prepared a Latin edition in 1649 and, above all, the 1659–1661 edition, expanded with explanations and examples from himself and his students. These editions consolidated analytic geometry in the seventeenth century, and its influence later reached Newton and Leibniz in the development of calculus.