Book: La Géométrie

Context and Purpose
Published in 1637 as one of three appendices to the Discourse on the Method, René Descartes’ La Géométrie reshaped mathematical practice by fusing algebra with classical geometry. Rather than treat geometric problems through synthetic constructions alone, Descartes shows how to translate them into equations, manipulate those equations symbolically, and then interpret the results back in geometric terms. The treatise lays the foundations of analytic geometry and supplies notational and conceptual tools that became standard.

Algebraic Notation and Geometric Magnitudes
A striking innovation is the systematic notation: letters x, y, z serve as unknowns, while a, b, c denote given quantities; exponents are indicated with superscripts (a^2, a^3, etc.), streamlining the expression of powers. Descartes treats line segments as magnitudes that can be added, multiplied, and divided relative to a chosen unit segment, thereby creating a geometric algebra. Constructions show how to realize these operations with straightedge and compass, making algebraic manipulation and geometric construction interchangeable.

Coordinates and Loci
Descartes introduces the idea of locating points by measured distances along fixed lines, using directed segments and a chosen origin. By relating two varying lengths attached to perpendicular lines, he represents curves with equations in two variables. This move turns classical locus problems into algebraic ones: a curve is defined by an equation, and conversely, an equation defines a curve. The degree of the equation provides a new way to classify curves, shifting attention from specific constructions to general families governed by algebraic relations.

Solving Equations by Curves
A central theme is to reduce geometric problems to polynomial equations and then solve them via intersections of appropriately chosen curves. Straight lines and circles suffice for quadratics; more complex equations can be addressed by introducing additional curves. Descartes permits certain nonclassical instruments and curves when necessary, expanding the repertoire beyond the Euclidean restriction to ruler and compass. His analysis links the solvability of a problem to the degree of the resulting equation, clarifying what can be constructed with given means.

Rule of Signs and the Nature of Roots
La Géométrie presents Descartes’ rule of signs: the number of positive real roots of a polynomial does not exceed the number of sign changes in the ordered coefficients, and the number of negative real roots can be bounded by applying the rule to the polynomial with x replaced by −x. He interprets solutions geometrically with directed lengths, thereby accommodating negative roots as meaningful magnitudes. He also acknowledges “imaginary” solutions, recognizing that algebraic equations may have roots beyond real magnitudes even if their geometric interpretation is less direct.

Tangents, Normals, and Extrema
Descartes develops a method of normals to find tangents to algebraic curves. By constructing a circle whose center lies on the normal at a point of the curve and enforcing a double intersection, he derives algebraic conditions that determine the slope. This procedure also yields tools for locating maxima and minima, anticipating differential techniques by casting questions about contact and extremality into eliminable algebraic relations.

Pappus’ Problem and Generality
A showcase application is the generalized Pappus problem: determine the locus of points whose distances to several given lines satisfy a fixed proportional relation. Descartes translates the geometric constraints into equations whose degrees depend on the number of lines, giving a unified treatment that surpasses earlier special cases. The solution illustrates the power of his method to subsume classical geometry under an algebraic framework.

Legacy
La Géométrie inaugurates analytic geometry, standardizes symbolic algebra, and reframes geometric problem-solving through equations, degrees, and loci. Its synthesis of notation, coordinate methods, and algorithmic procedures for curves, tangents, and roots established a lasting template for modern mathematics.