VI.11 René Descartes
b. La Haye (now “Descartes”), France, 1596; d. Stockholm, 1650
Algebra; geometry; analytic geometry; foundations of mathematics
In 1637 Descartes published La Géométrie as an “essay” appended to his philosophical treatise Discours de la Méthode. It remained his only mathematical publication. No single early modern text shaped the development of mathematics between 1650 and 1700 as strongly as La Géométrie. It was the founding text of analytic geometry and it paved the way for the merging of algebra and geometry that made possible the development of the integral and differential calculus about fifty years later.
Descartes was educated at the Jesuit College at La Flèche. He spent his life mostly outside France, traveling through Europe in his early twenties and living in the Netherlands from 1628 until 1649; he then left for Sweden, invited by Queen Christina to her court. From an early age his interest in mathematics was tightly linked to his primary philosophical preoccupation: the certainty of knowledge. In a letter of 1619 he sketched a method, clearly inspired by arithmetic and geometry, for solving all problems in natural philosophy. Shortly afterward, his ideas grew into a passionate conviction that he could and should develop a philosophy along these problem-solving and mathematics-inspired lines. La Géométrie grew out of the mathematical part of his philosophical program; it was not a textbook on analytic geometry. Descartes offered little in the way of general principles, explaining his ideas by means of examples.
Descartes used a classical problem, Pappus’s problem, in order to explain coordinates and equations of curves, and showed that the defining property of a curve could be written as an equation. He introduced coordinates x and y, using oblique as well as rectangular coordinate axes, which he always adjusted to the problem at hand. He also introduced the now very common usage of employing x, y, and z for unknowns and a, b, and c for indeterminate fixed quantities.
For Descartes, a geometrical problem required a geometrical answer. The equation was at best an algebraic reformulation of the problem; the answer had to be a construction of the curve or of individual points. If, as in the particular case of Pappus’s problem in four lines, the equation was quadratic, then for any fixed value of y the x-coordinate was a root of a quadratic equation. Earlier in the book Descartes had shown how such a root could be constructed (using ruler and compass). Thus, the curve could be constructed “pointwise” by choosing a series of values for y and constructing the corresponding xs and points on the curve. Pointwise construction could not provide the whole curve. Therefore in Pappus’s problem Descartes used the equation to show that the solution curves were conic sections, and explained how to determine the nature of the conic, the location of its axes, and the values of its parameters. This was an impressive result; it was, in fact, the first classification of an algebraically defined class of curves.
René Descartes
A further influential result in La Géométrie, and the one of which Descartes himself said he was most proud, was his method to determine the normal (and thus also the tangent) at a given point on a curve with a given equation. It was a pre-calculus forerunner of differentiation.
There are three important differences between how Descartes treated curves and their equations and how they are treated in modern analytic geometry: he employed oblique as well as rectangular axes; he did not consider the equation as defining a curve—rather it represented a problem, namely to construct the curve itself, as well as its axes, tangents, etc.; and he did not consider the plane itself as a collection of points characterized by pairs of real numbers—for him the xs and ys were not dimensionless numbers but the lengths of line segments. (The term “Cartesian plane” for 
2 is therefore anachronistic.)
Descartes supposed (too optimistically) that his procedures could be extended to polynomial equations of any degree (usually connected to Pappus’s problem with more than four lines) and that therefore he had shown how, in principle, all geometrical construction problems could be solved. For higher-order constructions he needed new algebraic techniques. The relevant section in La Géométrie constituted the first general theory of polynomial equations and their roots. It contained his “sign rule” about the number of positive and negative roots of a polynomial, various transformation rules, and methods to check equations for reducibility. He gave no proofs; his results were based on a conviction that polynomials could essentially be written as products of linear factors x − xi, in which the roots xi could be positive, negative, or “imaginary.”
It appears, then, that analytic geometry was not the primary goal of La Géométrie. Rather, its aim was to provide a universal method for solving geometrical problems, and to do so Descartes had to answer two urgent methodological questions. The first was how to solve geometrical problems not constructible by ruler and compass, and the second was how to use algebra as an analytic, i.e., solution-finding, tool in geometry.
For the first of these, Descartes allowed successively more complicated curves as means of construction. It was Descartes’s conviction that algebra, through the equations of these curves, could guide him to choose, among all such construction curves, the most appropriate for the problem, in particular the simplest, i.e., that of lowest degree.
The second question addressed serious conceptual difficulties that were felt at the time about using algebra in geometry. The transfer of algebraic operations to geometry was indeed problematic because multiplication in geometry was generally interpreted dimensionally: for example, a product of two lengths had to represent an area, and a product of three a volume. But until then algebra had dealt mostly with numbers and had routinely used products of more than three factors. Thus, a consistent and unrestricted geometrical interpretation of the operations of algebra was needed. Descartes did indeed provide such a reinterpretation. He introduced a unit line segment in such a way that multiplication no longer raised the dimension and inhomogeneous terms could be allowed in equations.
By 1637 he had given up on his earlier attempts to link philosophy and mathematics. Yet the preoccupation with certainty remained. As his concept of construction involved the use of curves, he had to consider which curves could be understood by the human mind with sufficient clarity to be acceptable in geometry. His answer was that all algebraic curves were acceptable (he called these “geometrical curves”) and all others were not (these he called “mechanical”). Few seventeenth-century mathematicians followed Descartes in this strict demarcation of geometry. This is typical of the reception of Descartes’s La Géométrie: the philosophical and methodological aspects of the book were largely ignored by his mathematical readers, but the technical mathematical aspects were eagerly accepted and used.
Further Reading
Bos, H. J. M. 2001. Redefining Geometrical Exactness: Descartes’ Transformation of the Early Modern Concept of Construction. New York: Springer.
Cottingham, J., ed. 1992. The Cambridge Companion to Descartes. Cambridge: Cambridge University Press.
Shea, W. R. 1991. The Magic of Numbers and Motion: The Scientific Career of René Descartes. Canton, MA: Watson Publishing.
Henk J. M. Bos