DESCARTES, RENÉ DU PERRON (b. La Haye, Touraine, France, 31 March 1596; d. Stockholm, Sweden, 11 February 1650), natural philosophy, scientific method, mathematics, optics, mechanics, physiology.

    in the form

(x - a) (x - b) ... (x - s) = 0,

where a,b, ... ,s are the roots of the equation, Descartes states and offers an intuitive proof of the fundamental theorem of algebra (first stated by Albert Girard in 1629) that an nth degree equation has exactly n roots. The proof rests simply on the principle that every root must appear in one of the binomial factors of P(x) and that it requires n such factors to achieve xn as the highest power of x in that polynomial. Descartes is therefore prepared to recognize not only negative roots (he gives as a corollary the law of signs for the number of negative roots) but also “imaginary” solutions to complete the necessary number.10 In a series of examples, he then shows how to alter the signs of the roots of an equation, to increase them (additively or multiplicatively), or to decrease them. Having derived from the factored form of an equation its elementary symmetric functions,11 Descartes uses them to eliminate the term containing xn - 1 in the equation. This step paves the way for the general solution of the cubic and quartic equations (material dating back to Descartes's earliest studies) and leads to a general discussion of the solution of equations, in which the first method outlined is that of testing the various factors of the constant term, and then other means, including approximate solution, are discussed.

The Géométrie represented the sum of mathematical knowledge to which Descartes was willing to commit himself in print. The same philosophical concepts that led to the brilliant new method of geometry also prevented him from appreciating the innovative achievements of his contemporaries. His demand for strict a priori deduction caused him to reject Fermat's use of counterfactual assumptions in the latter's method of maxima and minima and rule of tangents.12 His demand for absolute intuitive clarity in concepts excluded the infinitesimal from his mathematics. His renewed insistence on Aristotle's rigid distinction between “straight” and “curved” led him to reject from the outset any attempt to rectify curved lines.

Despite these hindrances to adventurous speculation, Descartes did discuss in his correspondence some problems that lay outside the realm of his Géométrie. In 1638, for example, he discussed with Mersenne, in connection with the law of falling bodies, the curve now expressed by the polar equation ? = aλθ (logarithmic spiral)13 and undertook the determination of the normal to, and quadrature of, the cycloid. Also in 1638 he took up a problem posed by Florimond Debeaune: (in modern terms) to construct a curve satisfying the differential equation a(dy/dx) = x - y. Descartes appreciated Debeaune's quadrature of the curve and was himself able to determine the asymptote y = x - a common to the family, but he did not succeed in finding one of the curves itself.14

By 1638, however, Descartes had largely completed his career in mathematics. The writing of the Meditations (1641), its defense against the critics, and the composition of the magisterial Principia philosophiae (1644) left little time to pursue further the mathematical studies begun in 1618.

Optics.

In addition to presenting his new method of algebraic geometry, Descartes's Géométrie also served in book II to provide rigorous mathematical demonstrations for sections of his Dioptrique published at the same time. The mathematical derivations pertain to his theory of lenses and offer, through four “ovals,” solutions to a generalized form of the anaclastic problem.15 The theory of lenses, a topic that had engaged Descartes since reading Kepler's Dioptrica in 1619, took its form and direction in turn from Descartes's solution to the more basic problem of a mathematical derivation of the laws of reflection and refraction, with which the Dioptrique opens.

Background to these derivations was Descartes's theory of light, an integral part of his overall system of cosmology.16 For Descartes light was not motion (which takes time) but rather a “tendency to motion,” an impulsive force transmitted rectilinearly and instantaneously by the fine particles that fill the interstices between the visible macrobodies of the universe. His model for light itself was the blind man's cane, which instantaneously transmits impulses from the objects it meets and enables the man to “see.” To derive the laws of reflection and refraction, however, Descartes required another model more amenable to mathematical description. Arguing that “tendency to motion” could be analyzed in terms of actual motion, he chose the model of a tennis ball striking a flat surface. For the law of reflection the surface was assumed to be perfectly rigid and immobile. He then applied two fundamental principles of his theory of collision: first, that a body in motion will continue to move in the same direction at the same speed unless acted upon by contact with another body; second, that a body can lose some or all of its motion only by transmitting it directly to another. Descartes measured motion by the product of the magnitude of the body and the speed at which it travels. He made a distinction, however, between the speed of a body and its “determination” to move in a certain direction.17 By this distinction, it might come about that a body impacting with another would lose none of its speed (if the other body remained unmoved) but would