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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.
DESCARTES: Mathematics and Physics.
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