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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.
Mathematics.
and lays the foundation of the new analytic geometry
contained therein, to wit, that given a line x and a
polynomial P(x) with rational coefficients it is possible
to construct another line y such that y = P(x).
Algebra
thereby becomes for Descartes the symbolic
method for realizing the second goal of his “true
mathematics,” the analysis and classification of problems.
The famous “Problem of Pappus,” called to
Descartes's attention by Jacob Golius in 1631, provides
the focus for Descartes's exposition of his new
method. The problem states in brief: given n coplanar
lines, to find the locus of a point such that, if it is
connected to each given line by a line drawn at a fixed
angle, the product of n/2 of the connecting lines bears
a given ratio to the product of the remaining n/2 (for
even n; for odd n, the product of (n + 1)/2 lines bears
a given ratio to k times the product of the remaining
(n - 1)/2, where k is a given line segment). In carrying
out the detailed solution for the case n = 4,
Descartes also achieves the classification of the solutions
for other n.
Implicit in Descartes's solution is the analytic geometry
that today bears his name. Taking lines AB,
AD, EF, GH as the four given lines, Descartes assumes
that point C lies on the required locus and draws the
connecting lines CB, CD, CF, CH. To apply algebraic
analysis, he then takes the length AB, measured from
the fixed point A, as his first unknown, x, and length
BC as the second unknown, y. He thus imagines the
locus to be traced by the endpoint C of a movable
ordinate BC maintaining a fixed angle to line AB (the
axis) and varying in length as a function7 of the length
AB. Throughout the Géométrie, Descartes chooses
his
axial system to fit the problem; nowhere does the now
standard—and misnamed—“Cartesian coordinate
system” appear.
FIGURE 2
The goal of the algebraic derivation that follows this
basic construction is to show that every other connecting
line may be expressed by a combination of
the two basic unknowns in the form αx + βy +
γ,
where α, β, γ derive from the data. From this last
result it follows that for a given number n of fixed
lines the power of x in the equation that expresses
the ratio of multiplied connecting lines will not exceed
n/2 (n even) or (n - 1)/2 (n odd); it will often
not
even be that large. Hence, for successive assumed
values of y, the construction of points on the locus
requires the solution of a determinate equation in x
of degree n/2, or (n - 1)/2; e.g., for five or fewer lines,
one need only be able to solve a quadratic equation,
which in turn requires only circle and straightedge for
its constructive solution.
Thus Descartes's classification of the various cases
of Pappus' problem follows the order of difficulty of
solving determinate equations of increasing degree.8
Solution of such equations carries with it the possibility
of constructing any point (and hence all points)
of the locus sought. The direct solvability of algebraic
equations becomes in book II Descartes's criterion for
distinguishing between “geometrical” and
“nongeometrical”
curves; for the latter (today termed “transcendental
curves”) by their nature allow the direct
construction of only certain of their points. For the
construction of the loci that satisfy Pappus' problem
for n ? 5, i.e., the conic sections, Descartes relies on
the construction theorems of Apollonius' Conics and
contents himself with showing how the indeterminate
equations of the loci contain the necessary parameters.
Descartes goes on to show in book II that the equation
of a curve also suffices to determine its geometrical
properties, of which the most important is
the normal to any point on the curve. His method of
normals—from which a method of tangents follows
directly—takes as unknown the point of intersection
of the desired normal and the axis. Considering a
family of circles drawn about that point, Descartes
derives an equation P(x) = 0, the roots of which are
the abscissas of the intersection points of any circle
and the curve. The normal is the radius of that circle
which has a single intersection point, and Descartes
finds that circle on the basis of the theorem that, if
P(x) = 0 has a repeated root at x = a, then
P(x) = (x - a)2R(x),
where R(a) ? 0. Here a is the
abscissa of the given point on the curve, and the
solution follows from equating the coefficients of like
powers of x on either side of the last equation. Descartes's
method is formally equivalent to Fermat's
method of maxima and minima and, along with the
latter, constituted one of the early foundations of the
later differential calculus.
The central importance of determinate equations
and their solution leads directly to book III of the
Géométrie with its purely algebraic theory of
equations.
Entirely novel and original, Descartes's theory
begins by writing every equation in the form
P(x) = 0, where P(x) is an algebraic polynomial
with
real coefficients.9 From the assertion, derived inductively,
that every such equation may also be expressed