1631, Gool had called Descartes's attention to this problem from
Pappus. It was considered a difficult problem by contemporary
mathematicians. The object was to find the locus of a point P
from which line segments AP, BP, CP, DP may be drawn in any
prescribed manner whatsoever, relative to four straight lines a,
b, c, d, so that AP • BP = CP • DP. Descartes chose a fixed origin
O on the line a, set OA = x, the obliquely placed "applicate"
AP = y. Then, by means of a relationship based on similarity,
he expressed BP, CP, DP as linear functions of x and
y. He eliminated the constant term of the resulting equa-
tion by a suitable choice of O and obtained

Avoiding infinitesimal (hence, "approximate") analysis, Des-
cartes solved the problems of normals to an algebraic curve as
follows (purely algebraic method). He set the applicates per-
pendicular to the axis, took the point P ₀ (x ₀, y ₀ ), on the curve,
the point M (t, 0) on the axis and let the circle whose center
was M and which passed through P ₀ intersect the curve again
at P(x, y). Then he substituted the value

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