352 12. MODERN GEOMETRIESto look at the Æ as a variable and the A and Å as constants, exactly the reverse
of what Fermat intended. If we make the replacements Æ é—> c, D >—> a, A >—> x,
Å »-» y, this equation becomes c^2 - aa; = fey, and now only the exponent looks
strange, the result of Fermat's adherence to the Euclidean niceties of dimension.
Fermat illustrated the claim of Apollonius that a locus was determined by the
condition that the sum of the pairwise products of lines from a variable point to
given lines is given. His example was the case of two lines, where it is the familiar
rectangular hyperbola that we have now seen used many times for various purposes.
Fermat wrote its equation as ae — z^2. He showed that the graph of any quadratic
equation in two variables is a conic section.1.2. Descartes. Fermat's work on analytic geometry was not published in his
lifetime, and therefore was less influential than it might have been. As a result, his
contemporary Rene Descartes is remembered as the creator of analytic geometry,
and we speak of "Cartesian" coordinates, even though Fermat was more explicit
about their use.
Rene Descartes is remembered not only as one of the most original and creative
modern mathematicians, but also as one of the leading voices in modern philoso-
phy and science. Both his scientific work on optics and mechanics and his geometry
formed part of his philosophy. Like Plato, he formed a grand project of integrating
all of human knowledge into a single system. Also like Plato, he recognized the
special place of mathematics in such a system, Àç his Discourse on Method, pub-
lished at Leyden in 1637, he explained that logic, while it enabled a person to make
correct judgments about inferences drawn through syllogisms, did not provide any
actual knowledge about the world, what we would call empirical knowledge. In
what was either a deadpan piece of sarcasm or a sincere tribute to Ramon Lull
(mentioned above in Chapter 8), he said that in the art of "Lully" it enabled a
person to speak fluently about matters on which he is entirely ignorant. He seems
to have agreed with Plato that mathematical concepts are real objects, not mere
logical relations among words, and that they are perceived directly by the mind. In
his famous attempt at doubting everything, he had brought himself back from utter
skepticism by deducing the principle that whatever he could clearly and distinctly
perceive with his mind must be correct.
As Davis and Hersh (1986) have written, the Discourse on Method was the
fruit of a decade and a half of hard work and thinking on Descartes' part, following
a series of three vivid dreams on the night of November 10, 1619, when he was
a 23-year-old soldier of fortune. The link between Descartes' philosophy and his
mathematics lies precisely in the matter of "clear and distinct perception." For
there seems to be no other area of thought in which human ideas are so clear and
distinct. As Grabiner (1995, p. 84) says, when Descartes attacked, for example, a
locus problem, the answer had to be "it is this curve, it has this equation, and it
can be constructed in this way." Descartes' Geometric, which contains his ideas on
analytic geometry, was published as the last of three appendices to the Discourse.
What Descartes meant by "clear and distinct" ideas in mathematics is shown
in a method of generating curves given in his Geometrie that appears mechanical,
but can be stated in pure geometric language. A pair of lines intersecting at a fixed
point Y coincide initially (Fig. 1). The point A remains fixed on the horizontal
line. As the oblique line rotates about Õ, the point B, which remains fixed on
it, describes a circle. The tangent at  intersects the horizontal line at C, and