- PRELUDE TO THE CALCULUS 465
FIGURE 2. The folium of Descartes. Descartes and Fermat con-
sidered only the loop in this curve.In this paper Fermat asserted, "And this method never fails " This assertion
provoked an objection from Descartes,^1 who challenged Fermat with the curve of
Fig. 2, now known as the folium of Descartes, having equation a;^3 + y^3 = Saxy.
Descartes did not regard curves such as the spiral and the quadratrix as admis-
sible in argument, since they are generated by two motions whose relationship to
each other cannot be determined exactly. A few such curves, however, were to prove
a very fruitful source of new constructions and applications. One of them, which
had first been noticed in the early sixteenth century by an obscure mathematician
named Charles Bouvelles (ca. 1470-ca. 1553), is the cycloid, the curve generated
by a point on a circle (called the generating circle) that rolls without slipping along
a straight line. We have already mentioned this curve, assuming that the reader
will have heard of it, in Section 3 of Chapter 12. It is easily pictured by imagining
a painted spot on the rim of a wheel as the wheel rolls along the ground. Since
the linear velocity of the rim relative to its center is exactly equal to the linear
velocity of the center, it follows that the point is at any instant moving along the
bisector of the angle formed by a horizontal line and the tangent to the generating
circle. In this way, given the generating circle, it is an easy matter to construct
the tangent to the cycloid. This result was obtained independently around 1638 by
Descartes, Fermat, and Gilles Personne de Roberval (1602-1675), and slightly later
by Evangelista Torricelli (1608-1647), a pupil of Galileo Galilei (1564-1642).
1.2. Lengths, areas, and volumes. Seventeenth-century mathematicians had
inherited two conceptually different ways of applying infinitesimal ideas to find
areas and volumes. One was to regard an area as a "sum of lines." The other
was to approximate the area by a sum of regular figures and try to show that the
approximation got better as the individual regular figures got smaller. The rigorous
version of the latter argument, the method of exhaustion, was tedious and of limited
application.
Cavalieri's principle. In the "sum of lines" approach, a figure whose area or volume
was required was sliced into parallel sections, and these sections were shown to be
equal or proportional to corresponding sections of a second figure whose area or(^1) There was little love lost between Descartes and Fermat, since Fermat had dismissed Descartes'
derivation of the law of refraction. (Descartes assumed that light traveled faster in denser media;
Fermat assumed that it traveled slower. Yet they both arrived at the same law! For details, see
Indorato and Nastasi, 1989) Descartes longed for revenge, and even though he eventually ended
the controversy over Fermat's methods with the equivalent of, "You should have said so in the
first place, and we would never have argued...," he continued to attack Fermat's construction of
the tangent to a cycloid.