Pierre
de Fermat (1601-1665) was born in a village in southwestern France,
obtained a degree from the University of Toulouse, then studied law at
the University of Orleans and practiced as a lawyer in Toulouse until his
death. He devoted much of his free time away from the courtroom to
the study of mathematics, which was his favorite hobby. He fed his
interests in mathematics by reading the classic works of Euclid, Apollonius,
Archimedes, Pappus, and Diophantus, but was especially influenced by the
work of his countryman, François Viéte. Another way
in which Fermat was able to keep in touch with what was going on in the
mathematical world was through his written correspondence. He was
fortunate to have started writing early in his career to Marin
Mersenne (1588-1648), a priest who lived in Paris and made it a point
to maintain correspondence on mathematical topics with dozens of people
across Europe; in effect he acted as a clearinghouse of information on
the latest developments in the science. The treatise we read here
appeared in this correspondence between Fermat and Mersenne. (It
is worth noting that although the letter to Mersenne in which this text
is dated 1638, the treatise may have been written as many as nine years
earlier for the benefit of another friend of Fermat.)

Galileo studied physical motion by developing the concept of an instantaneous
rate of change of one quantity with respect to another (velocity as the
rate of change of distance with respect to time and acceleration as the
rate of change of velocity with respect to time). The instantaneous
rate is the ratio of related infinitesimal values of the quantities: for
instance, if at a time *t*, an object has covered a distance *d*,
then by the "next" moment of time, an infinitesimally small duration of
time D*t
*has
elapsed, during which the object has proceeded across an infinitesimally
small distance D*d*.
Thus, the rate of change measures the velocity of the object (*v*
= D*d*/D*t*)
at that moment. We have identified this quantity as the derivative
of the distance with respect to the time.

Fermat's work also is related to the concept of derivative, but from a
considerably different direction. Whereas Galileo's method is geometric,
Fermat's is algebraic. Galileo's primary concern is the mathematical
description of motion. Fermat, on the other hand, is interested in
two problems: the first is the problem of optimiziation, that is, given
some quantity *y* dependent on another quantity *x* so that *y*
= *f*(*x*), Fermat asks how one can determine the value(s) of
*x*
for which *y* takes a maximum or minimum value. Second, he has
also contrived "a general method" for finding the tangent line to a curve
at a given point. It turns out that the same method covers both problems,
and that this method is essentially the computation of the derivative *f*´(*x*).

Neither Galileo nor Fermat knew that what they were doing was related;
it would have to wait for Newton and Leibniz at the end of the seventeenth
century and their successors in the eighteenth to see these connections.

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last modified 11/20/02

Copyright (c) 2000. Daniel E. Otero