Pierre de Fermat

Introduction: tangent lines and the derivative

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 Dt has elapsed, during which the object has proceeded across an infinitesimally small distance Dd.  Thus, the rate of change measures the velocity of the object (v = Dd/Dt) 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.