quiet and unaggressive life. Fermat has left the impress of his genius upon all branches of mathematics then known. A great contribution to geometry was his De maximis et minimis. About twenty years earlier, Kepler had first observed that the
increment of a variable, as, for instance, the ordinate of a curve, is evanescent for values very near a maximum or a minimum value of the variable. Developing this idea, Fermat obtained his rule for maxima and minima. He substituted
for in the given function of and then equated to each other the two consecutive values of the function and divided the equation by . If be taken 0, then the roots of this equation are the values of , making the function a maximum or a minimum. Fermat was in possession of this rule in 1629. The main difference between it and the rule of the differential
calculus is that it introduces the indefinite quantity instead of the infinitely small . Fermat made it the basis for his method of drawing tangents.
Owing to a want of explicitness in statement, Fermat's method of maxima and minima, and of tangents, was severely attacked by his great contemporary, Descartes, who could never be brought to render due justice to his merit. In the ensuing dispute, Fermat found two zealous defenders in Roberval and Pascal, the father; while Mydorge, Desargues, and
Hardy supported Descartes.
Since Fermat introduced the conception of infinitely small differences between consecutive values of a function and arrived at the principle for finding the maxima and minima, it was maintained by Lagrange, Laplace, and Fourier, that
Fermat may be regarded as the first inventor of the differential calculus. This point is not well taken, as will be seen
from the words of Poisson, himself a Frenchman, who rightly
says that the differential calculus "consists in a system of rules proper for finding the differentials of all functions, rather than in the use which may be made of these infinitely small variations in the solution of one or two isolated problems."
A contemporary mathematician, whose genius excelled even that of the great Fermat, was Blaise Pascal (1623–1662). He
was born at Clermont in Auvergne. In 1626 his father retired to Paris, where he devoted himself to teaching his son, for he would not trust his education to others. Blaise Pascal's genius for geometry showed itself when he was but twelve years old. His father was well skilled in mathematics, but did not wish his son to study it until he was perfectly acquainted with Latin and Greek. All mathematical books were hidden out of his sight. The boy once asked his father what mathematics treated of, and was answered, in general, "that it was the method of making figures with exactness, and of finding out what proportions they relatively had to one another." He was at the same time forbidden to talk any more about it, or ever to think of it. But his genius could not submit to be confined within these bounds. Starting with the bare fact that mathematics taught