While Descartes was formulating the basis of modern analytic geometry, the subject was also occupying the attention of another French mathematical genius, Fermat. Fermat’s claim to priority rests on a letter written to Roberval in September, 1636, in which it is stated that the ideas of the writer were even then seven years old. The details of the work appear in the posthumously published paper Isogoge ad locus planos et solidos. Here we find the equation of a general straight line and of a circle and a discussion of hyperbolas, ellipses and parabolas. In a work on tangents and quadratures, completed before 1637, Fermat defined many new curves analytically. Where Descartes suggested a few new curves generated by mechanical motion, Fermat proposed many new ones, defined by algebraic equations. The curves x to the power of m multiply by y to the power of n is equal to a, y to the power of n is equal to a multiply by x to the power of m, and r to the power of n is equal to a multiply by omega are still known as hyperbolas, parabolas and spirals of Fermat. Fermat also proposed, among others, the cubic curve later known as the witch of Agnesi, named after Maria Gaetana Agnesi (1718-1799 AD), a versatile woman distinguished as a mathematician, linguist, philosopher and somnambulist. Thus, where to a large extent Descartes began with a locus and then found its equation, Fermat started with the equation and then studied the locus. These are the two inverse aspects of the fundamental principle of analytic geometry. Fermat’s work is written in Viete’s notation and thus has an archaic look when compared with Descartes’ more modern symbolism.

There is a seemingly reliable report that Fermat was born at Beaumont de Lomagne, near Toulouse on August 17, 1601. It is known that he died at Castres or Toulouse on January 12, 1665. His tombstone, originally in the church of the Augustines in Toulouse and later moved to the local museum, gives the above date of death and Fermat’s age at death as fifty-seven years. Due to this conflicting data, Fermat’s dates are usually listed as (1601?-1665). Indeed, for various reasons, Fermat’s birth year, as given by different writers, ranges from 1590 to 1608.

Fermat was the son of a leather merchant and received his early education at home. At the age of thirty he obtained the post of councilor for the local parliament at Toulouse and there discharged his duties with modesty and punctiliousness. Working as a humble and retiring lawyer, he devoted the bulk of his leisure time to the study of mathematics. Although he published very little during his lifetime, he was in scientific correspondence with many leading mathematicians of his day and, in this way, considerably influenced his contemporaries. He enriched so many branches of mathematics with so many important contributions that he has been called the greatest French mathematician of the seventeenth century.

Of Fermat’s varied contributions to mathematics, the most outstanding is the founding of the modern theory of numbers. In this field, Fermat possessed extraordinary intuition and ability. It was probably the Latin translation of Diophantus’ Arithmetica, made by Bachet de Meziriac in 1621, that first directed Fermat’s attention to number theory. Many of Fermat’s contributions to the field occur as marginal statements made in his copy of Bachet’s work. In 1670, five years after his death, these notes were incorporated in a new, but unfortunately carelessly printed, edition of the Arithmetica, brought out by his son Clement-Samuel. Many of the unproved theorems announced by Fermat have later been shown to be correct.

In 1879, a paper was found in the library at Leyden, among the manuscripts of Christian Huygens, in which Fermat describes a general method by which he may have many of his discoveries. The method is known as Fermat’s method of infinite descent and is particularly useful in establishing negative results. In brief, the method is this. To prove that a certain relation connecting positive integers is impossible, assume, on the contrary, that the relation can be satisfied by some particular set of positive integers. From this assumption, show that the same relation then holds for another set of still positive integers, and so on ad infinitum. Since the positive integers cannot be decreased in magnitude indefinitely, it follows that the assumption at the start is untenable, and therefore the original relation is impossible.