Fermat was one of the greatest amateur mathematicians. He was born on August 20, 1601 in Beaumont-de-Lomagne, France, to a prosperous merchant family. Pierre's financially secure family situation and his mother's high parliamentary social status made the study of law a logical choice for Fermat. He received the degree of Bachelor of Civil Laws from the University of Orléans in 1631, and from that time he rose steadily through the provincial government hierarchy to attain the high position of King's Counsellor in 1648, a position he held for 17 years.
Fermat's interest in mathematics, always as a hobby and not a profession, blossomed during a stay in Bordeaux, France, toward the end of the 1620s, where he studied the works of François Viète. Fermat assimilated the new symbolic algebra and theory of equations that was the basis of Viète's work. Fermat viewed his own contributions to mathematics as a continuation of the Viètan tradition. The essence of his approach was to use algebra as a formal language to unite the worlds of geometry and arithmetic, or number theory, by reducing a complex problem into a set of problems for which solutions were already known. This " analytic art," as it was known to Viète and Fermat, was similar to the techniques of analytic geometry that were being developed by other mathematicians at that same time, notably René Descartes, and was revived by Leonhard Euler in the 1700s.
Fermat refused to publish his work except in the form of challenges to other mathematicians of problems and theorems to be solved. Most of what is known of Fermat's methods comes from the many correspondences he maintained with others in the field and from copious notes and annotations scribbled in the margins of Fermat's books. One particular claim, scrawled by Fermat in the margin of a book by Diophantus, may be one of the greatest marginal notes in the history of mathematics. In the note, Fermat claimed that the equation xn + yn = zn (n > 2) has no positive integer solutions for x, y, and z, but wrote that the margin of the book was too small to contain his "simple proof". Attempts to find a proof of this statement frustrated amateur and professional mathematicians alike for over 300 years until the Princeton mathematician Andrew Wiles finally proved in 1995 that Fermat had indeed been correct.
Though Fermat's reputation as a mathematician of the highest rank was established within his own lifetime, the depth of Fermat' s contribution to mathematics was not appreciated for over a hundred years after his death and became apparent only when his work was finally compiled and published by Charles Henry and Paul Tennery in the nineteenth century.
One of Fermat's notable contributions to mathematics was his celebrated, but brief, correspondence with Blaise Pascal in which the two laid down the basic principles of modern probability theory. In an exchange of letters in 1654, the two men discussed different methods of solving a problem that a gambler had posed to Pascal, namely, if a game of chance should be stopped before it came to its conclusion, how should the stakes on the table be divided among the players? Pascal and Fermat concluded that the distribution to each player was some ratio of the total number of possible outcomes. The mathematical treatment of this concept laid the foundation for future explorations of probability theory by later mathematicians.
Fermat involved himself in several scientific disputes with René Descartes, including a disagreement over the properties of light and its propagation through dense media that result in refraction. In his attempt to refute Descartes's law of refraction, Fermat produced a mathematical derivation of the law by assuming two postulates: one, that the speed of light is finite and varies according to the density of the medium through which it travels and, two, that "nature operates by the simplest and most expeditious ways and means." Fermat employed the second postulate, known as Fermat's principle, in a geometric analysis that showed how light travels by the path of least duration. Fermat 's method was elegant in that it arrived at Descartes's law of refraction through a purely mathematical process.
Because he rarely published his work, Fermat's general isolation from the world of science grew steadily as he aged. When he died on January 12, 1665, Fermat's passing was not noted so much as a tragedy in the world of science as it was considered the loss of a great, learned gentleman.
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