Fermat is regarded by some mathematicians as the inventor of differential calculus. He created modern number theory, and, together with Blaise Pascal, developed the theory of probability. Fermat's wide range of accomplishments is especially remarkable considering that mathematics was for him only a hobby.
Pierre Fermat was baptized and most likely born on August 20, 1601, in Beaumont-de-Lomagne. His father, Dominique Fermat, was bourgeois second consul of Beaumont-de-Lomagne and a prosperous leather merchant. His mother, Claire de Long, came from a prominent family of parliamentary lawyers. Fermat had three siblings, a brother, Clement, and two sisters, Louise and Marie. His early education was most likely received at the Franciscan monastery in Beaumont. During the 1620s he attended the University of Toulouse, then moved for a few years to Bordeaux, where he undertook an informal study of mathematics and made contact with students of François Viète, the noted French algebraist. Fermat then entered the University of Orleans, where he received a Bachelor of Civil Law. In 1631 he began practicing law, purchasing positions as councilor of the Parliament of Toulouse and Commissioner of Requests. It was then that he added "de" to his name, an indication of his social standing afforded by his offices. In the same year Fermat married his fourth cousin, Louise de Long. Together they had five children, Clement-Samuel, Jean, Claire, Catherine and Louise. In Fermat's profession as a jurist, he rose gradually through the ranks, being named to the criminal court in 1638 and promoted in 1648 to a King's councilorship. He retained the latter post until his death in 1665.
Within a few years of taking his first parliamentary post, Fermat began corresponding with several prominent Parisian mathematicians, including Marin Mersenne , Gille Personne de Roberval, and Étienne Pascal, father of probability theorist Blaise Pascal. In his first communications he described his work on geostatics. He proposed that Galileo had been incorrect in stating that a freely falling cannonball should follow a semicircular path. In the course of proving the path would be spiral, Fermat developed a new method of quadrature for curves. He also posed several analytical problems to the Parisian group, which he claimed to have already solved on his own. Roberval and Mersenne found those and subsequent problems difficult and eventually requested that Fermat describe the techniques by which he had derived their solution. In response, Fermat sent a paper called Method for Determining Maxima and Minima and Tangents to Curves Lines, along with his restoration of the Greek mathematician Apollonius's Plane Loci. These papers, received in Paris in 1636, set out the fundamentals of differential calculus. In 1637 Fermat wrote a manuscript which the Parisian mathematicians circulated called An Introduction to Plane and Solid Loci. At about the same time René Descartes had sent Mersenne the galley proofs of his Discourse on the Method and accompanying Essays. The group in Paris quickly realized that Fermat and Descartes had independently developed the principles of analytic geometry, deriving the same basic technique for treating geometric locus problems algebraically. While Fermat's treatise arrived first in Paris, Descartes had laid his foundation earlier. A bitter and protracted argument arose between the two men, ignited by issues of priority, but eventually focusing on the subject of maxima and minima. The dispute engaged most of the mathematicians in Paris and forced Fermat to prove the generality of his methods. Descartes finally admitted defeat regarding the mathematical controversy, but Fermat's reputation was damaged. While he had established himself as one of the best mathematicians in Europe, he was resented by some of his colleagues for communicating his work in piecemeal and sending problems in the form of challenges. He was even accused of posing problems that had no solutions, in order to expose his rivals.
In the 1640s Fermat began his most important work, the development of modern number theory. Civic duties, however, prevented him from any meaningful mathematical correspondence between 1643 and 1654. In 1648 Fermat was occupied with the Fronde, a civil war, and in 1649, the Spanish raid on Languedoc. In 1651 the plague struck Toulouse, and Fermat became so ill that in 1653 his death was mistakenly reported. During a decade of isolation from the mathematical community he focused attention on developing a method of determining whether numbers were prime, and if not, on finding their divisors. The culmination of this work is today known as Fermat's Theorem. It states that if n is any whole number and p any prime, then np - nis divisible by p. Fermat then went on to explore the concept of decomposition of primes of various forms into sums of their squares.
In the spring of 1654, Fermat received a letter from Blaise Pascal, asking for advice on a problem involving the consecutive throws of a die. The question to be resolved was how to divide the stakes in a dice game between two players, when their game was prematurely interrupted. The exchange that ensued between Pascal and Fermat over the course of just a few months helped lay the foundations of probability theory. By August 1654 Fermat was trying to divert Pascal's attention to his work on the theory of numbers. But Pascal, like most of his French contemporaries, saw little importance in this topic. In 1656 Fermat began to correspond about number theory with the Dutch physicist and astronomer Christiaan Huygens, expositor of the wave theory of light and inventor of the pendulum clock. Fermat described to him his method of infinite descent or "reduction analysis," in which larger problems are broken down into groups of problems more readily solvable. Though Huygens admired the seminal contributions of Fermat to calculus and analytical geometry, he believed Fermat's number theory had no practical application and that Fermat had become out of touch with important mathematical questions. Fermat tried to engage the two English mathematicians John Wallis and William Brouncker in discussion of his new theories, also to little avail. Finally, between 1658 and 1662 Fermat turned to the topic of optics. Using the method of maxima and minima, he investigated the laws of reflection and refraction. In the course of his work he discovered that light travels by the path of least duration, a concept now known as Fermat's Principle.
http://investinginknowledge.com/info/2010/06/fermat-pascal-letters/
http://www.amazon.com/Unfinished-Game-Pascal-Fermat-Seventeenth-Century/dp/0465018963
http://science.larouchepac.com/fermat/
http://science.larouchepac.com/
http://mathforum.org/isaac/problems/prob1.html
http://mathforum.org/
http://www.socsci.uci.edu/~bskyrms/bio/readings/pascal_fermat.pdf
http://en.wikipedia.org/wiki/Problem_of_points
