Theory of Numbers

Fermat's interest in the theory of numbers was probably stimulated by a translation of Diophantus' work on the property of whole numbers. It was published again in 1670 by Fermat's son, this time with the addition of notes jotted down by Fermat on the margins of his copy of the translated work, which his son had painstakingly retrieved. Among his many letters and scribbled notes, there was another theorem that every number is either a square or the sum of two, three or four squares. He arrived at his proof after long attempts to break down the solution into a number of smaller and minor solutions. In doing so, he came upon several other important results. He used a technique of infinite descent which he himself described as, "I have at last brought this under my method and I prove that, if a given number were not of this nature, there would exist a number smaller than it which would not be so either, and again a third number smaller than the second, etc ad infinitum; whence we infer that all numbers are of the nature indicated". Fermat worked out several important theorems in this manner, but left one particular theorem to which he claimed to have found a proof, but did not write down. This came to be known as Fermat's last theorem.

Analytical Geometry

In geometry, Fermat appears to have been inspired by Appollonius, particularly a famous problem that he had proposed –the four-line theorem. Fermat worked out his proof of the three-line theorem, making use of analytical methods, and using co-ordinates to represent equations. He also worked out the equations of the curves known as conic sections and was very proficient with co-ordinate methods in three dimensions. In other works of geometry, Fermat constructed a tangent to a curve using infinitesimals. He found a method of calculating the length of a curve (involving the method of tangents) by first solving the problem of areas. Another of his important contributions was his method of maxima and minima, which was used for finding curves of gravity.

Light Transmission

Hero of Alexandria had shown that light travelling between two points and undergoing a reflection in the process, follows the shortest path. But light passing through a bowl of water appears bent. Fermat showed that such refracted (bent) light had to be measured by optical distance -the sum of the products of the distances and the corresponding refractive index and this is always a minimum. Fermat's idea was that light travels more slowly in more optically dense media and in this he was correct. This concept was to play an important role in the derivation of geometrical optics later.

Pierre de Fermat was born on August 17, 1601, at Beaumont-de-Lamagne near Montaubon in France. There is some mystery regarding his exact date of birth as his tombstone puts his birth as 1608, whereas he was said to have been baptized in 1601. He was the son of Dominique Fermat, a leather merchant and his mother was Claire de Long. Fermat did his early schooling at home, and was then sent to Toulouse to train in law and become a magistrate. There is record of his installation at Toulouse on May 14, 1631. In 1648, he was promoted to King's Counselor at the parliament of Toulouse, a post that he held until his death in 1665.

Considering that Fermat devoted only part of his energies to mathematics, it is remarkable how much he was able to accomplish. But his habit of making hasty notes instead of systematic proof statements resulted in many of his great theories going unnoticed. Fermat did important work in the foundation of a theory of probability, which grew out of his great researches into the theory of numbers. His work in this area remained unequalled not only in his own era but probably for all time. But again, because of a lack of published material, his name is far less known than many other mathematicians of his genre.

Very little is known of Fermat's personal characteristics except his rather exasperating habit of scribbling notes at the margins of the books that he would work on. His son had immense difficulty in retrieving information about his father's work and it was only 5 years after his death (1665) that he compiled them and published them as a book. Among those to whom he conveyed his theorems in the form of letters was Rene Descartes. Fermat was in fact ahead of Descartes in many of his solutions and even Isaac Newton relied on many of his works in laying the foundations of Calculus. Fermat was married to his mother's cousin, Louis de Long in 1631 and had three sons and two daughters.