The French mathematician Pierre de Fermat (1601-1665) played an important part in the foundation and development of analytic geometry, the calculus of probabilities, and especially the theory of numbers.

# Pierre de Fermat

Pierre de Fermat was born on Aug. 17, 1601, at Beaumont-de-Lamagne near Montaubon. There is some doubt as to the precise date of his birth. He is said to have been baptized on Aug. 20, 1601, but his tombstone puts his birth as 1608, and others have stated 1595. He was the son of Dominique Fermat, a leather merchant, and Claire de Long.

It was decided that Fermat should be trained as a magistrate, and he was sent to Toulouse. The general lines on which he was educated can only be guessed, and so far as his career as jurist is concerned, there is a record of his installation at Toulouse on May 14, 1631. In 1648 he was promoted to king's counselor in the Parliament of Toulouse, a post which he held until his death on Jan. 12, 1665. In 1631 he married his mother's cousin, Louise de Long; they had three sons and two daughters.

## Theory of Numbers

It was perhaps C. G. Bachet's translation (1621) of Diophantus of Alexandria that stimulated Fermat's interest in the theory of numbers. That same edition was republished in 1670, with the addition of Fermat's notes edited by his son, who tells of the immense difficulties of collecting his father's writings, because they were only known from letters and notes in which Fermat usually stated theorems without proof. Much of Fermat's notes on Diophantus's problems were taken from the margin of his copy of Bachet's work.

As a specimen of Fermat's genius, there is his 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 multitude of minor solutions. In so doing, he also found many lesser but still important results. His final technique made use of his method of infinite descent, which may be faintly appreciated from a quotation. After saying that the theorem alluded to was beyond the power of René Descartes, by his own admission, Fermat goes on: "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." Later, to prove an even more general proposition, he had to prove first five lesser theorems, and there again he made use of the same technique. The first of these theorems is worth singling out for comment. It is the theorem that every prime number of the form 4n + 1 is the sum of two squares (thus 4 x 1 + 1 = 12 + 22;4x2 + 1 = 02 + 32;4x3 + 1 = 22 + 32; and so on). The great mathematician Leonhard Euler was the first to supply and publish a proof of this (1770).

There is one theorem, however, which has never been proved, and this has become something of a legend as Fermat's "Last Theorem," which states that there is no solution in integers of the equation *x*n + *y*n = *z*n (*xyz* ≠ 0, *n* >2). There is a method in Diophantus for dividing a given square into two squares. Against this proposition in the margin of Bachet's edition Fermat wrote the following note:
"On the other hand, it is impossible to separate a cube into two cubes, a fourth power into two fourth powers, or, generally, any power except a square into two powers with the same exponent (i.e., of the same degree). I have discovered a truly marvelous proof of this, which however, the margin is not large enough to contain." Did Fermat have a proof of what, for 3 centuries, others have failed to prove? Euler proved it when the exponent (*n*) was equal to 3 or 4. Others have extended enormously the range of related theorems which are provable, but none has found a "truly marvelous proof" such as Fermat claimed to have found.

## Analytical Geometry

Some of Fermat's first original mathematics appears to have been inspired by a famous problem of Apollonius. The crucial problem was this: if from a point in a plane four fixed lines are drawn, and four other lines through a point *P* cross the first four, all at the given angle; and if, furthermore, the distance along the four variable lines from *P* to where they cross the others is known as *a, b, c,* and *d;* then if *a* · *c* is in constant ratio to *b* · *d,* the point *P* moves on a conic section (an ellipse, parabola, hyperbola, circle, or pair of straight lines). This theorem, which is not easy to write more concisely in simple language, may be called the "four-line theorem." If two of the fixed lines coincide, then the path of P is still a conic, but the problem, the "three-line problem," is easier.

Fermat provided his own proof of the three-line theorem, and in doing so he made use of analytical methods, determining points in a plane by two coordinates, showing that if the coordinates are related by equations like 2*x* + 3*y* = 5, then the point lies on a straight line, and so on. He also worked out the equations of the curves known as conic sections, and he was quite familiar with coordinate methods in three dimensions.

Other work of a similar character by Fermat relates to the problem of constructing 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 a problem of areas. Almost as important was his method of maxima and minima. This was first published in 1638 and was used for finding centers of gravity.

## Principle of Light Transmission

It was shown by Hero of Alexandria that light traveling between two points, and undergoing a reflection in the process, follows the shortest path. For example, to follow the shortest path, light passing through a bowl of water would travel in a straight line; but observation shows that this is not true. Fermat demonstrated that such refracted (bent) light must be measured by optical distance, and this is always a minimum. The optical distance is the sum of the products of the distances and the corresponding refractive indexes. Fermat's conception involved the (correct) belief that light travels more slowly in more optically dense media. As subsequently developed, it was of great importance to the derivation of geometrical optics and also influenced J. Bernoulli in founding the calculus of variations.

Fermat did important work in the foundation of a theory of probability, which grew out of his early researches into the theory of numbers. In this theory he was without equal in his century and perhaps in any century. By any standards he was a great mathematician, but his name is less often encountered than it would have been had his personal life been better known.

## Further Reading on Pierre de Fermat

L. J. Mordell, *Three Lectures on Fermat's Last Theorem* (1921), gives a short résumé of Fermat's life. Eric Temple Bell, *Men of Mathematics* (1937), contains a chapter on Fermat. See also James R. Newman, ed., *The World of Mathematics* (4 vols., 1956), and Joseph Frederick Scott, *A History of Mathematics: From Antiquity to the Beginning of the Nineteenth Century* (1958).