Pierre de Fermat
Dana Pellegrino, History of Mathematics Research Paper, Spring 2000
Pierre de Fermat was one of the most brilliant and productive mathematicians of his time, making many contributions to the differential and integral calculus, number theory, optics, and analytic geometry, as well as initiating the development of probability theory in correspondence with Pascal. In this paper, we shall examine some of Fermat's contributions to the world of mathematics, paying specific attention to his work in number theory and in optics.
Pierre de Fermat was born on August 17, 1601 in Beaumont-de-Lomagne, France, and died on January 12, 1665 in Castres. He was the son of a prosperous leather merchant, and became a lawyer and magistrate (Singh, page 35). While not much is known of this French mathematician's early life and education, it is known that Fermat attended the University of Toulouse before moving to Bordeaux in the second half of the 1620s. He was educated at home and began his first serious mathematical researches in Bordeaux. He was also in contact with Beaugrand, and it was at this time that Fermat produced important work on maxima and minima (World Book). He communicated this work to Etienne d'Espagnet, who shared his mathematical interests.
From Bordeaux Fermat went to Orléans, where he studied law at the University. He received a degree in civil law and at the age of thirty was inducted as the "commissioner of requests." By 1631, Fermat was a lawyer and government official in Toulouse, and was promoted to a king's councillorship in the parliament of Toulouse in 1648. "Fermat's offices made him a member of that social class also and entitled him to add the de' to his name, which he did from 1631 on" (Mahoney, page 16). The office he now held entitled him to change his name from Pierre Fermat to Pierre de Fermat, as "de" is the mark of nobility in France.
Fermat was extremely preoccupied with mathematics. He kept up his mathematical friendship with Beaugrand after he moved to Toulouse but there he gained a new mathematical friend in Carcavi. Fermat met Carcavi in a professional capacity since both were councillors in Toulouse. Since they both shared a love of mathematics, Fermat told Carcavi about his mathematical discoveries (Singh, page 35).
Kepler's interest in the shape of wine casks was one of Fermat's first influences. Kepler began to investigate the volumes for specific choices of dimensions. His observations showed that as the maximum value was approached from eithe r direction, the change in volume (given a fixed change in dimension) became increasingly small (Kline, page 347). Fermat chose to expand on this concept by finding a maximum or minimum value of a function.
Fermat believed that if a function f(x) had a maximum or minimum at x, then taking e to be extremely small, it can be said that the value of f(x-e) was approximately equal to that of f(x) (Eves, page 3 25). Therefore, his method involved what can be considered a "pseudo-equality," (Edwards, page 123), in which he carefully set f(x) = f(x-e). Before Fermat could attempt to find the roots of the equation, enabling him to find the maximu m and/or minimum, he needed to correct it's "pseudo-ness" by letting e "assume the value of zero" (Eves, page 325).
Through his work on the properties of curves, Fermat contributed to the development of calculus. His study of curves and equations prompted him to generalize the equation for the ordinary parabola ay=x^{2}, and that for the rectangular hyperbola xy=a^{2}, to the form a^{n-1}y=x^{n}. The curves determined by this equation are known as the parabolas or hyperbolas of Fermat according as n is positive or negative (Kolata). He similarly generalized the Archimedean spiral, r=aQ. In the 1630s, these curves then directed him to an algorithm, or rule of mathematical procedure, that was equivalent to differentiation. This procedure enabled hi m to find tangents to curves and locate maximum, minimum, and inflection points of polynomials (Kolata).
His main contribution was finding the tangents of a curve as well as its points of extrema. He believed that his tangent-finding method was an extension of his method for locating extrema (Rosenthal, page 79). For any equation, Fermat 's method for finding the tangent at a given point actually finds the subtangent for that specific point (Eves, page 326). Fermat found the areas bounded by these curves through a summation process. "The creators of calculus, including Fermat, reli ed on geometric and physical (mostly kinematical and
dynamical) intuition to get them ahead: they looked at what passed in their imaginations for the graph of a continuous curve..." (Bell, page 59). This process is now called integral calculus.
Fermat founded formulas for areas bounded by these curves through a summation process that is now used for the same purpose in integral calculus. Such a formula is:
A= x^{n}dx = a^{n+1 }/ (n + 1)
It is not known whether or not Fermat noticed that differentiation of x^{n}, leading to na^{n-1}, is the inverse of integrating x^{n}. Through skillful transformations, he handled problems involving more general algebraic curves. Fermat applied his analysis of infinitesimal quantities to a variety of other problems, including the calculation of centers of gravity and finding the length of curves (Mahoney, pages 47, 156, 204-205).
Fermat was unable to notice what is now considered the Fundamental Theorem of Calculus, however, his work on this subject aided in the development of differential calculus (Parker, page 304). Additionally, he contributed to the law of refraction by disagreeing with his contemporary, the philosopher and amateur mathematician, René Descartes. Fermat claimed that Descartes had incorrectly deduced his law of refraction since it was deep-seated in his assumptions. As a result, Desc artes was irritated and attacked Fermat's method of maxima, minima, and tangents (Mahoney, pages 170-195). Fermat differed with Cartesian views concerning the law of refraction, published by Descartes in 1637 in La Dioptrique. Descartes attempted to justify the sine law through an assumption that light travels more rapidly in the denser of the two media involved in the refraction. (Mahoney, page 65). Twenty years later, Fermat noted th at this appeared to be in conflict with the view of the Aristotelians that nature always chooses the shortest path.
"According to [Fermat's] principle, if a ray of light passes from a point A to another point B, being reflected and refracted in any manner during the passage, the path which it must take can be calculated...th e time taken to pass from A to B shall be an extreme" (Bell, page 63).
Applying his method of maxima and minima, Fermat made the assumption that light travels less rapidly in the denser medium and showed that the law of refraction is concordant with his "principle of least time." "From this principle, Fermat deduced the familiar laws of reflection and refraction: the angle of reflection; the sine of the angle of incidence (in refraction) is a constant number times the sine of the angle of refraction in passing from one medium to anot her" (Bell, page 63). His argument concerning the speed of light was found later to be in agreement with the wave theory of the 17th-century Dutch scientist Huygens, and was verified experimentally in 1869 by Fizeau.
In addition to the law of refraction, Fermat obtained the subtangent to the ellipse, cycloid, cissoid, conchoid, and quadratrix by making the ordinates of the curve and a straight line the same for two points whose abscissae were x and x - e. There is nothing to indicate that he was aware that the process was general, and it is likely that he never separated it his method from the context of the particular problems he was considering (Coolidge, page 458). The first definite statement of the method was due to Barrow, and was published in 1669. Fermat also obtained the areas of parabolas and hyperbolas of any order, and determined the centers of mass of a few simple laminae and of a paraboloid of revolution (Ball, pages 49, 77 , 108).
Fermat was also strongly influenced by Viète, who revived interest in Greek analysis. The ancient Greeks divided their geometric arguments into two categories: analysis and synthesis. While analysis meant "assuming the pro position in question and deducing from it something already known," synthesis is what we now call "proof" (Mahoney, page 30). Fermat recognized the need for synthesis, but he would often give an analysis of a theorem. He would then state that it could easily be converted to a synthesis.
Fermat's reputation as one of the leading mathematicians in the world came quickly, but attempts to get his work published failed mainly because he never really desired to put his work into a polished form. Fermat enjoyed producing his results but was not willing to do the clean-up work required to make them suitable for publication. Wanting to avoid controversy, Fermat would not allow his name to be put on his results when others published them, but instead, used the concealing initi als MPEAS. When Roberval offered to edit and publish some of his works, Fermat replied, "whatever of my works is judged worthy of publication, I do not want my name to appear there" (Prime Glossary). Except for a few isolated papers, Fe rmat published nothing in his lifetime, and gave no systematic exposition of his methods. Fermat enjoyed the pleasure of discovery more than any reputation it might gain him. Some of the most striking of his results were found after his death on loose s heets of paper or written in the margins of works which he had read and annotated, and are unaccompanied by any proof. Thus, it is relatively difficult to estimate the dates and originality of his work.
In 1640, while studying perfect numbers, Fermat wrote to Mersenne "if p is called prime, then 2p divides 2^{p}-2." Shortly thereafter, he expanded this into what is now called Fermat's Little Theorem. As usual, Fermat stated, "I would send you a proof, if I did not fear its being too long" (Prime Glossary). Fermat is best remembered for this work in number theory, in particular for Fermat's Last Theorem. His most famous statement was attached to this theorem, which states:
x^{n} + y^{n} = z^{n} has no non-zero integer solutions for x, y and z when n > 2.
(Kramer, page 508). In the margin of Arithmetica, Fermat wrote the following observation in Latin:
"Cubem autem in duos cubos, aut quadratoquadratum in duos quadratoquadratos, et generaliter nullam in infinitum ultra quadratum potestatem in duos eiusdem nominis fas dividere. [Translation:] It is impossible to write a cube as a sum of two cubes, a fourth power as a sum of fourth powers, and, in general, any power beyond the second sum as a sum of two similar powers" (Singh, page 61).
Fermat was out of touch with his scientific colleagues in Paris during the period from 1643 to 1654 due to many complications. Primarily, the pressure of work kept him from devoting so much time to mathematics. Secondly, the Fronde (a civil war in France) took place and greatly affected Toulouse from 1648. Finally, there was the plague of 1651, which had great consequences both on life in Toulouse and its near fatal consequences on Fermat himself. Fermat contracted the plague, and his death was incorrectly reported in 1653, but this was soon corrected. "I informed you earlier of the death of Fermat. He is alive, and we no longer fear for his health, even though we had counted him among the dead a short time ago" (Ball, page 109). Despite these interruptions, it was during this time that Fermat worked on number theory.
It is now believed that Fermat's `proof' was incorrect, although it is impossible to be completely certain. The truth of Fermat's assertion was proved by Andrew Wiles, a British mathematician employed at Princeton University. He temporarily withdrew this claim when a gap emerged later in 1993. In November of 1994, Wiles again claimed to have a correct proof, which has now been accepted, and has been published. Thirty years after first having encountered this problem, Wiles remembered how he felt the moment he was introduced to Fermat's Last Theorem. "It looked so simple, and yet all the great mathematicians in histor y couldn't solve it. Here was a problem that I, a ten-year-old, could understand and I knew from that moment that I would never let it go. I had to solve it" (Singh, page 6). Unsuccessful attempts to prove the theorem over a 300-year period led to the discovery of commutative ring theory and a wealth of other mathematical discoveries.
Fermat posed further problems. He asked specifically for a proof that the sum of two cubes cannot be a cube. This special case of Fermat's Last Theorem may indicate that Fermat realized by this time that his proof of the general result was incomplete. Another problem is to show that there are exactly two integer solutions of x^{2} + 4 = y^{3} and that the equation x^{2} + 2 = y^{3} has only one integer solution (Ball, pages 47-63). (See appendix for additional theories). He posed these problems directly to the English. Fermat had been hoping his specific problems would lead them to discover deeper theoretical results, just as he had done. However, the English failed to follow his lead.
Around this time, one of Descartes students was collecting his correspondence for publication and he turned to Fermat for help with the Fermat - Descartes correspondence. This led Fermat to look again at the arguments he had used twenty years earlier, and again at his objections to Descartes' optics. In particular, he had been unhappy with Descartes' description of the refraction of light. He now settled on a principle, which did in fact yield the sine law of refraction that Snell and Descartes had proposed (World Book). At this time, however, Fermat had deduced it from a fundamental property that he proposed, namely that light always follows the shortest possible path. Fermat's principle, one of the most basic properties of optics, was not well-liked with mathematicians at the time (Mahoney, pages 375-390).
In 1656, Fermat had started a correspondence with Huygens. This grew out of Huygens' interest in probability and Fermat soon manipulated the correspondence onto topics of number theory (Smith, page 424). Fermat tried hard, even though this topic did not interest Huygens. In New Account of Discoveries in the Science of Numbers sent to Huygens via Carcavi in 1659, he revealed more of his methods than he had done to others. Fermat described his method of infinite descent, also known as "reverse induction" (Mahoney, page 231), and gave an example on how it could be used to prove that ever y prime of the form 4k+1 could be written as the sum of two squares.
"Suppose some number of the form 4k+1 could not be written as the sum of two squares. Then there is a smaller number of the form 4k+1 that cannot be written as the sum of two squares" ( Aczel, page 44).
Continuing the argument will lead to a contradiction. What Fermat failed to explain in this letter is how the smaller number is constructed from the larger number.
One can assume that Fermat did know how to make this step, but again his failure to disclose the method made mathematicians lose interest. It was not until Euler took up these problems that the missing steps were filled in. Pierre de Fermat is known as an intriguing mathematician (Eves, page 354). His contributions to the mathematical world, mainly to calculus, the law of refraction, and number theory have been described. All in all, Fermat has been referred to as the greatest French mathematician of the first half of the seventeenth century (Boyer, page 154).
Taken directly from A Short Account of the History of Mathematics by W.W. Rouse Ball
(a) If p be a prime and a be prime to p then a^{(p-1)}-1 is divisible by p, that is, a^{(p-1)}º1 (mod p). A proof of this, first given by Euler, is well known. A more general theorem is that a^{F(n)}-1 º0 (mod n), where a is prime to n and F(n) is the number of integers less than n and prime to it.
(b) An odd prime can be expressed as the difference of two square integers in one and only one way. Fermat's proof is as follows. Let n be the prime, and suppose it equal to x^{2} - y^{2}, that is, to (x + y)(x - y). Now, by hypothesis, the only integral factors of n are n and unity, hence x + y = n and x - y = 1. Solving these equations we get x = (n + 1) and y = (n - 1).
(c) He gave a proof of the statement made by Diophantus that the sum of the squares of two integers cannot be of the form 4n - 1; and he added a corollary which I take to mean that it is impossible that the product of a square and a prime of the form 4n - 1 [even if multiplied by a number prime to the latter], can be either a square or the sum of two squares. For example, 44 is a multiple of 11 (which is of the form 4 W 3 - 1) by 4, hence it cannot be expressed as the sum of two squares. He also stated that a number of the form a2 + b2, where a is prime to b, cannot be divided by a prime of the form 4n - 1.
(d) Every prime of the form 4n + 1 is expressible, and that in one way only, as the sum of two squares. This problem was first solved by Euler, who showed that a number of the form (4n + 1) can be always expressed as the sum of two squares.
(e) If a, b, c, be integers, such that a^{2} + b^{2} = c^{2}, then ab cannot be a square. Lagrange gave a proof of this.
(f) The determination of a number x such that x^{2}n + 1 may be a square, where n is a given integer which is not a square. Lagrange gave a solution of this.
(g) There is only one integral solution of the equation x^{2} + 2 = y^{3}; and there are only two integral solutions of the equation x2 + 4 = y3. The required solutions are evidently for the first equation x = 5, and for the second equation x = 2 and x = 11. This question was issued as a challenge to the English mathematicians Wallis and Digby.
(h) No integral values of x, y, z can be found to satisfy the equation x^{n} + y^{n} = z^{n}, if n is an integer greater than 2.
Sources: