Pierre de Fermat

French mathematician

• Born: August 17, 1601
• Birthplace: Beaumont-de-Lomagne, France
• Died: January 12, 1665
• Place of death: Castres, France

Fermat made several pivotal discoveries in the foundations of analytical geometry, differential calculus, and probability theory, serving as an intellectual catalyst for René Descartes, Gottfried Wilhelm Leibniz, and Sir Isaac Newton. His main achievements, however, were in number theory, in which he established the basis of the modern theory and formulated two fundamental theorems that still bear his name.

Early Life

Pierre de Fermat (pyehr deh fehr-mah) was born in a provincial village northwest of Toulouse in the Gascony region of southern France. His father, Dominique, was a well-to-do leather merchant and petty official; his mother, Claire, née de Long, belonged to a prominent family of jurists. Pierre, his brother Clement, and his two sisters acquired primary and secondary education at the local monastery of Grandselve. Pierre then attended the University of Toulouse, from which, having decided on a legal career, he entered the University of Law at Orléans, where he earned a bachelor of civil laws degree in 1631. Shortly before graduation, he purchased an office in the parlement of Toulouse; shortly after, he married a distant cousin, Louise de Long, and settled down to a long and apparently uneventful career as a civil official and legislator. For the next thirty-four years, he fathered five children, served capably in office, and overtly did little to distinguish himself. Few records remain of his life, beyond the normal transactions of the bourgeois.

The single remaining portrait of Fermat, apparently done when he was around forty-five, shows a round, somewhat fleshy face, with arched brows, a large straight nose, and a small, rather delicate mouth. The large eyes, the most prominent feature of his face, seem unfocused, as though staring at something deep within. On the whole, he looks remote, withdrawn, aloof, and a bit patrician—a proper image for a provincial jurist.

He looked undistinguished largely because he wanted to. His life spanned a turbulent period in French history, when distinction often led to disgrace or at least to difficulty. Fermat avoided this adroitly, and he therefore gained stability and a measure of leisure, allowing him to pursue his real interest: mathematics. Mathematics had not yet become a profession, hence, it could be pursued as a hobby. Fermat became one of the greatest mathematical hobbyists of his or of any other time. His correspondence is filled with the most daring mathematical speculations ever recorded, all the more striking because he strenuously resisted publication or any kind of public recognition. Publication might have jeopardized his stability, his security, and his serenity.

Life’s Work

Fermat’s major achievements lie in the field of his great love, number theory; but he anticipated these with striking discoveries in other analytical areas, which, characteristically, he neglected to publish, thereby allowing others—notably René Descartes and Blaise Pascal —to gain credit that was properly his. Thus, for example, he anticipated the fundamental discovery underlying the differential calculus thirteen years before the birth of Sir Isaac Newton and seventeen years before that of Gottfried Wilhelm Leibniz; yet they are commonly given independent credit for that finding. He did this in a characteristic way.

The basic problem of differentiation is to determine the rate of change of a system at a particular instant in time. This is commonly represented by the attempt to draw the straight-line tangent to the graph of a continuous function—that is, to discover how to construct a line tangent to any point of a given curve. After Descartes had invented a coordinate system, constructing the graph itself was relatively easy. The difficulty lay in determining the tangent, for it changed at every point in the curve. The inventors of the calculus solved this simply by visualizing what would happen to a given tangent as it approached a given point—that is, by seeing how the tangent changed as the distance between it and the point dwindled to nothing. Once they had seen this in their imagination, they could proceed to create algebraic or graphical means of specifying it; this was done by determining the limiting values of the y-component divided by the x-component as both approached zero simultaneously. This sounds complicated and is extremely difficult to visualize without graphic demonstration and some knowledge of trigonometry, but these problems had real physical applications that had to be determined before modern physics and the technology based on it could be carried out.

Fermat made a second discovery in this new differential calculus closely related to the first. Basic algebra presents equations in which one quantity is expressed in terms of another: y = 4t for example. This means that the value of y can be determined by calculating the value of 4t. From another point of view, the value of y depends on t, or y depends on t, or y is a function of t: y = f(t), in algebraic notation. In this particular instance, the function can be graphed as a straight line, and the calculations for applications are quite simple. When the graph is a complex curve, however, in many practical applications it is necessary to find the maximum and minimum values of the function. Fermat derived a way to do this simply, both graphically and algebraically. Beginning with his earlier observations about tangents, Fermat reasoned that the highest and lowest values of any function would be found at the highest and lowest points of the curve. Observation would show this easily on a graph. Furthermore, these tops and bottoms would occur only where the tangents became parallel to the horizontal axis—that is, where the equation of the tangent became zero. To find them, he had only to set the tangent equation equal to zero and calculate the point. This discovery, relatively simple, had vital and far-reaching effects.

Fermat himself occasionally dabbled in particular applications of his general theorems. In one case, he turned his attention to optics and the problems of determining how a ray of light will behave when it reflects from or is refracted through a surface. In the process of studying this, he discovered what has come to be called the principle of least time, which is the fundamental principle of quantum theory, particularly in its mathematical aspect of wave mechanics. A ray of light passes from point A to point B, undergoing several reflections and passing through several surfaces. Fermat proved that regardless of deviations, the path the ray must take can be calculated by a single factor: The time spent in passage must be a minimum. On the strength of this theorem, Fermat deduced that, in reflection, the angle of incidence equals the angle of reflection and, in refraction, the sine of the angle of incidence equals a constant multiple of the angle of refraction in moving from one medium to another.

Fermat was also an innovator in analytic geometry, anticipating Descartes in the process but characteristically refusing to declare his precedence by publishing his findings. In fact, he went beyond Descartes and made the crucial applications on which all further progress in the discipline depended. Fermat was the first to postulate a space of three dimensions, thereby laying the basis for modern multidimensional analytic geometry. Like most of his other discoveries, this marked a true turning point, for the great difficulty in this method of analysis is going from two to three dimensions. Moreover, in making this transition, Fermat also corrected Descartes in the classification of curves by degrees of equation. Descartes, assuming proprietary rights as the assumed inventor of the system, at first balked at accepting the corrections of an amateur but had to concede in the end. Yet true credit for this invention was denied Fermat for centuries. Similarly, not until 1934 did anyone discover that Newton had borrowed the fundamental theorem of the differential calculus from Fermat.

Any of these discoveries alone would have sufficed for the life’s work of any mathematical genius. For Fermat, however, these were mere incidents; his principal mathematical occupation was the theory of numbers, a field in which he made his major achievements. This field concerns itself with the most basic of all topics in mathematics: the simple whole numbers and their common relationships and properties. Although basic to mathematics and simple in the beginning, the problems presented here have led to the most abstract theories.

Fermat began by concentrating on prime numbers—those numbers greater than one that have no divisors other than 1 and the number itself: 2, 3, 5, 7, 11, and the like. In working with these numbers, Fermat routinely presented his theorems without proofs, or without proving them completely, or simply with hints about the methods he used to discover them. Furthermore, he sometimes happened to be completely—or partly—wrong. Something like that took place in his formulation of what came to be known as Fermat’s numbers: the series 3, 5, 17, 257, 65537. All of these numbers are found by the same process of raising 2 to a further power of 2 itself raised to a sequential power of 2. Fermat asserted that every number so found is a prime. He was right for the first five numbers, but the next two that follow are not primes. Thereafter in the sequence, there seems to be no general rule, though that could not be determined until the development of modern computers. The amazing point is this: Fermat was wrong, but these numbers still turned out to have significant applications in physics.

Fermat’s greatest accomplishments in number theory are found in two theorems that still bear his name: Fermat’s theorem and his last theorem. The first can be stated simply: If n is any whole number and p is any prime, then n^p–n is divisible by p. Typically, he gave this without proof, and one was not presented until fifty years after his death. Yet the proof depends on only two facts: that a given whole number can be made only by multiplying primes, and that if a prime divides a product of two numbers then it divides at least one of them. Yet it also depends on the use of the principle of mathematical induction, which was first formulated during Fermat’s lifetime. That Fermat had formulated this principle independently is clear from this theorem and from his description of a method he called “infinite descent,” already suggested in the account given of his method of tangents.

Fermat’s so-called last theorem, which states that there are no solutions to equations of the type xⁿ + yⁿ = zⁿ when n is greater than 2, grew out of his fascination with the kind of equations called Diophantine—equations with two or more unknowns requiring whole number solutions. Fermat accomplished much with these equations. For example, he asserted that the equation y³ = x² + 2 has only one solution: y = 3, x = 5. As usual, he gave no proof; yet he must have had one, since one eventually emerged.

Significance

The significance of someone such as Fermat is difficult to state directly or simply, since he worked solely in the area of abstract mathematics and produced few tangible or readily measurable results. It is even more difficult with him than with other mathematicians because he refused publicity; at his death, few could have been aware that one of the world’s truly seminal minds was passing away. Furthermore, many of his discoveries were paralleled by other workers; it would be easy to dismiss him as interesting but not particularly significant, but such a judgment would not do him justice.

So many of his discoveries were pivotal, providing a necessary impetus to the opening of several new and rich fields of inquiry. While it is true that others paralleled some of his work, in most cases he provided the catalyst. In analytic geometry, for example, Descartes preceded him in print, but Fermat made the absolutely necessary transition to the third dimension; he also corrected Descartes’s formulation of the degrees of the equations. Without these contributions, Cartesian analysis could not have become a formidable instrument in the development of mechanics and the incipient engineering of the Industrial Revolution. Similarly, Newton and Leibniz could not have begun differential calculus without Fermat’s work on tangents and slopes. Finally, whole areas of analysis in physics remain indebted to Fermat.

Bibliography

Beiler, Albert H. Recreations in the Theory of Numbers: The Queen of Mathematics Entertains. 2d ed. New York: Dover, 1966. Beiler provides a solid introduction to the problems that Fermat attacked and his methods of solution. This work is less technical in approach than many books in number theory, but it requires some knowledge of advanced mathematics.

Bell, Eric T. Men of Mathematics. New York: Simon & Schuster, 1986. An excellent general introduction to the major figures in classical mathematics. Bell’s discussion of Fermat covers all major topics with style and wit.

Burton, David M. The History of Mathematics: An Introduction. Boston: Allyn & Bacon, 1984. A standard text, written for readers with some knowledge of advanced mathematics. Burton re-creates the process of problem solving, which is particularly good for understanding Fermat.

Kline, Morris. Mathematical Thought from Ancient to Modern Times. New York: Oxford University Press, 1972. An excellent general history of mathematics, with a clear, succinct account of Fermat’s contributions.

Krizek, Michal, Florian Luca, and Lawrence Somer. Seventeen Lectures on Fermat Numbers. New York: Springer, 2001. These lectures provide an overview of the properties of Fermat numbers and their various mathematical applications.

Mahoney, Michael Sean. The Mathematical Career of Pierre de Fermat (1601-1665). Princeton, N.J.: Princeton University Press, 1973. The quality of writing and clarity of exposition makes this book a good work for general readers as well as historians of mathematics or science. The mathematical explanations are fully detailed and not overly technical.

Ribenboim, Paulo. Fermat’s Last Theorem for Amateurs. New York: Springer, 1999. Intended for students, teachers, and amateur mathematicians, the book explains the proofs related to Fermat’s last theorem.

Simmons, George Finlay. Calculus Gems: Brief Lives and Memorable Mathematics. New York: McGraw-Hill, 1992. This collection of biographies includes a chapter about Fermat’s discovery of analytic geometry and his founding of modern numbers theory.

Singh, Simon. Fermat’s Enigma: The Epic Quest to Solve the World’s Greatest Mathematical Problem. New York: Walker, 1997. Recounts how scientists and mathematicians during a 350-year period attempted to solve Fermat’s last theorem, concluding in Andrew Wiles’s eventual solution.