by E. T. Bell
As for Descartes and Fermat, each of them, entirely independently of the other, invented analytic geometry. They corresponded on the subject but this does not affect the preceding assertion. The major part of Descartes’ effort went to miscellaneous scientific investigations, the elaboration of his philosophy, and his preposterous “vortex theory” of the solar system—for long a serious rival, even in England, to the beautifully simple, unmetaphysical Newtonian theory of universal gravitation. Fermat seems never to have been tempted, as both Descartes and Pascal were, by the insidious seductiveness of philosophizing about God, man, and the universe as a whole; so, after having disposed of his part in the calculus and analytic geometry, and having lived a serene life of hard work all the while to earn his living, he still was free to devote his remaining energy to his favorite amusement—pure mathematics, and to accomplish his greatest work, the foundation of the theory of numbers, on which his undisputed and undivided claim to immortality rests.
It will be seen presently that Fermat shared with Pascal the creation of the mathematical theory of probability. If all these first-rank achievements are not enough to put him at the head of his contemporaries in pure mathematics we may ask who did more? Fermat was a born originator. He was also, in the strictest sense of the word, so far as his science and mathematics were concerned, an amateur. Without doubt he is one of the foremost amateurs in the history of science, if not the very first.
Fermat’s life was quiet, laborious, and uneventful, but he got a tremendous lot out of it. The essential facts of his peaceful career are quickly told. The son of the leather-merchant Dominique Fermat, second consul of Beaumont, and Claire de Long, daughter of a family of parliamentary jurists, the mathematician Pierre Fermat was born at Beaumont-de-Lomagne, France, in August, 1601 (the exact date is unknown; the baptismal day was August 20th). His earliest education was received at home in his native town; his later studies, in preparation for the magistracy, were continued at Toulouse. As Fermat lived temperately and quietly all his life, avoiding profitless disputes, and as he lacked a doting sister like Pascal’s Gilberte to record his boyhood prodigies for posterity, singularly little appears to have survived of his career as a student. That it must have been brilliant will be evident from the achievements and accomplishments of his maturity; no man without a solid foundation of exact scholarship could have been the classicist and littérateur that Fermat became. His marvelous work in the theory of numbers and in mathematics generally cannot be traced to his schooling; for the fields in which he did his greatest work, not having been opened up while he was a student, could scarcely have been suggested by his studies.
The only events worth noting in his material career are his installation at Toulouse, at the age of thirty (May 14, 1631), as commissioner of requests; his marriage on June 1st of the same year to Louise de Long, his mother’s cousin, who presented him with three sons, one of whom, Clément-Samuel, became his father’s scientific executor, and two daughters, both of whom took the veil; his promotion in 1648 to a King’s councillorship in the local parliament of Toulouse, a position which he filled with dignity, integrity, and great ability for seventeen years—his entire working life of thirty four years was spent in the exacting service of the state; and finally, his death at Castres on January 12, 1665, in his sixty fifth year, two days after he had finished conducting a case in the town of his death. “Story?” he might have said; “Bless you, sir! I have none.” And yet this tranquilly living, honest, even-tempered, scrupulously just man has one of the finest stories in the history of mathematics.
His story is his work—his recreation, rather—done for the sheer love of it, and the best of it is so simple (to state, but not to carry through or imitate) that any schoolboy of normal intelligence can understand its nature and appreciate its beauty. The work of this prince of mathematical amateurs has had an irresistible appeal to amateurs of mathematics in all civilized countries during the past three centuries. This, the theory of numbers as it is called, is probably the one field of mathematics in which a talented amateur today may hope to turn up something of interest. We shall glance at his other contributions first after a passing mention of his “singular erudition” in what many call the humanities. His knowledge of the chief European languages and literatures of Continental Europe was wide and accurate, and Greek and Latin philology are indebted to him for several important corrections. In the composition of Latin, French, and Spanish verses, one of the gentlemanly accomplishments of his day, he showed great skill and a fine taste. We shall understand his even, scholarly life if we picture him as an affable man, not touchy or huffy under criticism (as Newton in his later years was), without pride, but having a certain vanity which Descartes, his opposite in all respects, characterized by saying, “Mr. de Fermat is a Gascon; I am not. “The allusion to the Gascons may possibly refer to an amiable sort of braggadocio which some French writers (for example Rostand in Cyrano de Bergerac, Act II, Scene VII) ascribe to their men of Gascony. There may be some of this in Fermat’s letters, but it is always rather naïve and inoffensive, and nothing to what he might have justly thought of his work even if his head had been as big as a balloon. And as for Descartes it must be remembered that he was not exactly an impartial judge. We shall note in a moment how his own soldierly obstinacy caused him to come off a bad second-best in his protracted row with the “Gascon” over the extremely important matter of tangents.
Considering the exacting nature of Fermat’s official duties and the large amount of first-rate mathematics he did, some have been puzzled as to how he found time for it all. A French critic suggests a probable solution: Fermat’s work as a King’s councillor was an aid rather than a detriment to his intellectual activities. Unlike other public servants—in the army for instance—parliamentary councillors were expected to hold themselves aloof from their fellow townsmen and to abstain from unnecessary social activities lest they be corrupted by bribery or otherwise in the discharge of their office. Thus Fermat found plenty of leisure.
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We now briefly state Fermat’s part in the evolution of the calculus. As was remarked in the chapter on Archimedes, a geometrical equivalent of the fundamental problem of the differential calculus is to draw the straight line tangent to a given, unlooped, continuous arc of a curve at any given point. A sufficiently close description of what “continuous” means here is “smooth, without breaks or sudden jumps”; to give an exact, mathematical definition would require pages of definitions and subtle distinctions which, it is safe to say, would have puzzled and astonished the inventors of the calculus, including Newton and Leibniz. And it is also a fair guess that if all these subtleties which modern students demand had presented themselves to the originators, the calculus would never have got itself invented.
The creators of the calculus, including Fermat, relied 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,” pictured the process of drawing a straight line tangent to the curve at any point P on the curve by taking another point Q, also on the curve, drawing the straight line PQ joining P and Q, and then, in imagination, letting the point Q slip along the arc of the curve from Q to P, till Q coincided with P, when the chord PQ, in the limiting position just described, became the tangent PP to the curve at the point P—the very thing they were looking for.
The next step was to translate all this into algebraical or analytical language. Knowing the coordinates x, y of the point P on the graph, and those, say x + a, y + b, of Q, before Q started to slip along to coincidence with P, they inspected the graph and saw that the slope of the chord PQ was equal to b/a—obviously a measure of the “steepness” of the chord with relation to the x-axis (the line along which x-distances are measured); this “steepness” is precisely what is meant by slope. From this it was evident that the required slope of the tangent at P (after Q had slipped into coincidence with P) would be the limiti
ng value of b/a as both b and a approached the value zero simultaneously; for x + a, y + b, the coordinates of Q, ultimately become x, y, the coordinates of P. This limiting value is the required slope. Having the slope and the point P they could now draw the tangent.
This is not exactly Fermat’s process for drawing tangents but his own process was, broadly, equivalent to what has been described.
Why should all this be worth the serious attention of any rational or practical man? It is a long story, only a hint of which need be given here; more will be said when we discuss Newton. One of the fundamental ideas in dynamics is that of the velocity (speed) of a moving particle. If we graph the number of units of distance passed over by the particle in a unit of time against the number of units of time, we get a line, straight or curved, which pictures at a glance the motion of the particle, and the steepness of this line at any given point of it will obviously give us the velocity of the particle at the instant corresponding to the point; the faster the particle is moving, the steeper the slope of the tangent line. This slope does in fact measure the velocity of the particle at any point of its path. The problem in motion, when translated into geometry, is exactly that of finding the slope of the tangent line at a given point of a curve. There are similar problems in connection with tangent planes to surfaces (which also have important interpretations in mechanics and mathematical physics), and all are attacked by the differential calculus—whose fundamental problem we have attempted to describe as it presented itself to Fermat and his successors.
Another use of this calculus can be indicated from what has already been said. Suppose some quantity y is a “function” of another, t, written y = f(t), which means that when any definite number, say 10, is substituted for t, so that we get f(10)—“function f of 10”—we can calculate, from the algebraical expression off, supposed given, the corresponding value of y, here y =f(l0). To be explicit, suppose f(t) is that particular “function” of t which is denoted in algebra by t2, or t × t. Then, when t = 10, we get y = f( 10), and hence here y = 102, = 100, for this value of t; when t = ½, y = ¼, and so on, for any value of t.
All this is familiar to anyone whose grammar-school education ended not more than thirty or forty years ago, but some may have forgotten what they did in arithmetic as children, just as others could not decline the Latin mensa to save their souls. But even the most forgetful will see that we could plot the graph of y = f(t) for any particular form of f (whenf(t) is t2 the graph is a parabola like an inverted arch). Imagine the graph drawn. If it has on it maxima (highest) or minima (lowest) points—points higher or lower than those in their immediate neighborhoods—we observe that the tangent at each of these maxima or minima is parallel to the ¿-axis. That is, the slope of the tangent at such an extremum (maximum or minimum) of the f(t) we are plotting is zero. Thus if we were seeking the extrema of a given function f(t) we should again have to solve our slope-problem for the particular curve y = f(t) and, having found the slope for the general point t, y, equate to zero the algebraical expression of this slope in order to find the values of t corresponding to the extrema. This is substantially what Fermat did in his method of maxima and minima invented in 1628-29, but not made semipublic till ten years later when Fermat sent an account of it through Mersenne to Descartes.
The scientific applications of this simple device—duly elaborated, of course, to take account of far more complicated problems than that just described—are numerous and far reaching. In mechanics, for instance, as Lagrange discovered, there is a certain “function” of the positions (coordinates) and velocities of the bodies concerned in a problem which, when made an extremum, furnishes us with the “equations of motion” of the system considered, and these in turn enable us to determine the motion—to describe it completely—at any given instant. In physics there are many similar functions, each of which sums up most of an extensive branch of mathematical physics in the simple requirement that the function in question must be an extremum;I Hilbert in 1916 found one for general relativity. So Fermat was not fooling away his time when he amused himself in the leisure left from a laborious legal job by attacking the problem of maxima and minima. He himself made one beautiful and astonishing application of his principles to optics. In passing it may be noted that this particular discovery has proved to be the germ of the newer quantum theory—in its mathematical aspect, that of “wave mechanics”—elaborated since 1926. Fermat discovered what is usually called “the principle of least time.” It would be more accurate to say “extreme” (least or greatest) instead of “least.”II
According to this principle, if a ray of light passes from a point A to another point B, being reflected and refracted (“refracted,” that is, bent, as in passing from air to water, or through a jelly of variable density) in any manner during the passage, the path which it must take can be calculated—all its twistings and turnings due to refraction, and all its dodgings back and forth due to reflections—from the single requirement that the time taken to pass from A to B shall be an extremum (but see the preceding footnote).
From this principle Fermat deduced the familiar laws of reflection and refraction: the angle of incidence (in reflection) is equal to 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 another.
The matter of analytic geometry has already been mentioned; Fermat was the first to apply it to space of three dimensions. Descartes contented himself with two dimensions. The extension, familiar to all students today, would not be self-evident to even a gifted man from Descartes’ developments. It may be said that there is usually greater difficulty in finding a significant extension of a particular kind of geometry from space of two dimensions to three than there is in passing from three to four or five . . . , or n. Fermat corrected Descartes in an essential point (that of the classification of curves by their degrees). It seems but natural that the somewhat touchy Descartes should have rowed with the imperturbable “Gascon” Fermat. The soldier was frequently irritable and acid in his controversy over Fermat’s method of tangents; the equable jurist was always unaflfectedly courteous. As usually happens the man who kept his temper got the better of the argument. But Fermat deserved to win, not because he was a more skilful debater, but because he was right.
In passing, we should suppose that Newton would have heard of Fermat’s use of the calculus and would have acknowledged the information. Until 1934 no evidence to this effect had been published, but in that year Professor L. T. More recorded in his biography of Newton a hitherto unnoticed letter in which Newton says explicitly that he got the hint of the method of the differential calculus from Fermat’s method of drawing tangents.
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We now turn to Fermat’s greatest work, that which is intelligible to all, mathematicians and amateurs alike. This is the so-called “theory of numbers,” or “the higher arithmetic,” or finally, to use the unpedantic name which was good enough for Gauss, arithmetic.
The Greeks separated the miscellany which we lump together under the name “arithmetic” in elementary textbooks into two distinct compartments, logistica, and arithmetica, the first of which concerned the practical applications of reckoning to trade and daily life in general, and the second, arithmetic in the sense of Fermat and Gauss, who sought to discover the properties of numbers as such.
Arithmetic in its ultimate and probably most difficult problems investigates the mutual relationships of those common whole numbers 1, 2, 3, 4, 5, . . . which we utter almost as soon as we learn to talk. In striving to elucidate these relationships, mathematicians have been driven to the invention of subtle and abstruse theories in algebra and analysis, whose forests of technicalities obscure the initial problems—those concerning 1, 2, 3, . . . but whose real justification will be the solution of those problems. In the meantime the by-products of these apparently useless investigations amply repay those who undertake them by suggesting numerous powerful
methods applicable to other fields of mathematics having direct contact with the physical universe. To give but one instance, the latest phase of algebra, that which is cultivated today by professional algebraists and which is throwing an entirely new light on the theory of algebraic equations, traces its origin directly to attempts to settle Fermat’s simple Last Theorem (which will be stated when the way has been prepared for it).
We begin with a famous statement Fermat made about prime numbers. A positive prime number, or briefly a prime, is any number greater than 1 which has as its divisors (without remainder) only 1 and the number itself; for example 2, 3, 5, 7, IS, 17 are primes, and so are 257, 65537. But 4294967297 is not a prime, because it has 641 as a divisor, nor is the number 18446744073709551617, because it is exactly divisible by 274177; both 641 and 274177 are primes. When we say in arithmetic that one number has as divisor another number, or is divisible by another, we mean exactly divisible, without remainder. Thus 14 is divisible by 7, 15 is not. The two large numbers were displayed above with malice aforethought for a reason that will be apparent in a moment. To recall another definition, the nth power of a given number, say N, is the result of multiplying together n N’s, and is written Nn; thus 52 = 5 × 5 = 25; 84 = 8 × 8 × 8 × 8 = 4096. For uniformity N itself may be written as Nl. Again, such a pagoda as 235 means that we are first to calculate 35 ( = 243), and then “raise” 2 to this power, 2243; the resulting number has seventy four digits.
The next point is of great importance in the life of Fermat, also in the history of mathematics. Consider the numbers 3, 5, 17, 257, 65537. They all belong to one “sequence” of a specific kind, because they are all generated (from 1 and 2) by the same simple process, which will be seen from