Men of Mathematics (12 page)

For straight lines and circles this may not seem very exciting; we
knew how to do it all before in another, a Greek, way. Now comes the real power of the method.
We start with equations of any desired or suggested degree of complexity and interpret their algebraic and analytic properties geometrically.
Thus we have not only dropped geometry as our pilot; we have tied a sackful of bricks to his neck before pitching him overboard.
Henceforth algebra and analysis are to be our pilots to the unchartered seas of “space” and its “geometry.”
All that we have done can be extended, at one stride, to space of any number of dimensions; for the plane we need
two
coordinates, for ordinary “solid” space
three,
for the geometry of mechanics and relativity,
four
coordinates, and finally, for “space” as mathematicians like it, either
n
coordinates, or as many coordinates as there are of
all
the numbers 1, 2, 3, . . ., or as many as there are of
all
the points on a line. This is beating Achilles and the tortoise in their own race.

Descartes did not revise geometry; he created it.

It seems fitting that an eminent living mathematical fellow-countryman of Descartes should have the last word, so we shall quote Jacques Hadamard. He remarks first that the mere invention of coordinates was not Descartes' greatest merit, because that had already been done “by the ancients”—a statement which is exact only if we read the unexpressed intention into the unaccomplished deed. Hell is paved with the half-baked ideas of “the ancients” which they could never quite cook through with their own steam.

“It is quite another thing to recognize [as in the use of coordinates] a general method and to follow to the end the idea which it represents. It is exactly this merit, whose importance every real mathematician knows, that was preëminently Descartes' in geometry; it was thus that he was led to what . . . is his truly great discovery in the matter; namely, the application of the method of coordinates not only to translate into equations curves already defined geometrically, but, looking at the question from an exactly opposite point of view, to the
a priori
definition of more and more complicated curves and, hence, more and more general . . . .

“Directly, with Descartes himself, later, indirectly, in the return which the following century made in the opposite direction, it is the entire conception of the object of mathematical science that was revolutionized. Descartes indeed understood thoroughly the significance of what he had done, and he was right when he boasted that he had so far surpassed all geometry before him as Cicero's rhetoric surpasses the ABC.”

I
. Daughter of Frederick, Elector Palatine of the Rhine, and King of Bohemia, and a granddaughter of James I of England.

I have found a very great number of exceedingly beautiful theorems.

—P. F
ERMAT

N
OT ALL OF OUR DUCKS
can be swans; so after having exhibited Descartes as one of the leading mathematicians of all time, we shall have to justify the assertion, frequently made and seldom contradicted, that the greatest mathematician of the seventeenth century was Descartes' contemporary Fermat (1601?–1665). This of course leaves Newton (1642–1727) out of consideration. But it can be argued that Fermat was
at least
Newton's equal
as a pure mathematician,
and anyhow nearly a third of Newton's life fell into the eighteenth century, whereas the whole of Fermat's was lived out in the seventeenth.

Newton appears to have regarded his mathematics principally as an instrument for scientific exploration and put his main effort on the latter. Fermat on the other hand was more strongly attracted to pure mathematics although he also did notable work in the applications of mathematics to science, particularly optics.

Mathematics had just entered its modern phase with Descartes' publication of analytic geometry in 1637, and was still for many years to be of such modest extent that a gifted man could reasonably hope to do good work in both the pure and applied divisions.

As a pure mathematician Newton reached his climax in the invention of the calculus, an invention also made independently by Leibniz. More will be said on this later; for the present it may be remarked that Fermat conceived and applied the leading idea of the differential calculus thirteen years before Newton was born and seventeen before Leibniz was born, although he did not, like Leibniz, reduce his method to a set of rules of thumb that even a dolt can apply to easy problems.

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.

*  *  *

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.