Men of Mathematics (31 page)

As Arago points out, one source of Euler's great and immediate success as a teacher through his writings was his total lack of false
pride. If certain works of comparatively low intrinsic merit were demanded to clarify earlier and more impressive works, Euler did not hesitate to write them. He had no fear of lowering his reputation.

Even on the creative side Euler combined instruction with discovery. His great treatises of
1748, 1755
and
1768-70
on the calculus
(Introductio in analysin infinitorum; Institutiones calculi differentialis; Institutiones calculi integralis)
instantly became classic and continued for three-quarters of a century to inspire young men who were to become great mathematicians. But it was in his work on the calculus of variations (
Methodus inveniendi lineas curvas maximi minimive proprietate gaudentes
, 1744) that Euler first revealed himself as a mathematician of the first rank. The importance of this subject has been noted in previous chapters.

Euler's great step forward when he made mechanics
analytical
has already been remarked; every student of rigid dynamics is familiar with Euler's analysis of rotations, to cite but one detail of this advance. Analytical mechanics is a branch of pure mathematics, so that Euler was not tempted here, as in some of his other flights toward the practical, to fly off on the first tangent he saw leading into the infinite blue of pure calculation. The severest criticism which Euler's contemporaries made of his work was his uncontrollable impulse to calculate merely for the sake of the beautiful analysis. He may occasionally have lacked a sufficient understanding of the physical situations he attempted to reduce to calculation without seeing what they were all about. Nevertheless, the fundamental equations of fluid motion, in use today in hydrodynamics, are Euler's. He could be practical enough when it was worth his trouble.

One peculiarity of Euler's analysis must be mentioned in passing, as it was largely responsible for one of the main currents of mathematics in the nineteenth century. This was his recognition that unless an infinite series is
convergent
it is unsafe to use. For example, by long division we find

the series continuing indefinitely. In this put
x = ½
. Then

The study of
convergence
(to be discussed in the chapter on Gauss) shows us how to avoid absurdities like this. (See also the chapter on
Cauchy.) The curious thing is that although Euler recognized the necessity for caution in dealing with
infinite
processes, he failed to observe it in much of his own work. His faith in analysis was so great that he would sometimes seek a preposterous “explanation” to make a patent absurdity respectable.

But when all this is said, we must add that few have equalled or approached Euler in the mass of sound and novel work of the first importance which he put out. Those who love arithmetic—not a very “important” subject, possibly—will vote Euler a palm in Diophantine analysis of the same size and freshness as those worn by Fermat and Diophantus himself. Euler was the first and possibly the greatest of the mathematical universalists.

Nor was he merely a narrow mathematician: in literature and all of the sciences, including the biologic, he was at least well read. But even while he was enjoying his
Aeneid
Euler could not help seeing a problem for his mathematical genius to attack. The line “The anchor drops, the rushing keel is stay'd” set him to working out the ship's motion under such circumstances. His omnivorous curiosity even swallowed astrology for a time, but he showed that he had not digested it by politely declining to cast the horoscope of Prince Ivan when ordered to do so in 1740, pointing out that horoscopes belonged in the province of the court astronomer. The poor astronomer had to do it.

One work of the Berlin period revealed Euler as a graceful (if somewhat too pious) writer, the celebrated
Letters to a German Princess,
composed to give lessons in mechanics, physical optics, astronomy, sound, etc., to Frederick's niece, the Princess of Anhalt-Dessau. The famous letters became immensely popular and circulated in book form in seven languages. Public interest in science is not the recent development we are sometimes inclined to imagine it is.

Euler remained virile and powerful of mind to the very second of his death, which occurred in his seventy seventh year, on September 18, 1783. After having amused himself one afternoon calculating the laws of ascent of balloons—on his slate, as usual—he dined with Lexell and his family. “Herschel's Planet” (Uranus) was a recent discovery; Euler outlined the calculation of its orbit. A little later he asked that his grandson be brought in. While playing with the child and drinking tea he suffered a stroke. The pipe dropped from his hand, and with the words “I die,” “Euler ceased to live and calculate.”
^I

I
. The quotation is from Condorcet's
Eloge.

I do not know.
—J. L. L
AGRANGE

“L
AGRANGE IS THE LOFTY PYRAMID
of the mathematical sciences.” This was Napoleon Bonaparte's considered estimate of the greatest and most modest mathematician of the eighteenth century, Joseph-Louis Lagrange (
1736-1813),
whom he had made a Senator, a Count of the Empire, and a Grand Officer of the Legion of Honor. The King of Sardinia and Frederick the Great had also honored Lagrange, but less lavishly than the imperial Napoleon.

Lagrange was of mixed French and Italian blood, the French predominating. His grandfather, a French cavalry captain, had entered the service of Charles Emmanuel II, King of Sardinia, and on settling at Turin had married into the illustrious Conti family. Lagrange's father, once Treasurer of War for Sardinia, married Marie-Thérèse Gros, the only daughter of a wealthy physician of Cambiano, by whom he had eleven children. Of this numerous brood only the youngest, Joseph-Louis, born on January
25, 1736,
survived beyond infancy. The father was rich, both in his own right and his wife's. But he was also an incorrigible speculator, and by the time his son was ready to inherit the family fortune there was nothing worth inheriting. In later life Lagrange looked back on this disaster as the luckiest thing that had ever happened to him: “If I had inherited a fortune I should probably not have cast my lot with mathematics.”

At school Lagrange's first interests were in the classics, and it was more or less of an accident that he developed a passion for mathematics. In line with his classical studies he early became acquainted with the geometrical works of Euclid and Archimedes. These do not seem to have impressed him greatly. Then an essay by Halley (Newton's friend) extolling the superiority of the calculus over the synthetic geometrical methods of the Greeks fell into young Lagrange's hands. He was captivated and converted. In an incredibly short time he had
mastered entirely by himself what in his day was modern analysis. At the age of sixteen (according to Delambre there may be a slight inaccuracy here) Lagrange became professor of mathematics at the Royal Artillery School in Turin. Then began one of the most brilliant careers in the history of mathematics.

From the first Lagrange was an analyst, never a geometer. In him we see the first conspicuous example of that specialization which was to become almost a necessity in mathematical research. Lagrange's analytical preferences came out strongly in his masterpiece, the
Mécanique analytique
(Analytical Mechanics), which he had projected as a boy of nineteen at Turin, but which was published in Paris only in
1788
when Lagrange was fifty two. “No diagrams will be found in this work,” he says in the preface. But with a half-humorous libation to the gods of geometry he remarks that the science of mechanics may be considered as the geometry of a space of four dimensions—three Cartesian coordinates with one time-coordinate sufficing to locate a moving particle in both space and time, a way of looking at mechanics that has become popular since
1915
when Einstein exploited it in his general relativity.

Lagrange's analytical attack on mechanics marks the first complete break with the Greek tradition. Newton, his contemporaries, and his immediate successors found diagrams helpful in their study of mechanical problems; Lagrange showed that greater flexibility and incomparably greater power are attained if general analytical methods are employed from the beginning.

At Turin the boyish professor lectured to students all older than himself. Presently he organized the more able into a research society from which the Turin Academy of Sciences developed. The first volume of the Academy's memoirs was published in
1759,
when Lagrange was twenty three. It is usually supposed that the modest and unobtrusive Lagrange was responsible for much of the fine mathematics in these early works published by others. One paper by Foncenex was so good that the King of Sardinia put the supposed author in charge of the Department of the Navy. Historians of mathematics have sometimes wondered why Foncenex never lived up to his first mathematical success.

Lagrange himself contributed a memoir on maxima and minima (the calculus of variations, described in Chapters
4, 8)
in which he promises to treat the subject in a work from which he will deduce
the whole of mechanics, of both solids and fluids. Thus at twenty three—actually earlier—Lagrange had imagined his masterpiece, the
Mécanique analytique,
which does for general mechanics what Newton's law of universal gravitation did for celestial mechanics. Writing ten years later to the French mathematician D'Alembert (1717–1783), Lagrange says he regards his early work, the calculus of variations, thought out when he was nineteen, as his masterpiece. It was by means of this calculus that Lagrange unified mechanics and, as Hamilton said, made of it “a kind of scientific poem.”

When once understood the Lagrangian method is almost a platitude. As some have remarked the Lagrangian equations dominating mechanics are the finest example in all science of the art of getting something out of nothing. But if we reflect a moment we see that any scientific principle which is general to the extent of uniting a whole vast universe of phenomena
must
be simple: only a principle of the utmost simplicity can dominate a multitude of diverse problems which on even a close inspection appear to be individual and distinct.

In the same volume of Turin memoirs Lagrange took another long step forward: he applied the differential calculus to the theory of probability. As if this were not enough for the young giant of twenty three he advanced beyond Newton with a radical departure in the mathematical theory of sound, bringing that theory under the sway of the mechanics of systems of elastic particles (rather than of the mechanics of fluids), by considering the behavior of all the air particles in one straight line under the action of a shock transmitted along the line from particle to particle. In the same general direction he also settled a vexed controversy that had been going on for years between the leading mathematicians over the correct mathematical formulation of the problem of a vibrating string—a problem of fundamental importance in the whole theory of vibrations. At twenty three Lagrange was acknowledged the equal of the greatest mathematicians of the age—Euler and the Bernoullis.

Euler was always generously appreciative of the work of others. His treatment of his young rival Lagrange is one of the finest pieces of unselfishness in the history of science. When as a boy of nineteen Lagrange sent Euler some of his work the famous mathematician at once recognized its merit and encouraged the brilliant young beginner to continue. When four years later Lagrange communicated to Euler the true method for attacking the isoperimetrical problems (the calculus
of variations, described in connection with the Bernoullis), which had baffled Euler with his semi-geometrical methods for many years, Euler wrote to the young man saying that the new method had enabled him to overcome his difficulties. And instead of rushing into print with the long-sought solution, Euler held it back till Lagrange could publish his first, “so as not to deprive you of any part of the glory which is your due.”