by E. T. Bell
Five years after Euler’s return to St. Petersburg another disaster overtook him. In the great fire of 1771 his house and all its furnishings were destroyed, and it was only by the heroism of his Swiss servant (Peter Grimm, or Grimmon) that Euler escaped with his life. At the risk of his own life Grimm carried his blind and ailing master through the flames to safety. The library was burned, but thanks to the energy of Count Orloff all of Euler’s manuscripts were saved. The Empress Catherine promptly made good all the loss and soon Euler was back at work again.
In 1776 (when he was sixty nine) Euler suffered a greater loss in the death of his wife. The following year he married again. The second wife, Salome Abigail Gsell, was a half-sister of the first. His greatest tragedy was the failure (through surgical carelessness, possibly) of an operation to restore the sight of his left eye—the only one for which there was any hope. The operation was “successful” and Euler’s joy passed all bounds. But presently infection set in, and after prolonged suffering which he described as hideous, he lapsed back into darkness.
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In looking back over Euler’s enormous output we may be inclined at the first glance to believe that any gifted man could have done a large part of it almost as easily as Euler. But an inspection of mathematics as it exists today soon disabuses us. For the present state of mathematics with its jungles of theories is relatively no more complicated, when we consider the power of the methods now at our disposal, than what Euler faced. Mathematics is ripe for a second Euler. In his day he systematized and unified vast tracts cluttered with partial results and isolated theorems, clearing the ground and binding up the valuable things by the easy power of his analytical machinery. Even today much of what is learned in a college course in mathematics is practically as Euler left it—the discussion of conic sections and quadrics in three-space from the unified point of view provided by the general equation of the second degree, for example, is Euler’s. Again, the subject of annuities and all that grows out of it (insurance, old-age pensions, and so on) were put into the shape now familiar to students of the “mathematical theory of investment” by Euler.
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
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I. The quotation is from Condorcet’s Eloge.
CHAPTER TEN
A Lofty Pyramid
LAGRANGE
I do not know.—J. L. LAGRANGE
“LAGRANGE 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 Lag
range 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.”
Private letters, however flattering, could not have helped Lagrange. Realizing this, Euler went out of his way when he published his work (after Lagrange’s) to say how he had been held up by difficulties which, till Lagrange showed the way over them, were insuperable. Finally, to clinch the matter, Euler got Lagrange elected as a foreign member of the Berlin Academy (October 2, 1759) at the unusually early age of twenty three. This official recognition abroad was a great help to Lagrange at home. Euler and D’Alembert schemed to get Lagrange to Berlin. Partly for personal reasons they were eager to see their brilliant young friend installed as court mathematician at Berlin. After lengthy negotiations they succeeded, and the great Frederick, slightly outwitted in the whole transaction, was childishly (but justifiably) delighted.
Something must be said in passing about D’Alembert, Lagrange’s devoted friend and generous admirer, if only for the grateful contrast one aspect of his character offers to that of the snobbish Laplace, whom we shall meet later.
Jean le Rond d’Alembert took his name from the little chapel of St. Jean le Rond hard by Notre-Dame in Paris. An illegitimate son of the Chevalier Destouches, D’Alembert had been abandoned by his mother on the steps of St. Jean le Rond. The parish authorities turned the foundling over to the wife of a poor glazier, who reared the child as if he were her own. The Chevalier was forced by law to pay for his bastard’s education. D’Alembert’s real mother knew where he was, and when the boy early gave signs of genius, sent for him, hoping to win him over.
“You are only my stepmother,” the boy told her (a good pun in English, but not in French); “the glazier’s wife is my true mother.” And with that he abandoned his own flesh and blood as she had abandoned hers.
When he became famous and a great figure in French science D’Alembert repaid the glazier and his wife by seeing that they did not fall into want (they preferred to keep on living in their humble quarters), and he was always proud to claim them as his parents. Although we shall not have space to consider him apart from Lagrange, it must be mentioned that D’Alembert was the first to give a complete solution of the outstanding problem of the precession of the equinoxes. His most important purely mathematical work was in partial differential equations, particularly in connection with vibrating strings.
D’Alembert encouraged his modest y
oung correspondent to attack difficult and important problems. He also took it upon himself to make Lagrange take reasonable care of his health—his own was not good. Lagrange had in fact seriously impaired his digestion by quite unreasonable application between the ages of sixteen and twenty six, and all his life thereafter he was forced to discipline himself severely, especially in the matter of overwork. In one of his letters D’Alembert lectures the young man for indulging in tea and coffee to keep awake; in another he lugubriously calls Lagrange’s attention to a recent medical book on the diseases of scholars. To all of which Lagrange blithely replies that he is feeling fine and working like mad. But in the end he paid his tax.
In one respect Lagrange’s career is a curious parallel to Newton’s. By middle age prolonged concentration on problems of the first magnitude had dulled Lagrange’s enthusiasm, and although his mind remained as powerful as ever, he came to regard mathematics with indifference. When only forty five he wrote to D’Alembert, “I begin to feel the pull of my inertia increasing little by little, and I cannot say that I shall still be doing mathematics ten years from now. It also seems to me that the mine is already too deep, and that unless new veins are discovered it will have to be abandoned.”