Men of Mathematics

by E. T. Bell

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  After the death of Frederick the Great (August 17, 1786) resentment against non-Prussians and indifference to science made Berlin an uncomfortable spot for Lagrange and his foreign associates in the Academy, and he sought his release. This was granted on condition that he continue to send memoirs to the proceedings of the Academy for a period of years, to which Lagrange agreed. He gladly accepted the invitation of Louis XVI to continue his mathematical work in Paris as a member of the French Academy. On his arrival in Paris in 1787 he was received with the greatest respect by the royal family and the Academy. Comfortable quarters were assigned him in the Louvre, where he lived till the Revolution, and he became a special favorite of Marie Antoinette—then less than six years from the guillotine. Marie was about nineteen years younger than Lagrange, but she seemed to understand him and did what she could to lighten his overwhelming depression.

  At the age of fifty one Lagrange felt that he was through. It was a clear case of nervous exhaustion from long-continued and excessive overwork. The Parisians found him gentle and agreeable in conversation, but he never took the lead. He spoke but little and appeared distrait and profoundly melancholy. At Lavoisier’s gatherings of scientific men Lagrange would stand staring absently out of a window, his back to the guests who had come to do him honor, a picture of sad indifference. He said himself that his enthusiasm was extinct and that he had lost the taste for mathematics. If he were told that some mathematician was engaged on an important research he would say “So much the better; I began it; I shall not have to finish it.” The Mécanique analytique lay unopened on his desk for two years.

  Sick of everything smelling of mathematics Lagrange now turned to what he considered his real interests—as Newton had done after the Principia: metaphysics, the evolution of human thought, the history of religions, the general theory of languages, medicine, and botany. In this strange miscellany he surprised his friends with his extensive knowledge and the penetrating quality of his mind on matters alien to mathematics. Chemistry at the time was fast becoming a science—in distinction to the alchemy which preceded it, largely through the efforts of Lagrange’s close friend Lavoisier (1743-1794). In a sense which any student of elementary chemistry will appreciate Lagrange declared that Lavoisier had made chemistry “as easy as algebra.”

  As for mathematics, Lagrange considered that it was finished or at least passing into a period of decadence. Chemistry, physics, and science generally he foresaw as the future fields of greatest interest to first-class minds, and he even predicted that the chairs of mathematics in academies and universities would presently sink to the undistinguished level of those for Arabic. In a sense he was right. Had not Gauss, Abel, Galois, Cauchy, and others injected new ideas into mathematics the surge of the Newtonian impulse would have spent itself by 1850. Happily Lagrange lived long enough to see Gauss well started on his great career and to realize that his own forebodings had been unfounded. We may smile at Lagrange’s pessimism today, thinking of the era before 1800 at its brightest as only the dawn of the modern mathematics in the first hour of whose morning we now stand, wondering what the noon will be like—if there is to be any; and we may learn from his example to avoid prophecy.

  The Revolution broke Lagrange’s apathy and galvanized him once more into a living interest in mathematics. As a convenient point of reference we may remember July 14, 1789, the day on which the Bastille fell.

  When the French aristocrats and men of science at last realized what they were in for, they urged Lagrange to return to Berlin where a welcome awaited him. No objection would have been raised to his departure. But he refused to leave Paris, saying he would prefer to stay and see the “experiment” through. Neither he nor his friends foresaw the Terror, and when it came Lagrange bitterly regretted having stayed until it was too late to escape. He had no fear for his own life. In the first place as a half-foreigner he was reasonably safe, and in the second he did not greatly value his life. But the revolting cruelties sickened him and all but destroyed what little faith he had left in human nature and common sense. “Tu l‘as voulu” (“You wished it,” or “You would do it”), he would keep reminding himself as one atrocity after another shocked him into a realization of his error in staying to witness the inevitable horrors of a revolution.

  The grandiose schemes of the revolutionists for the regeneration of mankind and the reform of human nature left him cold. When Lavoisier went to the guillotine—as he no doubt would have deserved had it been merely a question of social justice—Lagrange expressed his indignation at the stupidity of the execution: “It took them only a moment to cause this head to fall, and a hundred years perhaps will not suffice to produce its like.” But the outraged and oppressed citizens had assured the tax-farmer Lavoisier that “the people have no need of science” when the great chemist’s contributions to science were urged as a common-sense reason that his head be left on his shoulders. They may have been right. Without the science of chemistry soap is impossible.

  Although practically the whole of Lagrange’s working life had been spent under the patronage of royalty his sympathies were not with the royalists. Nor were they with the revolutionists. He stood squarely and unequivocally on the middle ground of civilization which both sides had ruthlessly invaded. He could sympathize with the people who had been outraged beyond human endurance and wish them success in their struggle to gain decent living conditions. But his mind was too realistic to be impressed by any of the chimerical schemes put forth by the leaders of the people for the amelioration of human misery, and he refused to believe that the fabrication of such schemes was indubitable evidence of the greatness of the human mind as claimed by the enthusiastic guillotineers. “If you wish to see the human mind truly great,” he said, “enter Newton’s study when he is decomposing white light or unveiling the system of the world.”

  They treated him with remarkable tolerance. A special decree granted him his “pension,” and when the inflation by paper money reduced the pension to nothing, they appointed him on the committee of inventions to eke out his pay, and again on the committee for the mint. When the École Normale was established in 1795 (for an ephemeral first existence), Lagrange was appointed professor of mathematics. When the Normale closed and the great École Polytechnique was founded in 1797, Lagrange mapped out the course in mathematics and was the first professor. He had never taught before he was called upon to lecture to ill-prepared students. Adapting himself to his raw material, Lagrange led his pupils through arithmetic and algebra to analysis, seeming more like one of his pupils than their teacher. The greatest mathematician of the age became a great teacher of mathematics—preparing Napoleon’s fierce young brood of military engineers for their part in the conquest of Europe. The sacred superstition that a man who knows anything is incapable of teaching was shattered. Advancing far beyond the elements Lagrange developed new mathematics before his pupils’ eyes and presently they were taking part in the development themselves.

  Two works thus developed were to exercise a great influence on the analysis of the first three decades of the nineteenth century. Lagrange’s pupils found difficulty with the concepts of the infinitely small and the infinitely great permeating the traditional form of the calculus. To remove these difficulties Lagrange undertook the development of the calculus without the use of Leibniz’ “infinitesimals” and without Newton’s peculiar conception of a limit. His own theory was published in two works, the Theory of Analytic Functions (1797), and the Lessons on the Calculus of Functions (1801). The importance of these works is not in their mathematics but in the impulse they gave Cauchy and others to construct a satisfactory calculus. Lagrange failed completely. But in saying this we must remember that even in our own day the difficulties with which Lagrange grappled unsuccessfully have not been completely overcome. His was a notable attempt and, for its epoch, satisfactory. If our own lasts as long as his did we shall have done well enough.

  Lagrange’s most important work durin
g the period of the Revolution was his leading part in perfecting the metric system of weights and measures. It was due to Lagrange’s irony and common sense that 12 was not chosen as a base instead of 10. The “advantages” of 12 are obvious and continue to the present day to be set forth in impressive treatises by earnest propagandists who escape the circle-squaring fraternity only by a hairsbreadth. A base of 12 superimposed on the 10 of our number-system would be a hexagonal peg in a pentagonal hole. To bring home the absurdity of 12 even to the cranks, Lagrange proposed 11 as better yet—any prime number would have the advantage of giving all fractions in the system the same denominator. The disadvantages are numerous and obvious enough to anyone who understands short division. The committee saw the point and stuck to 10.

  Laplace and Lavoisier were members of the committee as first constituted, but after three months they were “purged” out of their seats with some others. Lagrange remained as president. “I do not know why they kept me,” he remarked, modestly unaware that his gift for silence had saved not only his seat but his head.

  In spite of all his interesting work Lagrange was still lonely and inclined to despondency. He was rescued from this twilight between life and death at the age of fifty six by a young girl nearly forty years his junior, the daughter of his friend the astronomer Lemonnier. She was touched by Lagrange’s unhappiness and insisted on marrying him. Lagrange gave in, and contrary to all the laws of whatever it may be that governs the way of a man with a maid, the marriage turned out ideal. The young wife proved not only devoted but competent. She made it her life to draw her husband out and reawaken his desire to live. For his part Lagrange gladly made many concessions and accompanied his wife to balls where he would never have thought of going alone. Before long he could not bear to have her out of his sight for long, and during her brief absences—shopping—he was miserable.

  Even in his new happiness Lagrange retained his curiously detached attitude to life and his perfect honesty about his own wishes. “I had no children by my first marriage,” he said; “I don’t know whether I shall have any by my second. I scarcely desire any.” Of all his successes the one he prized most highly, he said simply and sincerely, was having found so tender and devoted a companion as his young wife.

  Honors were showered on him by the French. The man who had been a favorite of Marie Antoinette now became an idol of the people who had put her to death. In 1796 when France annexed Piedmont, Talleyrand was ordered to wait in state on Lagrange’s father, still living in Turin, to tell him that “Your son, whom Piedmont is proud to have produced and France to possess, has done honor to all mankind by his genius.” When Napoleon turned to civil affairs between his campaigns he often talked with Lagrange on philosophical questions and the function of mathematics in a modern state, and conceived the highest respect for the gently-spoken man who always thought before he spoke and who was never dogmatic.

  Beneath his calm reserve Lagrange concealed an ironic wit which flashed out unexpectedly on occasion. Sometimes it was so subtle that coarser men—Laplace, for one—missed the point when it was directed at themselves. Once in defense of experiment and observation against mere woolgathering and vague theorizing Lagrange remarked “These astronomers are queer; they won’t believe in a theory unless it agrees with their observations.” Noticing his rapt forgetfulness at a musicale, someone asked him why he liked music. “I like it because it isolates me,” he replied. “I hear the first three measures; at the fourth I distinguish nothing; I give myself up to my thoughts; nothing interrupts me; and it is thus that I have solved more than one difficult problem.” Even his sincere reverence for Newton has a faint flavor of the same gentle irony. “Newton,” he declared, “was assuredly the man of genius par excellence, but we must agree that he was also the luckiest: one finds only once the system of the world to be established.” And again: “How lucky Newton was that in his time the system of the world still remained to be discovered!”

  Lagrange’s last scientific effort was the revision and extension of the Mécanique analytique for a second edition. All his old power returned to him although he was past seventy. Resuming his former habits he worked incessantly, only to discover that his body would no longer obey his mind. Presently he began to have fainting spells, especially on getting out of bed in the morning. One day his wife found him unconscious on the floor, his head badly cut by a fall against the edge of a table. Thereafter he moderated his pace but kept on working. His illness, which he knew to be grave, did not disturb his serenity; all his life Lagrange lived as a philosopher would like to live, indifferent to his fate.

  Two days before Lagrange died Monge and other friends called, knowing that he was dying and that he wished to tell them something of his life. They found him temporarily better, except for lapses of memory which obliterated what he had wished to tell them.

  “I was very ill yesterday, my friends,” he said. “I felt I was going to die; my body grew weaker little by little; my intellectual and physical faculties were extinguished insensibly; I observed the well-graduated progression of the diminution of my strength, and I came to the end without sorrow, without regrets, and by a very gentle decline. Oh, death is not to be dreaded, and when it comes without pain, it is a last function which is not unpleasant.”

  He believed that the seat of life is in all the organs, in the whole of the bodily machine, which, in his case, weakened equally in all its parts.

  “In a few moments there will be no more functions anywhere, death will be everywhere; death is only the absolute repose of the body.

  “I wish to die; yes, I wish to die, and I find a pleasure in it. But my wife did not wish it. In these moments I should have preferred a wife less good, less eager to revive my strength, who would have let me end gently. I have had my career; I have gained some celebrity in Mathematics. I never hated anyone, I have done nothing bad, and it would be well to end; but my wife did not wish it.”

  He soon had his wish. A fainting spell from which he never awoke came on shortly after his friends had left. He died early on the morning of April 10, 1813, in his seventy sixth year.

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  I. A ridiculous “proof” by a Spanish gentleman is funny enough to be quoted. The customary abbreviation for 1 × 2 × . . . × n is n! Now p −1 + 1 = p which is divisible by p. Put exclamation points throughout: (p −1)! + 1! = p!. The right side is again divisible by p; hence (p −1)! + 1 is divisible by p. Unfortunately this works equally well if p is not prime.

  CHAPTER ELEVEN

  From Peasant to Snob

  LAPLACE

  All the effects of nature are only the mathematical consequences of a small number of immutable laws.—P. S. LAPLACE

  THE MARQUIS PIERRE-SIMON DE LAPLACE (1749-1827) was not born a peasant nor did he die a snob. Yet to within small quantities of the second order his illustrious career is comprised within the limits indicated, and it is from this approximate point of view that he is of greatest interest as a specimen of humanity.

  As a mathematical astronomer Laplace has justly been called the Newton of France; as a mathematician he may be regarded as the founder of the modern phase of the theory of probability. On the human side he is perhaps the most conspicuous refutation of the pedagogical superstition that noble pursuits necessarily ennoble a man’s character. Yet in spite of all his amusing foibles—his greed for titles, his political suppleness, and his desire to shine in the constantly changing spotlight of public esteem—Laplace had elements of true greatness in his character. We may not believe all that he said about his unselfish devotion to truth for truth’s sake, and we may smile at the care with which he rehearsed his sententious last words—“What we know is not much; what we do not know is immense”—in an endeavor to telescope Newton’s boy playing on the seashore into a neat epigram, but we cannot deny that Laplace in his generosity to unknown beginners was anything but a shifty and ungrateful politician. To give one young man a helping hand up Laplace once cheated himself.

  V
ery little is known of Laplace’s early years. His parents were peasants living in Beaumont-en-Auge, Department of Calvados, France, where Pierre-Simon was born on March 23, 1749. The obscurity surrounding Laplace’s childhood and youth is due to his own snobbishness: he was thoroughly ashamed of his humble parents and did everything in his power to conceal his peasant origin.

  Laplace got his chance through the friendly interest of wealthy neighbors on the occasion, presumably, of his having shown remarkable talent in the village school. It is said that his first success was in theological disputations. If this is true it is an interesting prelude to the somewhat aggressive atheism of his maturity. He took to mathematics early. There was a military academy at Beaumont, which Laplace attended as an externe, and in which he is said to have taught mathematics for a time. One dubious legend states that the young man’s prodigious memory attracted more attention than his mathematical ability and was responsible for the cordial recommendations from influential people which he carried with him to Paris when, at the age of eighteen he wiped the mud of Beaumont off his boots forever and set out to seek his fortune. His own estimate of his powers was high, but not too high. With justified self-confidence young Laplace invaded Paris to conquer the mathematical world.

  Arriving in Paris, Laplace called on D’Alembert and sent in his recommendations. He was not received. D’Alembert was not interested in young men who came recommended only by prominent people. With remarkable insight for so young a man Laplace sensed what the trouble was. He returned to his lodgings and wrote D’Alembert a wonderful letter on the general principles of mechanics. This did the trick. In his reply inviting Laplace to call, D’Alembert wrote: “Sir, you see that I paid little enough attention to your recommendations; you don’t need any. You have introduced yourself better. That is enough for me; my support is your due.” A few days later, thanks to D’Alembert, Laplace was appointed professor of mathematics at the Military School of Paris.