Men of Mathematics (32 page)

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 young 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.”

When he wrote this Lagrange was ill and melancholic. Nevertheless it expressed the truth so far as he was concerned. D'Alembert's last letter (September, 1783), written a month before his death, reverses his early advice and counsels work as the only remedy for Lagrange's psychic ills: “In God's name do not renounce work, for you the strongest of all distractions. Goodbye, perhaps for the last time. Keep some memory of the man who of all in the world cherishes and honors you the most.”

Happily for mathematics Lagrange's blackest depression, with its inescapable corollary that no human knowledge is worth striving for,
was twenty glorious years in the future when D'Alembert and Euler were scheming to get Lagrange to Berlin. Among the great problems Lagrange attacked and solved before going to Berlin was that of the libration of the Moon. Why does the Moon always present the same face to the Earth—within certain slight irregularities that can be accounted for? It was required to deduce this fact from the Newtonian law of gravitation. The problem is an instance of the famous “Problem of Three Bodies”—here the Earth, Sun, and Moon—mutually attracting one another according to the law of the inverse square of the distances between their centers of gravity. (More will be said on this problem when we come to Poincaré.)

For his solution of the problem of libration Lagrange was awarded the Grand Prize of the French Academy of Sciences in 1764—he was then only twenty eight.

Encouraged by this brilliant success the Academy proposed a yet more difficult problem, for which Lagrange again won the prize in
1766.
In Lagrange's day only four satellites of Jupiter had been discovered. Jupiter's system (himself, the Sun, and his satellites) thus made a six-body problem. A
complete
mathematical solution is beyond our powers even today (1936) in a shape adapted to practical computation. But by using methods of approximation Lagrange made a notable advance in explaining the observed inequalities.

Such applications of the Newtonian theory were one of Lagrange's major interests all his active life. In 1772 he again captured the Paris prize for his memoir on the three-body problem, and in 1774 and
1778
he had similar successes with the motion of the Moon and cometary perturbations.

The earlier of these spectacular successes induced the King of Sardinia to pay Lagrange's expenses for a trip to Paris and London in 1766. Lagrange was then thirty. It had been planned that he was to accompany Caraccioli, the Sardinian minister to England, but on reaching Paris Lagrange fell dangerously ill—the result of an over-generous banquet of rich Italian dishes in his honor—and he was forced to remain in Paris. While there he met all the leading intellectuals, including the Abbé Marie, who was later to prove an invaluable friend. The banquet cured Lagrange of his desire to live in Paris and he eagerly returned to Turin as soon as he was able to travel.

At last, on November 6, 1766, Lagrange was welcomed, at the
age of thirty, to Berlin by Frederick, “the greatest King in Europe,” as he modestly styled himself, who would be honored to have at his court “the greatest mathematician.” The last, at least, was true. Lagrange became director of the physico-mathematical division of the Berlin Academy, and for twenty years crowded the transactions of the Academy with one great memoir after another. He was not required to lecture.

At first the young director found himself in a somewhat delicate position. Naturally enough the Germans rather resented foreigners being brought in over their heads and were inclined to treat Frederick's importations with a little less than cool civility. In fact they were frequently quite insulting. But in addition to being a mathematician of the first rank Lagrange was a considerate, gentle soul with the rare gift of knowing when to keep his mouth shut. In letters to trusted friends he could be outspoken enough, even about the Jesuits, whom he and D'Alembert seem to have disliked, and in his official reports to academies on the scientific work of others he could be quite blunt. But in his social contacts he minded his own business and avoided giving even justifiable offense. Until his colleagues got used to his presence he kept out of their way.

Lagrange's constitutional dislike of all disputes stood him in good stead at Berlin. Euler had blundered from one religious or philosophical controversy to another; Lagrange, if cornered and pressed, would always preface his replies with his sincere formula “I do not know.” Yet when his own convictions were attacked he knew how to put up a spirited, reasoned defense.

On the whole Lagrange was inclined to sympathize with Frederick who had sometimes been irritated by Euler's tilting at philosophical problems about which he knew nothing. “Our friend Euler,” he wrote to D'Alembert, “is a great mathematician, but a bad enough philosopher.” And on another occasion, referring to Euler's effusion of pious moralizing in the celebrated
Letters to a German Princess,
he dubs the classic “Euler's commentary on the Apocalypse”—incidentally a backhand allusion to the indiscretion which Newton permitted himself when he had lost his taste for natural philosophy. “It is incredible,” Lagrange said of Euler, “that he could have been so flat and childish in metaphysics.” And for himself, “I have a great aversion to disputes.” When he did philosophize in his letters it was with an unexpected touch of cynicism which is wholly absent from
the works he published, as when he remarks, “I have always observed that the pretensions of all people are in exact inverse ratio to their merits; this is one of the axioms of morals.” In religious matters Lagrange was, if anything at all, agnostic.

Frederick was delighted with his prize and spent many friendly hours with Lagrange, expounding the advantages of a regular life. The contrast Lagrange offered to Euler was particularly pleasing to Frederick. The King had been irritated by Euler's too obvious piety and lack of courtly sophistication. He had even gone so far as to call poor Euler a “lumbering cyclops of a mathematician,” because Euler at the time was blind in only one of his eyes. To D'Alembert the grateful Frederick overflowed in both prose and verse. “To your trouble and to your recommendation,” he wrote, “I owe the replacement in my Academy of a mathematician blind in one eye by a mathematician with two eyes, which will be especially pleasing to the anatomical section.” In spite of sallies like this Frederick was not a bad sort.

*  *  *

Shortly after settling in Berlin Lagrange sent to Turin for one of his young lady relatives and married her. There are two accounts of how this happened. One says that Lagrange had lived in the same house with the girl and her parents and had taken an interest in her shopping. Having an economical streak in his cautious nature, Lagrange was scandalized by what he considered the girl's extravagance and bought her ribbons himself. From there on he was dragooned into marrying her.

The other version can be inferred from one of Lagrange's letters—certainly the strangest confession of indifference ever penned by a supposedly doting young husband. D'Alembert had joked his friend: “I understand that you have taken what we philosophers call the fatal plunge. . . . A great mathematician should know above all things how to calculate his happiness. I do not doubt then that after having performed this calculation you found the solution in marriage.”

Lagrange either took this in deadly earnest or set out to beat D'Alembert at his own game—and succeeded. D'Alembert had expressed surprise that Lagrange had not mentioned his marriage in his letters.

“I don't know whether I calculated ill or well,” Lagrange replied, “or rather, I don't believe I calculated at all; for I might have done
as Leibniz did, who, compelled to reflect, could never make up his mind. I confess to you that I never had a taste for marriage, . . . but circumstances decided me to engage one of my young kinswomen to take care of me and all my affairs. If I neglected to inform you it was because the whole thing seemed to me so inconsequential in itself that it was not worth the trouble of informing you of it.”

The marriage was turning out happily for both when the wife declined in a lingering illness. Lagrange gave up his sleep to nurse her himself and was heartbroken when she died.

He consoled himself in his work. “My occupations are reduced to cultivating mathematics, tranquilly and in silence.” He then tells D'Alembert the secret of the perfection of all his work which has been the despair of his hastier successors. “As I am not pressed and work more for my pleasure than from duty, I am like the great lords who build: I make, unmake, and remake, until I am passably satisfied with my results, which happens only rarely.” And on another occasion, after complaining of illness brought on by overwork, he says it is impossible for him to rest: “My bad habit of rewriting my memoirs several times till I am passably satisfied is impossible for me to break.”

Not all of Lagrange's main efforts during his twenty years at Berlin went into celestial mechanics and the polishing of his masterpiece. One digression—into Fermat's domain—is of particular interest as it may suggest the inherent difficulty of simple-looking things in arithmetic. We see even the great Lagrange puzzled over the unexpected effort his arithmetical researches cost him.

“I have been occupied these last few days,” he wrote to D'Alembert on August
15, 1768,
“in diversifying my studies a little with certain problems of Arithmetic, and I assure you I found many more difficulties than I had anticipated. Here is one, for example, at whose solution I arrived only with great trouble. Given any positive integer
n
which is not a square, to find a square integer,
x
²
,
such that
nx
²
+ 1 shall be a square. This problem is of great importance in the theory of squares [today,
quadratic forms
, to be described in connection with Gauss] which [squares] are the principal object in Diophantine analysis. Moreover I found on this occasion some very beautiful theorems of Arithmetic, which I will communicate to you another time if you wish.”

The problem Lagrange describes has a long history
going
back to Archimedes and the Hindus. Lagrange's classic memoir on making
nx
²
+ 1 a square is a landmark in the theory of numbers. He was also the first to prove some of Fermat's theorems and that of John Wilson (1741-1793), who had stated that if
p
is any prime number, then if all the numbers 1, 2, . . . up to
p—
1 are multiplied together and 1 be added to the result, the sum is divisible by
p.
The like is not true if
p
is not prime. For example, if
p
= 5, 1 × 2 × 3 × 4 + 1 = 25. This can be proved by elementary reasoning and is another of those arithmetical super-intelligence tests.
^I