Men of Mathematics

by E. T. Bell

  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 occupati
ons 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, x2, such that nx2 + 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 nx2 + 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

  In his reply D’Alembert states his belief that Diophantine analysis may be useful in the integral calculus, but does not go into detail. Curiously enough, the prophecy was fulfilled in the 1870’s by the Russian mathematician, G. Zolotareff.

  Laplace also became interested in arithmetic for a while and told Lagrange that the existence of Fermat’s unproved theorems, while one of the greatest glories of French mathematics, was also its most conspicuous blemish, and it was the duty of French mathematicians to remove the blemish. But he prophesied tremendous difficulties. The root of the trouble, in his opinion, is that discrete problems (those dealing ultimately with 1, 2, 3, . . .) are not yet attackable by any general weapon such as the calculus provides for the continuous. D’Alembert also remarks of arithmetic that he found it “more difficult than it seems at first.” These experiences of mathematicians like Lagrange and his friends may imply that arithmetic really is hard.

  Another letter of Lagrange’s (February 28, 1769) records the conclusion of the matter. “The problem I spoke of has occupied me much more than I anticipated at first; but finally I am happily finished and I believe I have left practically nothing to be desired in the subject of indeterminate equations of the second degree in two unknowns.” He was too optimistic here; Gauss had yet to be heard from—his father and mother had still seven years to go before meeting one another. Two years before the birth of Gauss (in 1777), Lagrange looked back over his work in a pessimistic mood: “The arithmetical researches are those which have cost me most trouble and are perhaps the least valuable.”

  When he was feeling well Lagrange seldom lapsed into the error of estimating the “importance” of his work. “I have always regarded mathematics,” he wrote to Laplace in 1777, “as an object of amusement rather than of ambition, and I can assure you that I enjoy the works of others much more than my own, with which I am always dissatisfied. You will see by that, if you are exempt from jealousy by your own success, I am none the less so by my disposition.” This was in reply to a somewhat pompous declaration by Laplace that he worked at mathematics only to appease his own sublime curiosity and did not give a hang for the plaudits of “the multitude”—which, in his case, was partly balderdash.

  A letter of September 15, 1782, to Laplace is of great historical interest as it tells of the finishing of the Mécanique analytique: “I have almost completed a Treatise on Analytical Mechanics, founded solely on the principle or formula in the first section of the accompanying memoir; but as I do not know when or where I can get it printed, I am not hurrying with the finishing touches.”

  Legendre undertook the editing of the work for the press and Lagrange’s old friend the Abbé Marie finally persuaded a Paris publisher to risk his reputation. This canny individual consented to proceed with the printing only when the Abbé agreed to purchase all stock remaining unsold after a certain date. The book did not appear until 1788, after Lagrange had left Berlin. A copy was delivered into his hands when he had grown so indifferent to all science and all mathematics that he did not even bother to open the book. For all he knew at the time the printer might have got it out in Chinese. He did not care.

  * * *

  One investigation of Lagrange’s Berlin period is of the highest importance in the development of modern algebra, the memoir of 1767 On the Solution of Numerical Equations and the subsequent additions dealing with the general question of the algebraic solvability of equations. Possibly the greatest importance of Lagrange’s researches in the theory and solution of equations is the inspiration they proved to be to the leading algebraists of the early nineteenth century. Time after time we shall see the men who finally disposed of a problem which had baffled algebraists for three centuries or more returning to Lagrange for ideas and inspiration. Lagrange himself did not resolve the central difficulty—that of stating necessary and sufficient conditions that a given equation shall be solvable algebraically, but the germ of the solution is to be found in his work.

  As the problem is one of those major things in all algebra which can be simply described we may glance at it in passing; it will recur many times as a leading motive in the work of some of the great mathematicians of the nineteenth century—Cauchy, Abel, Galois, Hermite, and Kronecker, among others.

  First it may be emphasized that there is no difficulty whatever in solving an algebraic equation with numerical coefficients. The labor may be excessive if the equation is of high degree, say

  3x101 – 17.3x70 + x – 11 = 0,

  but there are many straightforward methods known whereby a root of such a numerical equation can be found to any prescribed degree of accuracy. Some of these are part of the regular school course in algebra. But in Lagrange’s day uniform methods for solving numerical equations to a preassigned degree of accuracy were not commonplace—if known at all. Lagrange provided such a method. Theoretically it did what was required, but it was not practical. No engineer faced with a numerical equation today would dream of using Lagrange’s method.

  The really significant problem arises when we seek an algebraic solution of an equation with literal coefficients, say ax2 + bx + c = 0, or ax3 + bx2 + cx + d = 0, and so on for degrees higher than the third. What is required is a set of formulas expressing the unknown x in terms of the given a, b, c, . . . , such that if any one of these expressions for x be put in the lefthand side of the equation, that side shall reduce to zero. For an equation of degree n the unknown x has precisely n values. Thus for the above quadratic (second degree) equation,

  are the two values which when substituted for x will reduce ax2 + bx + c to zero. The required values of x in any case are to be expressed in terms of a, b, c, . . . by means of only a finite number of additions, subtractions, multiplications, divisions, and extracti
ons of roots. This is the problem. Is it solvable? The answer to this was not given till about twenty years after Lagrange’s death, but the clue is easily traced to his work.

  As a first step toward a comprehensive theory Lagrange made an exhaustive study of all the solutions given by his predecessors for the general equations of the first four degrees, and succeeded in showing that all of the dodges by which solutions had been obtained could be replaced by a uniform procedure. A detail in this general method contains the clue mentioned. Suppose we are given an algebraic expression involving letters a, b, c, . . .: how many different expressions can be derived from the given one if the letters in it are interchanged in all possible ways? For example, from ab + cd we get ad + cb by interchanging b and d. This problem suggests another closely related one, also part of the clue Lagrange was seeking. What interchanges of letters will leave the given expression invariant (unaltered)? Thus ab + cd becomes ba + cd under the interchange of a and b, which is the same as ab + cd since ab = ba. From these questions the theory of finite groups originated. This was found to be the key to the question of algebraic solvability. It will reappear when we consider Cauchy and Galois.

  Another significant fact showed up in Lagrange’s investigation. For degrees 2, 3, and 4 the general algebraic equation is solved by making the solution depend upon that of an equation of lower degree than the one under discussion. This works beautifully and uniformly for equations of degrees 2, 3, and 4, but when a precisely similar process is attempted on the general equation of degree 5,

  ax5 + bx4 + cx3 + dx2 + ex + f = 0,

  the resolvent equation, instead of being of degree less than 5 turns out to be of degree 6. This has the effect of replacing the given equation by a harder one. The method which works for 2, 3, 4 breaks down for 5, and unless there is some way round the awkward 6 the road is blocked. As a matter of fact we shall see that there is no way of avoiding the difficulty. We might as well try to square the circle or trisect an angle by Euclidean methods.