Men of Mathematics (33 page)

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

3
x
¹⁰¹
– 17.3
x
⁷⁰
+
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
ax
²
+ bx + c
= 0, or
ax
³
+ bx
²
+ 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
ax
²
+ 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 extractions 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,

ax
⁵
+
bx
⁴
+
cx
³
+
dx
²
+
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.

*  *  *

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.