Men of Mathematics (105 page)

(the arrow is read “is replaced by”) which leave the value of
F(z)
unchanged as just explained
form a group:
the result of the successive performance of two substitutions in the set,

is in the set; there is an “identity substitution” in the set, namely
z → z
(here
a
= 1, b = 0,
c
= 0,
d =
l); and finally each substitution has a unique “inverse”—that is, for each substitution in the set there is a single other one which, if applied to the first, will produce the identity substitution. In summary, using the terminology of previous chapters, we see that
F(z)
is
a function which is invariant under an infinite group of linear fractional transformations.
Note that the infinity of substitutions is a
denumerable
infinity, as first stated: the substitutions can be counted off 1, 2, 3, . . . , and are
not
as numerous as the points on a line. Poincaré actually constructed such functions and developed their most important properties in a series of papers in the 1880's. Such functions are called
automorphic.

Only two remarks need be made here to indicate what Poincaré achieved by this wonderful creation. First, his theory includes that of the elliptic functions as a detail. Second, as the distinguished French mathematician Georges Humbert said, Poincaré found two memorable propositions which “gave him the keys of the algebraic cosmos”:

Two automorphic functions
^III
invariant under the same group are connected by an algebraic equation;

Conversely, the coordinates of a point on any algebraic curve can be expressed in terms of automorphic functions, and hence by uniform functions of a single parameter (variable).

An algebraic curve is one whose equation is of the type
P
(x, y)
= 0, where
P
(x, y)
is a polynomial in
x
and
y.
As a simple example, the equation of the circle whose center is at the origin—(0, 0)—and whose
radius is
a,
is
x
²
+
y
²
= a
²
.
According to the second of Poincaré's “keys,” it must be possible to express
x, y
as automorphic functions of a single parameter, say
t.
It is; for if
x = a
cos
t
and
y = a
sin
t,
then, squaring and adding, we get rid of
t
(since cos
²
t
sin
²
t
= l), and find
x
²
+ y
²
= a
²
.
But the trigonometric functions cos
t,
sin
t
are special cases of elliptic functions, which in turn are special cases of automorphic functions.

The creation of this vast theory of automorphic functions was but one of many astonishing things in analysis which Poincaré did before he was thirty. Nor was all his time devoted to analysis; the theory of numbers, parts of algebra, and mathematical astronomy also shared his attention. In the first he recast the Gaussian theory of binary quadratic forms (see chapter on Gauss) in a geometrical shape which appeals particularly to those who, like Poincaré, prefer the intuitive approach. This of course was not all that he did in the higher arithmetic, but limitations of space forbid further details.

Work of this caliber did not pass unappreciated. At the unusually early age of thirty two (in
1887)
Poincaré was elected to the Academy. His proposer said some pretty strong things, but most mathematicians will subscribe to their truth: “[Poincaré's] work is above ordinary praise and reminds us inevitably of what Jacobi wrote of Abel—that he had settled questions which, before him, were unimagined. It must indeed be recognized that we are witnessing a revolution in Mathematics comparable in every way to that which manifested itself, half a century ago, by the accession of elliptic functions.”

To leave Poincaré's work in pure mathematics here is like rising from a banquet table after having just sat down, but we must turn to another side of his universality.

*  *  *

Since the time of Newton and his immediate successors astronomy has generously supplied mathematicians with more problems than they can solve. Until the late nineteenth century the weapons used by mathematicians in their attack on astronomy were practically all immediate improvements of those invented by Newton himself, Euler, Lagrange, and Laplace. But all through the nineteenth century, particularly since Cauchy's development of the theory of functions of a complex variable and the investigations of himself and others on the convergence of infinite series, a huge arsenal of untried weapons had been accumulating from the labors of pure mathematicians. To
Poincaré, to whom analysis came as naturally as thinking, this vast pile of unused mathematics seemed the most natural thing in the world to use in a new offensive on the outstanding problems of celestial mechanics and planetary evolution. He picked and chose what he liked out of the heap, improved it, invented new weapons of his own, and assaulted theoretical astronomy in a grand fashion it had not been assaulted in for a century. He
modernized
the attack; indeed his campaign was so extremely modern to the majority of experts in celestial mechanics that even today, forty years or more after Poincaré opened his offensive, few have mastered his weapons and some, unable to bend his bow, insinuate that it is worthless in a practical attack. Nevertheless Poincaré is not without forceful champions whose conquests would have been impossible to the men of the pre-Poincaré era.

Poincaré's first (1889) great success in mathematical astronomy grew out of an unsuccessful attack on “the problem of
n
bodies.” For
n
= 2 the problem was completely solved by Newton; the famous “problem of three bodies”
(n
= 3) will be noticed later; when
n
exceeds 3 some of the reductions applicable to the case
n
= 3 can be carried over.

According to the Newtonian law of gravitation two particles of masses m,
M
at a distance
D
apart attract one another with a force proportional to
Imagine
n
material particles distributed in any manner in space; the masses, initial motions, and the mutual distances of all the particles are assumed known at a given instant. If they attract one another according to the Newtonian law,
what will be their positions and motions (velocities) after any stated lapse of time?
For the purposes of mathematical astronomy the stars in a cluster, or in a galaxy, or in a cluster of galaxies, may be thought of as material particles attracting one another according to the Newtonian law. The “problem of
n
bodies” thus amounts—in one of its applications—to asking what will be the aspect of the heavens a year from now, or a billion years hence, it being assumed that we have sufficient observational data to describe the general configuration
now.
The problem of course is tremendously complicated by radiation—the masses of the stars do not remain constant for millions of years; but a complete, calculable solution of the problem of
n
bodies in its Newtonian form would probably give results of an accuracy sufficient for all human
purposes—the human race will likely be extinct long before radiation can introduce observable inaccuracies.

This was substantially the problem proposed for the prize offered by King Oscar II of Sweden in 1887. Poincaré did not solve the problem, but in 1889 he was awarded the prize anyhow by a jury consisting of Weierstrass, Hermite, and Mittag-Leffler for his general discussion of the differential equations of dynamics and an attack on the problem of three bodies. The last is usually considered the most important case of the
n
-body problem, as the Earth, Moon, and Sun furnish an instance of the case
n
= 3. In his report to Mittag-Leffler, Weierstrass wrote, “You may tell your Sovereign that this work cannot indeed be considered as furnishing the complete solution of the question proposed, but that it is nevertheless of such importance that
its publication will inaugurate a new era in the history of Celestial Mechanics.
The end which His Majesty had in view in opening the competition may therefore be considered as having been attained.” Not to be outdone by the King of Sweden, the French Government followed up the prize by making Poincaré a Knight of the Legion of Honor—a much less expensive acknowledgment of the young mathematician's genius than the King's 2500 crowns and gold medal.

As we have mentioned the problem of three bodies we may now report one item from its fairly recent history; since the time of Euler it has been considered one of the most difficult problems in the whole range of mathematics. Stated mathematically, the problem boils down to solving a system of nine simultaneous differential equations (all linear, each of the second order). Lagrange succeeded in reducing this system to a simpler. As in the majority of physical problems, the solution is not to be expected
infinite
terms;
if a solution exists at all
it will be given by
infinite series.
The solution will “exist” if these series satisfy the equations (formally) and moreover
converge
for certain values of the variables. The central difficulty is to prove the convergence. Up till 1905 various special solutions had been found, but the existence of anything that could be called general had not been proved.

In 1906 and 1909 a considerable advance came from a rather unexpected quarter—Finland, a country which sophisticated Europeans even today consider barely civilized, especially for its queer custom of paying its debts, and which few Americans thought advanced beyond the Stone Age till Paavo Nurmi ran the legs off the United
States. Excepting only the rare case when all three bodies collide simultaneously, Karl Frithiof Sundman of Helsingfors, utilizing analytical methods due to the Italian Levi-Civita and the French Painlevé, and making an ingenious transformation of his own,
proved
the existence of a solution in the sense described above. Sundman's solution is not adapted to numerical computation, nor does it give much information regarding the actual motion, but that is not the point of interest here: a problem which had not been known to be solvable was proved to be so. Many had struggled desperately to prove this much; when the proof was forthcoming, some, humanly enough, hastened to point out that Sundman had done nothing much because he had not solved some problem other than the one he had. This kind of criticism is as common in mathematics as it is in literature and art, showing once more that mathematicians are as human as anybody.

Poincaré's most original work in mathematical astronomy was summed up in his great treatise
Les méthodes nouvelles de la mécanique céleste
(New methods of celestial mechanics; three volumes, 1892, 1893, 1899). This was followed by another three-volume work in 1905-1910 of a more immediately practical nature,
Leçons de mécanique céleste,
and a little later by the publication of his course of lectures
Sur les figures d'équilibre d'une masse fluide
(On the figures of equilibrium of a fluid mass), and a historical-critical book
Sur les hypothèses cosmogoniques
(On cosmological hypotheses).

Of the first of these works Darboux (seconded by many others) declares that it did indeed start a new era in celestial mechanics and that it is comparable to the
Mécanique céleste
of Laplace and the earlier work of D'Alembert on the precession of the equinoxes. “Following the road in analytical mechanics opened up by Lagrange,” Darboux says, “. . . Jacobi had established a theory which appeared to be one of the most complete in dynamics. For fifty years we lived on the theorems of the illustrious German mathematician, applying them and studying them from all angles, but without adding anything essential. It was Poincaré who first shattered these rigid frames in which the theory seemed to be encased and contrived for it vistas and new windows on the external world. He introduced or used, in the study of dynamical problems, different notions: the first, which had been given before and which, moreover, is applicable not solely to mechanics, is that of
variational equations,
namely, linear differential equations that determine solutions of a problem infinitely near to a
given solution; the second, that of
integral invariants,
which belong entirely to him and play a capital part in these researches. Further fundamental notions were added to these, notably those concerning so-called ‘periodic' solutions, for which the bodies whose motion is studied return after a certain time to their initial positions and original relative velocities.”