Men of Mathematics (104 page)

Thus young Poincaré, like Gauss, was overwhelmed by the host of ideas which besieged his mind but, unlike Gauss, his motto was not “Few, but ripe.” It is an open question whether a creative scientist who hoards the fruits of his labor so long that some of them go stale does more for the advancement of science than the more impetuous man who scatters broadcast everything he gathers, green or ripe, to fall where it may to ripen or rot as wind and weather take it. Some believe one way, some another. As a decision is beyond the reach of objective criteria everyone is entitled to his own purely subjective opinion.

Poincaré was not destined to become a mining engineer, but during his apprenticeship he showed that he had at least the courage of a real engineer. After a mine explosion and fire which had claimed sixteen victims he went down at once with the rescue crew. But the calling was uncongenial and he welcomed the opportunity to become a professional mathematician which his thesis and other early work opened up to him. His first academic appointment was at Caen on December 1, 1879, as Professor of Mathematical Analysis. Two years later he was promoted (at the age of twenty seven) to the University of Paris where, in 1886, he was again promoted, taking charge of the course in mechanics and experimental physics (the last seems rather strange, in view of Poincaré's exploits as a student in the laboratory). Except for trips to scientific congresses in Europe and a visit to the United States in 1904 as an invited lecturer at the St. Louis Exposition, Poincaré
spent the rest of his life in Paris as the ruler of French mathematics.

*  *  *

Poincaré's creative period opened with the thesis of
1878
and closed with his death in
1912
—when he was at the apex of his powers. Into this comparatively brief span of thirty four years he crowded a mass of work that is sheerly incredible when we consider the difficulty of most of it. His record is nearly five hundred papers on
new
mathematics, many of them extensive memoirs, and more than thirty books covering practically all branches of mathematical physics, theoretical physics, and theoretical astronomy as they existed in his day. This leaves out of account his classics on the philosophy of science and his popular essays. To give an adequate idea of this immense labor one would have to be a second Poincaré, so we shall presently select two or three, of his most celebrated works for brief description, apologizing here once for all for the necessary inadequacy.

Poincaré's first successes were in the theory of differential equations, to which he applied all the resources of the analysis of which he was absolute master. This early choice for a major effort already indicates Poincaré's leaning toward the applications of mathematics, for differential equations have attracted swarms of workers since the time of Newton chiefly because they
are
of great importance in the exploration of the physical universe. “Pure” mathematicians sometimes like to imagine that all their activities are dictated by their own tastes and that the applications of science suggest nothing of interest to them. Nevertheless some of the purest of the pure drudge away their lives over differential equations that first appeared in the translation of physical situations into mathematical symbolism, and it is precisely these practically suggested equations which are the heart of the theory. A particular equation suggested by science may be generalized by the mathematicians and then be turned back to the scientists (frequently without a solution in any form that they can use) to be applied to new physical problems, but first and last the motive is scientific. Fourier summed up this thesis in a famous passage which irritates one type of mathematician, but which Poincaré endorsed and followed in much of his work.

“The profound study of nature,” Fourier declared, “is the most fecund source of mathematical discoveries. Not only does this study, by offering a definite goal to research, have the advantage of excluding vague questions and futile calculations, but it is also a sure means
of molding analysis itself and discovering those elements in it which it is essential to know and which science ought always to conserve. These fundamental elements are those which recur in all natural phenomena.” To which some might retort: No doubt, but what about arithmetic in the sense of Gauss? However, Poincaré followed Fourier's advice whether he believed in it or not—even his researches in the theory of numbers were more or less remotely inspired by others closer to the mathematics of physical science.

The investigations on differential equations led out in
1880,
when Poincaré was twenty six, to one of his most brilliant discoveries, a generalization of the elliptic functions (and of some others). The nature of a (uniform) periodic function of a single variable has frequently been described in preceding chapters, but to bring out what Poincaré did, we may repeat the essentials. The trigonometric function sin
z
has the period
2π
, namely, sin
(z + 2Π)
= sin
z;
that is, when the variable
z
is increased by 2π, the sine function of
z
returns to its initial value. For an elliptic function, say
E(z),
there are
two
distinct periods, say
pi
and
p
₂
,
such that
E(z + p
₁
) = E(z), E(z
+
p
₂
) = E(z).
Poincaré found that
periodicity
is merely a special case of a more general property: the value of certain functions is restored when the variable is replaced by any one of a
denumerable
infinity of linear fractional transformations of itself, and all these transformations form a group. A few symbols will clarify this statement.

Let
z
be replaced by
Then, for a
denumerable infinity
of sets of values of
a, b, c, d,
there are uniform functions of
z,
say
F(z)
is one of them, such that

Further, if
a
₁
, b
₁
, c
₁
, d
₁
,
and
a
₂
, b
₂
, c
₂
, d
₂
are any two of the sets of values of
a, b, c, d,
and if
z
be replaced first by
and then, in this,
z
be replaced by
giving, say,
then not only do we have

but also

Further the set of all the substitutions