if f3 
NATURE 
[SEPTEMBER 18, 1913 
by differential equations; and our knowledge of even 
the possibilities of the solutions of differential 
equations consists largely, save for some special types, 
of that kind of ignorance which, in the nature of the 
case, can form no idea of its own extent. There are 
subjects the whole content of which is an excuse 
for a desired solution of a differential equation; there 
are infinitely laborious methods of arithmetical com- 
putation held in high repute of which the same must 
be said. And yet I stand here to-day to plead with 
you for tolerance of those who feel that the prosecu- 
tion of the theoretic studies, which alone can alter 
this, is a justifiable aim in life! Our hope and belief 
is that over this vast domain of differential equations 
the theory of functions shall one day rule, as already 
it largely does, for example, over linear differential 
equations. 
Theory of Numbers. 
In concluding this table of contents, I would also 
refer, with becoming brevity, to the modern develop- 
ments of theory of numbers. Wonderful is the fas- 
cination and the difficulty of these familiar objects of 
thought—ordinary numbers. We know how the great 
Gauss, whose lynx eye was laboriously turned upon 
all the physical science of his time, has left it on 
record that in order to settle the law of a plus or 
minus sign in one of the formule of his theory of 
numbers he took up the pen every week for four 
years. In these islands perhaps our imperial necessi- 
ties forbid the hope of much development of such a 
theoretical subject. But in the land of Kummer and 
Gauss and Dirichlet the subject to-day claims the 
allegiance of many eager minds. And we can reflect 
that one of the latest triumphs has been with a 
problem known by the name of our English senior 
wrangler, Waring—the problem of the representation 
of a number by sums of powers. 
Ladies and gentlemen, I have touched only a few 
of the matters with which pure mathematics is con- 
cerned. Each of those I have named is large enough 
for one man’s thought; but they are interwoven and 
interlaced in indissoluble fashion and form one mighty 
whole, so that to be ignorant of one is to be weaker 
in all. I am not concerned to depreciate other pur- 
suits, which seem at first sight more practical; I 
wish only, indeed, as we all do, it were possible for 
one man to cover the whole field of scientific research; 
and I vigorously resent the suggestion that those who 
follow these studies are less careful than others of 
the urgent needs of our national life. But pure mathe- 
matics is not the rival, even less is it the handmaid, 
of other branches of science. Properly pursued, it is 
the essence and soul of them all. It is not for them; 
they are for it; and its results are for all time. No 
man who has felt its fascination can be content to be 
ignorant of any manifestation of regularity and law, 
or can fail to be stirred by all the need of adjustment 
of our actual world. 
And if life is short, if the greatest magician, join- 
ing with the practical man, reminds us that, like 
this vision, 
The cloud-capp'd towers, the gorgeous palaces, 
The solemn temples, the great globe itself, 
Yea, all which it inherit, shall dissolve 
And... leave not a rack behind, 
we must still believe that it is best for us to try to 
reach the brightest light. And all here must believe 
it; for else—no fact is more firmly established—we 
shall not study science to any purpose. 
But that is not all I want to say, or at least to | 
indicate. I.have dealt so far only with proximate 
motives; to me it seems demonstrable that a physical | 
science that is Gonscientious requires the cultivation | 
NO. 2290, VOL. 92] 
of pure mathematics; and the most mundane of 
reasons seem to me to prompt the recognition of the 
zsthetic outlook as a practical necessity, not merely 
a luxury, in a successful society Nor do I want to 
take a transcendental ground. Every schoolboy, I 
suppose, knows the story of the child born, so small, 
if I remember aright, that he could be put into a 
quart pot, in a farmhouse on the borders of Lincoln- 
shire—it was the merest everyday chance. By the 
most incalculable of luck his  brain-stuff was so 
arranged, his parts so proportionately tempered, that 
he became Newton, and taught us the laws of the 
planets. It was the blindest concurrence of physical 
circumstances; and so is all our life. Matter in cer- 
tain relations to itself, working by laws we can 
examine in the chemical laboratory, produces all 
these effects, produces even that state of brain which 
accompanies the desire to speak of the wonder of it 
all. And the same laws will inevitably hurl all into 
confusion and darkness again; and where will all our 
joys and fears, and all our scientific satisfaction, be 
then? 
As students of science, we have no right to shrink 
from this point of view; we are pledged to set aside 
prepossession and dogma, and examine what seems 
possible, wherever it may lead. Even life itself may 
be mechanical, even the greatest of all things, even 
personality, may some day be resoluble into the pro- 
perties of dead matter, whatever that is. We can all 
see that its coherence rises and falls with illness and 
health, with age and physical conditions. Nor, as 
it seems to me, can anything but confusion of thought 
arise from attempts to people our material world 
with those who have ceased to be material. 
An argument could perhaps be based on the diverg- 
ence, as the mathematician would say, of our com- 
prehension of the properties of matter. For though 
we seem able to summarise our past experiences with 
ever-increasing approximation by means of fixed laws, 
our consciousness of ignorance of the future is only 
increased thereby. Do we feel more, or less, com- 
petent to grasp the future possibilities of things, when 
we can send a wireless message 4000 miles, from 
Hanover to New Jersey? 
Our life is begirt with wonder, and with terror. 
Reduce it by all means to ruthless mechanism, if you 
can; it will be a great achievement. But it can make 
no sort of difference to the fact that the things for 
which we live are spiritual. The rose is no less sweet 
because its growth is conditioned by the food we 
supply to its roots. It is an obvious fact, and I 
ought to apologise for remarking it, were it not that 
so much of our popular science is understood by the 
hasty to imply an opposite conclusion. If a chemical 
analysis of the constituents of sea-water could take 
away from the glory of a mighty wave breaking in 
the sunlight, it would still be true that it was the 
mind of the chemist which delighted in finding the 
analysis. Whatever be its history, whatever its 
physical correlations, it is an undeniable fact that the 
mind of man has been evolved; I believe that is the 
scientific word. You may speak of a continuous_up- 
holding of our material framework from without; you 
may ascribe fixed qualities to something you call 
matter; or you may refuse to be drawn into any state- 
| ment. But anyway, the fact remains that the precious 
things of life are those we call the treasures of the 
mind. Dogmas and philosophies, it would seem, rise 
and fall. But gradually accumulating throughout the 
ages, from the earliest dawn of history, there is a body 
of doctrine, a reasoned insight into the relations of 
exact ideas, painfully won and often tested. And this 
remains the main heritage of man; his little beacon 
of light amidst the solitudes and darknesses of infinite 
space; or, if you prefer, like the shout of children at 
