7O 
invention is exhaustless and open to all. It is also a 
science. For the mind of man is one; to scale the 
peaks it spreads before the explorer is to open ever 
new prospects of possibility for the formulation of laws 
of nature. It resources have been tested by the 
experience of generations; to-day it lives and thrives 
and expands and wins the life-service of more workers 
than ever before. 
This, at least, is what I wanted to say, and I have 
said it with the greatest brevity I could command. 
But may I dare attempt to carry you further? If 
this seems fanciful, what will you say to the setting 
in which I would wish to place this point of view? 
And yet I feel bound to try to indicate something 
more, which may be of wider appeal. I said a word 
at starting as to the relations of science to those many 
to whom the message of our advanced civilisation is 
the necessity, above all things, of getting bread. 
Leaving this aside, I would make another reference. 
In our time old outlooks have very greatly changed; 
old hopes, disregarded perhaps because undoubted, 
have very largely lost their sanction, and given place 
to earnest questionings. Can anyone who watches 
doubt that the courage to live is in some danger of 
being swallowed up in the anxiety to acquire? May 
it not be, then, that it is good for us to realise, and to 
confess, that the pursuit of things that are beautiful, 
and the achievement of intellectual things that bring 
the joy of overcoming, is at least as demonstrably 
justifiable as the many other things that fill the lives 
of men? May it not be that a wider recognition of 
this would be of some general advantage at present ? 
Is it not even possible that to bear witness to this 
is one of the uses of the scientific spirit? Moreover, 
though the pursuit of truth be a noble aim, is it so 
new a profession; are we so sure that the ardour to 
set down all the facts without extenuation is, un- 
assisted, so continuing a purpose? May science itself 
not be wise to confess to what is its own sustaining 
force? 
Such, ladies and gentlemen, in crude, imperfect 
phrase, is the apologia. If it does not differ much 
from that which workers in other ways would make, 
it does, at least, try to represent truly one point of 
view, and it seems to me specially applicable to the 
case of pure mathematics. But you may ask: What, 
then, is this subject? What can it be about if it is 
not primarily directed to the discussion of the laws of 
natural phenomena? What kind of things are they 
that can occupy alone the thoughts of a lifetime? 
I propose now to attempt to answer this, most in- 
adequately, by a bare recital of some of the broader 
issues of present interest—though this has difficulties, 
because the nineteenth century was of unexampled 
fertility in results and suggestions, and I must be 
as little technical as possible. 
Precision of Definitions. 
First, in regard to two matters which illustrate how 
we are forced by physical problems into abstract in- 
quiries. It is a constantly recurring need of science 
to reconsider the exact implication of the terms 
employed; and as numbers and functions are inevit- 
able in all measurement, the precise meaning of 
number, of continuity, of infinity, of limit, and so 
on, are fundamental questions; those who will receive 
the evidence can easily convince themselves that these 
notions have many pitfalls. Such an imperishable 
monument as Euclid’s theory of ratio is a familiar 
sign that this has always been felt. The last century 
has witnessed a vigorous inquiry into these matters, 
and many of the results brought forward appear to 
be new; nor is the interest of the matter by any means 
exhausted. I may cite, as intelligible to all, such a 
NO. 2290, VOL. 92] 
NATURE 
[SEPTEMBER 18, 1913 
fact as the construction of a function which is con- 
tinuous at all points of a range, yet possesses no 
definite differential coefficient at any point. Are we 
sure that human nature is e only continuous 
variable in the concrete world, assuming it be con- 
tinuous, which can possess such a vacillating char- 
acter? Or I may refer to the more elementary fact 
that all the rational fractions, infinite in number, 
which lie in any given range, can be enclosed in 
intervals the aggregate length of which is arbitrarily 
small. Thus we could take out of our life all the 
moments at which we can say that our age is a 
certain number of years, and days, and fractions of 
a day, and still have appreciably as long to live; this 
would be true, however often, to whatever exactness, 
we named our age, provided we were quick enough 
in naming it. Though the recurrence of these in- 
quiries is part of a wider consideration of functions 
of complex variables, it has been associated also with 
the theory of those series which Fourier used so 
boldly, and so wickedly, for the conduction of heat. 
Like all discoverers, he took much for granted. Pre- 
cisely how much is the problem. This problem has 
led to the precision of what is meant by a function of | 
real variables, to the question of the uniform con- 
vergence of an infinite series, as you may see in early 
papers of Stokes, to new formulation of the conditions 
of integration and of the properties of multiple 
integrals, and so on. And it remains still incom- 
pletely solved. 
Calculus of Variations. 
Another case in which the suggestions of physics 
have caused grave disquiet to the mathematicians is 
the problem of the variation of a definite integral. 
No one is likely to underrate the grandeur of the aim 
of those who would deduce the whole physical history 
of the world from the single principle of least action. 
Everyone must be interested in the theorem that a 
potential function, with a given value at the boundary 
of a volume, is such as to render a certain integral, 
representing, say, the energy, a minimum. But in 
that proportion one desires to be sure that the logical 
processes employed are free from objection. And, 
alas! to deal only with one of the earliest problems 
of the subject, though the finally sufficient conditions 
for a minimum of a simple integral seemed settled 
long ago, and could be applied, for example, to New- 
ton’s celebrated problem of the solid of least resist- 
ance, it has since been shown to be a general fact 
that such a problem cannot have any definite solution 
at all. And, although the principle of Thomson and 
Dirichlet, which relates to the potential problem re- 
ferred to, was expounded by Gauss, and accepted by 
Riemann, and remains to-day in our standard treatise 
on natural philosophy, there can be no doubt that, 
in the form in which it was originally stated, it 
proves just nothing. Thus a new investigation has 
been necessary into the foundations of the principle. 
There is another problem, closely connected with this 
subject, to which I would allude: the stability of the 
solar system. For those who can make pronounce- 
ments in regard to this I have a feeling of envy; for 
their methods, as yet, I have a quite other feeling. 
The interest of this problem alone is sufficient to 
justify the craving of the pure mathematician for 
powerful methods and unexceptionable rigour. 
Non-Euclidean Geometry. 
But I turn to another matter. It is an old view, I 
suppose, that geometry deals with facts about which 
there can be no two opinions. You are familiar with 
the axiom that, given a straight line and a point, one 
and only one straight line can be drawn through the 
point parallel to the given straight line. According to 
