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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, unassisted, 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 inadequately, 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 inquiries. It is a constantly recurring need of science to 
reconsider the exact implication of the terms employed; and as numbers and 
functions are inevitable 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. 1 
may cite, as intelligible to all, such a fact as the construction of a function which 
is continuous at all points of a range, yet possesses no definite differential co- 
efficient at any point. Are we sure that human nature is the only continuous 
variable in the concrete world, assuming it be continuous, which can possess 
such a vacillating character? 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 whose aggregate length 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 inquiries 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. Precisely 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 convergence of an infinite series, as you may see in early papers 
of Stokes, to new formulation of the conditions of integration and of the pro- 
perties of multiple integrals, and so on. And it remains still incompletely 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. nd, alas! 
1913. BB 
