372 TRANSACTIONS OF SECTION A. 
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 whose whole content is an excuse for a desired 
solution of a differential equation; there are infinitely laborious methods of 
arithmetical computation 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 prosecution 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 developments of theory of numbers. Wonderful is the 
fascination 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 necessities 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 concerned. 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 pursuits, 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 Mathematics 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, joining 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 conscientious requires the cultivation of Pure Mathematics; and the most 
mundane of reasons seem to me to prompt the recognition of the esthetic 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 Lincolnshire—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. 1t was the blindest concurrence of physical 
circumstances; and so is all our life. Matter in certain relations to itself, 
