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TRANSACTIONS OF SECTION A. 723: 
eyen by whole subjects (such as spherical harmonics), which will be entirely 
due to the concrete sciences. Of course, it will sometimes happen that a 
differential equation or an integral has already been considered in connection 
with some other theory, or a whole body of analysis or geometry will suddenly be 
found to admit of a physical interpretation ; but, after all, even the pure sciences 
themselves exert but an indirect effect upon the perfection of mathematical 
formulz and processes, and we must be prepared to find that in general the 
requirements of physics have to be met by special analytical researches. Having 
now endeavoured to consider the proposed question impartially, and from a cold 
and rational standpoint, I cannot refrain from adding that, in spite of all I have 
said, I believe that every mathematician must cherish in his heart the conviction 
that at any moment some special analysis, devised in connection with a branch of 
pure mathematics, may bear wonderful fruit in one of the applied sciences, 
giving short and complete solutions of problems which could hitherto be 
treated only by prolix and cumbrous methods. For example, it is difficult to 
believe that the present unwieldy and imperfect treatment of the Lunar Theory is 
the most satisfactory that can be devised. We cannot but hope that some happy 
discovery in pure mathematics may replace the clumsy and tedious series of our 
day by simple and direct analytical methods exactly suited to the problem in 
question. In the different branches of pure mathematics, we find not infrequently 
that researches connected with one subject incidentally throw a flood of light 
= pe another, and that we are thus led to solutions of problems and explanations 
of mysteries which would never have yielded to direct attack in the complete 
absence of any guide to the proper path to be pursued. So, too, in the Lunar 
Theory, if the direct attack should fail to supply any better treatment of the sub- 
ject, we cannot but hope that some day the development of a new branch of 
mathematics, entirely unconnected with dynamics, may supply the key to the 
required method. It should be remembered also that dynamics, which differs 
from the pure sciences only by the inclusion of the laws of motion, is but little 
removed from them in the character of its more general problems. 
It would seem at first sight as if the rapid expansion of the region of mathe- 
matics must be a source of danger to its future progress. Not only does the area 
widen, but the subjects of study increase rapidly in number, and the work of the 
mathematician tends to become moreand more specialised. It is of course merely 
a brilliant exaggeration to say that no mathematician is able to understand the work 
of any other mathematician, but it is certainly true that it is daily becoming more 
and more difficult for a mathematician to keep himself acquainted, even in a 
general way, with the progress of any of the branches of mathematics except those 
which form the field of his own labours. I believe, however, that the increasing 
extent of the territory of mathematics will always be counteracted by increased 
facilities in the means of communication. Additional knowledge opens to us new 
principles and methods which may conduct us with the greatest ease to results 
which previously were most difficult of access; and improvements in notation 
may exercise the most powerful effects both in the simplification and accessibility 
of a subject. It rests with the worker in mathematics not only to explore new 
truths, but to devise the language by which they may be discovered and expressed ; 
and the genius of a great mathematician displays itself no less in the notation he 
invents for deciphering his subject than in the results attained. There are some 
theories in which the notation seems to arise so simply and naturally out of the 
subject itself, that it is difficult to realise that it could have required any creative 
power to produce it; but it may well have happened that in these very cases it 
was the discovery of the appropriate notation which gave the subject its first 
real start, and rendered it amenable to effective treatment. When the prin- 
ciples that underlie a theory have been well grasped, the proper notation almost 
necessarily suggests itself, if it has not been already discovered ; but some sort of 
provisional notation is required in the early stages of a theory in order to make 
any progress at all, and the mathematician who first gains a real insight into the 
nature of a subject is almost sure to be the first to seize upon the right notation. 
Ihave great faith in the power of well-chosen notation to simplify complicated 
