TRANSACTIONS OF SECTION A. 725 
desire to see the system of instruction which has found favour in this country so: 
modified that pure mathematics could be carried on by narrow specialists. I 
should be sorry, for example, that a student, after learning algebra and differential 
calculus, should pass directly to the theory of curves, and devote himself to 
research in this field without ever having acquired a general knowledge of other 
branches of mathematics or of any of its applications. Every person who proposes: 
to engage in mathematical research should be equipped at starting upon his career 
with some knowledge of at least all the subjects included in the first part of the 
mathematical tripos. From what I have said in an earlier portion of this address 
it may be inferred that, from the point of view of the pure mathematician, I think 
that the course of study, and some of the text-books, are capable of improvement ;. 
but I am satisfied that a general mathematical training such as the tripos requires 
is of the greatest possible value to every student, and that without it he cannot 
even make a good decision as to the class of subjects to which he is likely to devote 
his labour with the best effect. If the student were brought by the shortest 
possible route to the frontier of one of the subjects, where a fruitful field of 
research was pointed out to him, there is no doubt that the amount of mathe- 
matical literature produced might be greatly increased; but I am sure that the 
advantage to science would not be proportional to this increased amount. I am 
convinced that no one should devote himself to the abstract sciences unless he 
feels strongly drawn to them by his tastes. These subjects are treated by means of" 
a powerful symbolic language, and it is the business of the investigator to 
discriminate between equations and formule which represent valuable facts in 
nature, and those which are merely symbolic relations, deducible from others that 
are more fundamental, and having no special significance in the subject itself. 
The mathematician requires tact and good taste at every step of his work, and he 
has to learn to trust to his own instinct to distinguish between what is really 
worthy of his efforts and what is not; he must take care not to be the slave of his- 
symbols, but always to have before his mind the realities which they merely serve 
to express. Tor these and other reasons it seems to me of the highest importance 
that a mathematician should be trained in no narrow school; a wide course of 
reading in the first few years of his mathematical study cannot fail to influence for 
good the character of the whole of his subsequent work. 
Before leaving this part of my subject I should like to say a few words upon 
the subject of accuracy of form in the presentation of mathematical results. In 
other branches of science, where quick publication seems to be so much desired, 
there may possibly be some excuse for giving to the world slovenly or ill-digested 
work, but there is no such excuse in mathematics. The form ought to be as perfect as 
the substance and the demonstrations as rigorous as those of Euclid. The mathe- 
matician has to deal with the most exact facts of nature, and he should spare no: 
effort to render his interpretation worthy of his subject, and to give to his work its 
highest degree of perfection. ‘ Pauca sed matura’ was Gauss’s motto. 
The Universities are the natural home of mathematics, and to them we chiefly 
owe its cultivation andencouragement. There is, however, one other much younger 
body whose services to our science should not be passed over in any survey of its 
present state—I mean the London Mathematical Society. Twenty-five years ago, 
upon its foundation, I think the most sanguine mathematician would scarcely have 
ventured to predict that it would so soon take the position that it has among the 
scientific institutions of the world. The continuous interest taken by its members in 
its meetings, and the number and value of the papers published by it, show how 
steadily the flame of mathematical inquiry is burning among us. I do not presume to- 
assert that the interest taken in the pure sciences can be regarded as an index of the 
energy and power of a nation, but it is certain that mathematical research flourishes 
only in a vigorous community. The search after abstract truth for its own 
sake, without the smallest thought of practical application or return in any form, 
and the yearning desire to explore the unknown, are signs of the vitality of a 
people, which are among the first to disappear when decay begins. 
In conclusion, I will refer in some detail to one special subject—the 
Theory of Numbers. It is much to be regretted that this great theory, perhaps the 
