September ii, 1890] 



NATURE 



467 



we cannot point with pride to any distinct school of the pure 

 sciences corresponding to the Cambridge school of mathematical 

 physics, and I am afraid that the old saying that we have 

 generals without armies is as true as ever. For this there is no 

 immediate rertiedy ; a school must grow up gradually of itself, 

 as the study of mathematical physics has grown up at Cambridge. 

 I certainly should not wish, even if it were possible, to obtain 

 more recruits for the pure sciences at the expense of the applied, 

 nor do I 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 mathe- 

 matics or of any of its applications. Every person who proposes 

 to engage in mathematical research should be equipped at start- 

 ing upon his career with some knowledge of at least all the sub- 

 jects 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 

 mathematical 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 formulae 

 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. For 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 presenta- 

 tion 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 demonstra- 

 tions as rigorous as those of Euclid. The mathematician 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 

 i^ive 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 and encouragement. 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 mathe- 

 matician would scarcely have ventured to predict that it would 

 so soon take the position that it has among the scientific institu- 

 tions of the world. The continuous interest taken by its mem- 

 bers 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 sub- 

 ject — the Theory of Numbers. It is much to be regretted that 

 this great theory, perhaps the greatest and most perfect of all 

 the mathematical theories, should have been so little cultivated in 

 this country, and that no portion of it should ever have been in- 

 cluded in an ordinary course of mathematical study. It may be 

 said to date from the year 1801, when Gauss published his *' Dis- 

 quisitiones Arithmeticae," so that it is nearly thirty years older than 

 the Theory of Elliptic Functions, to which we may assign the date 

 1829, the year in which Jacobi's "Fundamenta Nova 'appeared. 

 But the latter theory has already found a congenial home among 

 us, while the former is nowhere systematically studied, and is 

 still without a text-book. The chapters in books upon Algebra 

 which bear the title " Theory of Numbers " give a misleading idea 

 of the nature of the subject, the results there given being mainly 

 introductory lemmas of the simplest kind. The theory has 

 nothing to do with arithmetic in the ordinary sense of the word, 

 or numerical tables, or the representation of numbers by figures 

 in the decimal system or otherwise. All its results are actual 

 truths of the most fundamental kind, which must exist in rerun 

 natiird. Its prhicipal branches are the theory of forms and the 

 so-called complex theories. Such a proposition as that every 

 prime number, which when divided by 4 leaves remainder I, can 

 always be expressed as the sum of two squares, and that this can 

 be done in one way only, affords a good example of a very simple 

 result in the theory of forms. It is entirely independent of any 

 method of representing numbers, and merely asserts that if we 

 have 5, 13, 17, 29, &c., things — let us say marbles, to fix the 

 ideas— we can always succeed in so arranging them as to form 

 them into two squares, and that for each number we can do this 

 in but one way. Simple as such a theorem is to enunciate and 

 comprehend, the demonstration is far from easy. This is charac- 

 teristic of the whole subject ; simple propositions, which we can 

 easily discover by trial, and of the universal truth of which we 

 can feel but little doubt, require for their demonstration a refined 

 and intricate analysis, founded upon the most difficult and im- 

 aginative conceptions which mathematics has as yet attained to 

 in its struggles to grapple with the actual problems of the worlds 

 of thought and matter. 



The theory of quantity consists of two distinct branches — one 

 relating to discrete quantity, and the other to continuous quantity. 

 To the latter branch belong algebra and all the ordinary subjects 

 of pure mathematics ; the former bears the name of the theory 

 of numbers. Its truths are of the most absolute kind, involving 

 only the notions of number and arrangement ; in fact, if we 

 imagine all the exact sciences ranged in order, it naturally takes 

 its place at one end of the series. Different sciences appeal to 

 different intellects with very different force, but there are some 

 minds over which the absolute character of the fundamental 

 truths that belong to this theory and the absolute precision of its 

 methods exercise the strongest fascination, and excite an interest 

 which neither the truths of geometry nor the most important dis- 

 coveries depending upon the constitution of matter are capable 

 of producing. 



Many of the greatest masters of the mathematical sciences 

 were first attracted to mathematical inquiry by problems relating 

 to numbers, and no one can glance at the periodicals of the 

 present day which contain questions for solution without noticing 

 how singular a charm such problems still continue to exert. This 

 interest in numbers seems implanted in the human mind, and it 

 is a pity that it should not have freer scope in this country. The 

 methods of the theory of numbers are peculiar to itself, and are 

 not readily acquired by a student whose mind has for years been 

 familiarized with the very different treatment which is appropriate 

 to the theory of continuous magnitude ; it is therefore extremely 

 desirable that some portion of the theory should be included in 

 the ordinary course of mathematical instruction at our Univer- 

 sitie!:. From the moment that Gauss, in his wonderful treatise 

 of 1801, laid down the true lines of the theory, it entered upon a 

 new day, and no one is likely to be able to do useful work in 

 any part of the subject who is unacquainted with the principles 

 and conceptions with which he endowed it. 



Undoubted the subject is a difficult and intricate one even in 

 its elementary parts, but there can be but little doubt that when 

 the processes which are now only read by specialists on their 

 way to the border become more generally known and studied, 

 they will be found to admit of great simplification. It is in fact 

 a territory where there is quite as much scope for the mathe* 



NO. 1089, VOL. 42] 



