466 



NATURE 



[Skptember II, 1890 



seems to me that if the extension of the pure sciences could only 

 be justified by the value of their applications, it is very doubtful 

 whether a satisfactory plea for any further developments could 

 be sustained. As a rule each subject involves its own ideas and 

 its own special analysis, and it can only occasionally happen 

 that analytical methods devised for the expression and develop- 

 ment of one subject will be found to be appropriate for another. 

 It is obvious also that the chance of such applications becomes 

 les and less as we travel farther and farther from the ele- 

 mentary processes and methods which are common to all the 

 exact sciences. There is a general resemblance of style running 

 through much of the analysis required in the physical sciences, 

 but there is no such resemblance in the case of the pure sciences, 

 or between the pure and the physical sciences. It appears 

 likely therefore that, in the future, the mathematical obstacles 

 which present themselves in physical research will have to be 

 overcome, as heretofore, by means of investigations undertaken 

 for the purpose, and that analysis will continue to be enriched 

 by conceptions and results, and even by whole subjects (such as 

 spherical harmonies), 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 theoiy, or a whole body of analysis or geo- 

 metry will suddenly be found to admit of a physical inter- 

 pretation ; but, after all, even the pure sciences themselves 

 exert but an indirect effect upon the perfection of mathematical 

 formula; 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 

 l^roposed 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 upon another, 

 and that we are thus led to solutions of problems and explana- 

 tions 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 subject, 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 mathematics 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 mathe- 

 matician tends to become more and more specialized. 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 counter- 

 acted 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 ia 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 



NO. 1089, VOL. 42] 



in which the notation seems to arise so simply and naturally out 

 of the subject itself, that it is difficult to realize 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 

 principles 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. I have great faith in the power 

 of well-chosen notation to simplify complicated theories and to 

 bring remote ones near ; and I think it is safe to predict that the 

 increased knowledge of principles and the resulting improvements 

 in the symbolic language of mathematics will always enable us 

 to grapple satisfactorily with the difficulties arising from the mere 

 extent of the subject. 



Quite distinct from the theoretical question of the manner in 

 which mathematics will rescue itself from the perils to which it 

 is exposed by its own prolific nature is the practical problem of 

 finding means of rendering available for the student the results 

 which have been already accumulated, and making it possible 

 for a learner to obtain some idea of the present state of the 

 various departments of mathematics. This is a problem which 

 is common to all rapidly moving branches of science, although 

 the difficulties are increased in the case of mathematics by its 

 wide extent and the comparative smallness of the audience 

 addressed. The great mass of mathematical literature will be 

 always contained in Journals and Transactions, but there is no 

 reason why it should not be rendered far more useful and 

 accessible than at present by means of treatises or higher text- 

 books. The whole science suffers from want of avenues of 

 approach, and many beautiful branches of mathematics are re- 

 garded as difficult and technical merely because they are not 

 easily accessible. Ten years ago I should have said that even a 

 bad treatise was better than none at all. I do not .'ay that now,, 

 but I feel very strongly that any introduction to a new subject 

 written by a competent person confers a real benefit on the whole 

 science. The number of excellent text-books of an elementary 

 kind that are published in this country makes it all the more to 

 be regretted that we have so few that are intended for the more 

 advanced student. As an example of the higher kind of text- 

 book, the want of which is so badly felt in many subjects, I may 

 mention the second part of Prof Chrystal's " Algebra " published 

 last year, which in a small compass gives a great mass of valuable 

 and fundamental knowledge that has hitherto been beyond the 

 reach of an ordinary student, though in reality lying so close at 

 hand. I may add that in any treatise or higher text-book it is 

 always desirable that references to the original memoirs should be 

 given, and, if possible, short historical notices also. I am sure 

 that no subject loses more than mathematics by any attempt to 

 dissociate it from its history. 



There is no more striking feature in the mathematical literature 

 of our day than the numerous republications in a collected form 

 of the writings of the greatest mathematicians. These collected 

 editions not only set before us as a whole the complete works of 

 the masters of our science, but they make it possible for others 

 besides those who reside in the vicinity of large libraries to 

 become acquainted with the principal contributions with which 

 it has been enriched in our century ; and, besides being of 

 immense advantage to the science at large, they even go some 

 way towards supplying the want of systematic introductions to- 

 the advanced subjects. Among these republications the collected 

 edition of Cayley's works, now in course of publication by the 

 University of Cambridge, is deserving of especial notice. By 

 undertaking this great work, not only in the lifetime of its author, 

 but while in the full vigour of his powers, the University has 

 secured the inestimable advantage of his own editorship, and 

 thus, under the very best auspices, the world is now being placed 

 in full possession of this grand series of memoirs, which already 

 cover a period of nearly fifty years. 



Although it may not be possible to contemplate the actual 

 position of pure mathematics in this country with any great 

 amount of enthusiasm, we may yet feel some satisfaction in 

 reflecting that there is more cause for congratulation at present 

 than there has been at any time in the last hundred and fifty 

 years, and that we are far removed from the state of affairs which 

 existed before the days of Cayley and Sylvester. Unfortunately, 



