NA TURE 



[September 12, 1901 



original research in a particular subdivision of his subject. It 

 will be sufficient to consider the subjects that come under the 

 purview of Section A, though it will be obvious that a similar train 

 of reasoning would have equal validity in connection with the 

 subjects included in any of the other sections. I take the word 

 "specialist" to denote a man who makes original discoveries 

 in some branch of science, and I deny that any other man has 

 the right, in the modern meaning of the word, to be called by 

 others, or to call himself, a specialist. I would not wish to be 

 understood to imply a belief that a truly scientific man is 

 necessarily a specialist ; I do believe that a scientific man of 

 high type is almost invariably an original discoverer in one or 

 more special branches of science ; but I can conceive that a man 

 may study the mutual relations of different sciences and of 

 different branches of the same science and may throw such 

 an amount of light upon the underlying principles as to be in 

 the highest degree scientific. I will now advance the proposi- 

 tion that, with this exception, all scientific workers are 

 specialists ; it is merely a question of degree. An extreme 

 specialist is that man who makes discoveries in only one branch, 

 perhaps a very narrow branch, of his subject. I shall consider 

 that in defending him I am a fortiori defending the man who 

 is a specialist, but not of this extreme character. 



A subject of study may acquire the reputation of being narrow 

 either because it has for some reason or other not attracted 

 workers and is in reality virgin soil only awaiting the arrival of 

 a husbandman with the necessary skill, or because it is an ex- 

 tremely difficult subject which has resisted previous attempts to 

 elucidate it. In the latter case, it is not likely that a scientific 

 man will obstinately persist in trying to force an entrance 

 through a bare blank wall. Either from weariness in striving 

 or from the exercise of his judgment he will turn to some other 

 subdivision which appears to give greater promise of success. 

 When the subject is narrow merely because it has been over- 

 looked, the specialist has a grand opportunity for widening it 

 and freeing it from the reproach of being narrow ; when it is 

 narrow from its inherent difficulty he has the opportunity of 

 exerting his full strength to pierce the barriers which close the 

 way to discoveries. In either case the specialist, before he 

 can determine the particular subject which is to engage 

 his thoughts, must have a fairly wide knowledge of the 

 whole of his subject. If he does not possess this he will most 

 likely make a bad choice of particular subjects, or, having made 

 a wise selection, he will lack an essential part of the mental 

 equipment necessary for a successful investigation. Again, 

 though the subject may be a narrow one, it by no means follows 

 that the appropriate or possible methods of research are pre- 

 scribed within narrow limits. I will instance the Theory 

 of Numbers, which, in comparatively recent times, was a 

 subject of small extent and of restricted application to 

 other branches of science. The problems that presented 

 themselves naturally, or were brought into prominence by 

 the imaginations of great intellects, were fraught with diffi- 

 culty. There seemed to be an absence, partial or complete, 

 of the law and order that investigators had been accustomed to 

 find in the wide realm of continuous quantity. The country to 

 be explored was found to be full of pitfalls for the unwary. 

 Many a lesson concerning the danger of hasty generalisation 

 had to be learnt and taken to heart. Many a false step had 

 to be retraced. Many a road which a first reconnaissance had 

 shown to be straight for a short distance was found, on further 

 exploration, to change suddenly its direction and to break up 

 into a number of paths which wandered in a fitful manner in 

 country of increasing natural difficulty. There were few vanish- 

 ing points in the perspective. Few, also, and insignificant 

 were the peaks from which a general notion could be gathered 

 of any considerable portion of the country. The surveying in- 

 struments were inadequate to cope with the physical characters 

 of the land. The province of the Theory of Numbers was 

 forbidding. Many a man returned empty-handed and batHed 

 from the pursuit, or else was drawn into the vortex of a kind 

 of maelstrom and had his heart crushed out of him. But early 

 in the last century the dawn of a brighter day was breaking. 

 A combination of great intellects — Legendre, Gauss, Eisenstein, 

 Stephen Smith, &c. — succeeded in adapting some of the exist- 

 ing instruments of research in continuous quantity to effective 

 use in discontinuous quantity. These adaptations are of so 

 difficult and ingenious a nature that they are to-day, at the 

 commencement of a new century, the wonder and, I may add, 

 the delight of beholders. True it is that the beholders are 



NO. 1663, VOL. 64] 



few. To attain to the point of vantage is an arduous task 

 demanding alike devotion and courage. I am reminded, to 

 take a geographical analogy, of the Hamilton Falls, near 

 Hamilton Inlet, in Labrador. I have been informed that 

 to obtain a view of this wonderful natural feature demands 

 so much time and intrepidity, and necessitates so many 

 collateral arrangements, that a few years ago only nine 

 white men had feasted their eyes on falls which are 

 finer than those of Niagara. The labours of the 

 mathematicians named have resulted in the formation of 

 a large body of doctrine in the Theory of Numbers. Much 

 that, to the superficial observer, appears to lie on the threshold 

 of the subject is found to be deeply set in it and to be only 

 capable of attack after problems at first sight much more com- 

 plicated have been solved. The mirage that distorted the 

 scenery and obscured the perspective has been to some extent 

 dissipated ; certain vanishing points have been ascertained ; 

 certain elevated spots giving extensive views have been either 

 found or constructed. The point I wish to urge is that these 

 specialists in the Theory of Numbers were successful for the 

 reason that they were not specialists at all in any narrow mean- 

 ing of the word. Success was only possible because of the wide ' 

 learning of the investigator ; because of his accurate knowledge 

 of the instruments that had been made effective in other 

 branches ; because he had grasped the underlying principles 

 which caused those instruments to be effective in particular 

 cases. I am confident that many a worker who has been the 

 mark of sneer and of sarcasm from the supposed extremely 

 special character of his researches would be found to have 

 devoted the larger portion of his time to the study of methods 

 which had been available in other branches, perhaps remote 

 from the one which was particularly attracting his attention. 

 He would be found to have realised that analogy is often the 

 finger-post that points the way to useful advance ; that his mind 

 had been trained and his work assisted by studying exhaustively 

 the successes and failures of his fellow-workers. But it is not 

 only existing methods that may be available in a special 

 research. 



Furthermore, a special study frequently creates new methods 

 which may be subsequently found applicable in ether branches. 

 The Theory of Numbers furnishes several beautiful illustrations 

 of this. Generally, the method is more important than the 

 immediate result. Though the result is the offspring of the 

 method, the method is the offspring of the search after 

 the result. The Law of (Quadratic Reciprocity, a corner- 

 stone of the edifice, stands out not only for the influence 

 it has exerted in many branches, but also for the number 

 of new methods to which it has given birth, which are 

 now a portion of the stock-in-trade of a mathematician. 

 Euler, Legendre, Gauss, Eisenstein, Jacobi, Kronecker, Poin- 

 care and Klein are great names that will be for ever associated 

 with it. Who can forget the work of H. J. S. Smith on homo- 

 geneous forms and on the five-square theorem, work which gave 

 rise to processes that have proved invaluable over a wide field, 

 and which supplied many connecting links between departments 

 which were previously in more or less complete isolation ? 



In this connection I will further mention two branches with 

 which I may claim to have a special acquaintance — the theory 

 of invariants and the combinatorial analysis. The theory of 

 invariants .was evolved by the combined efforts of Boole, Cayley, 

 Sylvester and Salmon, and has progressed during the last sixty 

 years with the cooperation, amongst others, of Aronhold, 

 Clebsch, Gordan, Brioschi, Lie, Klein, Poincare, Forsyth, 

 Hilbert, Elliott and Voung. It involves a principle which is 

 of wide significance in all the subject-matters of inorganic 

 science, of organic science, and of mental, moral and political 

 philosophy. In any subject of inquiry there are certain enti- 

 ties, the mutual relations of which under various conditions it 

 is desirable to ascertain. A certain combination of these enti- 

 ties may be found to have an unalterable value when the entities 

 are submitted to certain processes or are made the subjects of 

 certain operations. The theory of invariants in its widest 

 scientific meaning determines these combinations, elucidates 

 their properties, and expresses results when possible in terms 

 of them. Many of the general principles of political science 

 and economics can be expressed by means of invariantive rela- 

 tions connecting the factors which enter as entities into the 

 special problems. The great principle of chemical science which 

 asserts that when elementary or compound bodies combine with 

 1 one another the total vveight of the materials is unchanged. 



