NA TURE 



[August 26, 1909 



visible. We have here one of the most entrancing problems 

 in chemistry and physics it is possible to conceive. 



.Again, the specialisation prevalent in schools often pre- 

 vents students of science from acquiring sufficient Ijnow- 

 ledge of mathematics ; it is true that most of those who 

 study physics do some mathematics, but I hold that, in 

 general, they do not do enough, and that they are not 

 as efticient physicists as they would be if they had a 

 wider knowledge of that subject. There seems at present 

 a tendency in some quarters to discourage the use of 

 mathematics in physics ; indeed, one might infer, from 

 the statements of some writers in quasi-scientific journals, 

 that ignorance of mathematics is almost a virtue. If this 

 is so, then surely of all the virtues this is the easiest and 

 most prevalent. 



I do not for a moment urge that the physicist should 

 confine himself to looking at liis problems from the mathe- 

 matical point of view ; on the contrary, I tliink a famous 

 French malhematiciaji and physicist was guilty of only 

 slight exaggeration when he said that no discovery was 

 really important or properly understood by its author 

 unless and until he could explain it to the first man he 

 met in the street. 



But two points of view are better than one, and the 

 pliysicist who is also a mathematician possesses a most 

 powerful instrument for scientific research with which 

 many of the greatest discoveries have been made ; for ex- 

 ample, electric waves were discovered by mathematics 

 long before they were detected in the laboratory. He has 

 also at his command a language clear, concise, and 

 universal, and there is no better way of detecting ambigui- 

 ties and discrepancies in his ideas than by trying to express 

 them in this language. Again, it often happens that we 

 are not able to appreciate the full significance of some 

 physical discovery until we have subjected it to mathe- 

 matical treatment, when we find that the effect we have 

 discovered involves other effects which have not been 

 detected, and we are able by this means to duplicate the 

 discovery. Thus James Thomson, starting from the fact 

 that ice fioats on water, showed that it follows bv mathe- 

 matics that ice can be melted and water prevented from 

 freezing by pressure. This effect, which was at that time 

 unknown, was afterwards verified by his brother, Lord 

 Kelvin. Multitudes of similar duplication of physical dis- 

 coveries by mathematics could be quoted. 



I have been pleading in the interests of physics for a 

 greater study of mathematics by physicists.' I would also 

 plead for a greater study of physics by mathematicians in 

 the interest of pure mathematics. 



The history of pure mathematics shows that many of 

 the most important branches of the subject have arisen 

 from the attempts made to get a mathematical .solution of 

 a problem suggested by physics. Thus the differential 

 calculus arose from attempts to deal with the problem of 

 moving bodies. Fourier's theorem resulted from attempts 

 to deal with the vibrations of strings and the conduction 

 of heat; indeed, it would seem that the most fruitful 

 crop of scientific ideas is produced bv cross-fertilisation 

 between the mind and some definite fact, and that the 

 mind bv itself is comparatively unproductive. 



I think, if we could trace the origin of some of our 

 most comorehensive and important scientific ideas, it 

 would he found that they arose in the attempt to find an 

 explanation of some apparently trivia! and very special 

 phenomenon ; when once started the ideas grew to such 

 generality and importance that their modest origin could 

 hardly be suspected. Water vapour we know will refuse 

 to condense into rain unless there are particles of dust to 

 form nuclei ; so an idea before t.iking shape seems to re- 

 quire a nucleus of solid fact round which it can condense. 



1 have ventured to urge the closer union between mathe- 

 matics and physics, because 1 think of late years there 

 has been some tendency for these sciences to drift nnart, 

 and that the workers in applied mathematics are relatively 

 f"wer than they were some years aeo. This is no doubt 

 duf to soT:te extent to the remarkable developments made 

 m the Inst few years in experimental physics on the one 

 hind .and in the most abstract and metaphysical parts 

 of pure mathematics on the other. The fascination of 

 these has di;awn workers to the frontiers of these regions 

 who woMld otherwise have worked nearer the junction of | 



NO. 2078, VOL. Si] 



the two. In part, too, it may be due 10 the fact that 

 the problems with which the applied mathematician has 

 to deal are exceedingly difiicult, and many may have felt 

 that the problems presented by the older physics have been 

 worked over so often by men of the highest genius that 

 there was but little chance of any problem which they 

 could have any hope of solving being left. 



But the newer developments of physics have opened 

 virgin ground which has not yet been worked over, and 

 which offers problems to the mathematician of great 

 interest and novelty — problems which will suggest and 

 require new methods of attack, the development of which 

 will advance pure mathematics as well as physics. 



I have alluded to the fact that pure mathematicians 

 have been indebted to the study of concrete problems for 

 the origination of some of their most valuable conceptions ; 

 but though no doubt pure mathematicians are in many 

 ways very exceptional folk, yet in this respect they are 

 very human. Most of us need to tackle some definite 

 difficulty before our minds develop whatever powers they 

 may possess. This is true for even the youngest of us, 

 for our schoolbovs and schoolgirls, and I think the 

 moral to be drawn from it is that we should aim at 

 making the education in our schools as little bookish ainl 

 as practical and concrete as possible. 



I once had an illustration of the power of the concrete 

 in stimulating the mind which made a very lasting 

 impression upon me. One of my first pupils came to me 

 with the assurance from his previous teacher that he 

 knew little and cared less about mathematics, and that 

 he had no chance of obtaining a degree in that subject. 

 For some time I thought this estimate was correct, but 

 he happened to be enthusiastic about billiards, and when 

 we were reading that part of mechanics which deals wi;h 

 the collision of elastic bodies I pointed out that many of 

 the effects he was constantly observing were illustrations 

 of the subject we were studying. From that time he was 

 a changed man. He had never before regarded mathe- 

 matics as anvthing but a means of annoying innocent 

 undergraduates ; now, when he saw what important results 

 it v"ould obtain, he became enthusiastic about it, developed 

 very considerable mathematical ability, and, though he h.ad 

 already wasted two out of his three years at college, took 

 a good place in the Mathematical Tripos. 



It is possible to read books, to pass examinations with- 

 out the higher qualities of the mind being called into play. 

 Indeed, I doubt if there is any process in which the mind 

 is more quiescent than in reading without interest. I 

 might appeal to the widespread habit of reading in bed 

 as a prevention of insomnia as a proof of this. But it is 

 not possible for a boy to make a boat or for a girl to 

 cook a dinner without using their brains. With practical 

 things the difficulties have to be surmou.Tted, the boat 

 must be made watertight, the dinner must be cooked, 

 while in reading there is always the hope that the difficul- 

 ties which have been slurred over will not be set in the 

 examination. 



I think it was Helmholtz who said that often in the 

 course of a research more thought and energy were spent 

 in reducing a refractory piece of brass to order than in 

 devising the method or planning the scheme of campaign. 

 This constant need for thought and action gives to 

 original research in any branch of experimental science 

 great educational value even for those who will not become 

 professional men of science. I have had considerable ex- 

 perience with students beginning research in experimental 

 physics, and I have always been struck by the quite re- 

 markable improvement in judgment, independence of 

 thought and maturity produced by a year's research. 

 Research develops qualities which are apt to atrophy when 

 the student is preparing for examinations, and, quite apart 

 from the addition of new knowledge to our store, is of 

 the greatest importance as a means of education. 



It is the practice in many universities to make special 

 provision for the reception of students from other universi- 

 ties w^ho wish to do original research or to study the more 

 advanced parts of their subject, and considerable numbers 

 of such ■ students migrate from one university to another. 

 I think it would be a good thing if this practice were to 

 extend to students at an earlier stage in the'rr career; 

 especially should I lik* to see a considerable interchange 



