August 27, 1909] 



SCIENCE 



261 



attempts to deal with the vibrations of 

 strings and the conduction of heat; indeed, 

 it would seem that the most fniitful crop 

 of scientific ideas is produced by cross- 

 fertilization between the mind and some 

 definite fact, and that the mind by itself is 

 comparatively unproductive. 



I think, if we could trace the origin of 

 some of our most comprehensive and im- 

 portant scientific ideas, it would be found 

 that they arose in the attempt to find an 

 explanation of some apparently trivial and 

 very special phenomenon; when once 

 started the ideas grew to such generality 

 and importance that their modest origin 

 could hardlj' be suspected. Water vapor 

 we know will refuse to condense into rain 

 unless there are particles of dust to form 

 nuclei; so an idea before taking shape 

 seems to require a nucleus of solid fact 

 round which it can condense. 



I have ventured to urge the closer union 

 between mathematics and physics, because 

 I think of late yeai^s thei'e has been some 

 tendency for these sciences to drift apart, 

 and that the workers in applied mathemat- 

 ics are relatively fewer than they were 

 some years ago. This is no doubt due to 

 some extent to the remarkable develop- 

 ments made in the last few years in experi- 

 mental physics on the one hand and in the 

 most abstract and metaphysical parts of 

 pure mathematics on the other. The fas- 

 cination of these has drawn workers to the 

 frontiers of these regions who would other- 

 wise have worked nearer the junction of 

 the two. In part, too, it may be due to the 

 fact that the problems with which the ap- 

 plied mathematician has to deal are ex- 

 ceedingly difficult, 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 orig- 

 ination 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 pessess. 

 This is true for even the youngest of us, 

 for our school boys and school girls, 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 and as prac- 

 tical 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 sub- 

 ject. For some time I thought this esti- 

 mate was correct, but he happened to be 

 enthusiastic about billiards, and when we 

 were reading that part of mechanics which 

 deals with 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 mathematics as anything 

 but a means of annoying innocent under- 

 graduates; now, when he saw what impor- 

 tant results it could obtain, he became en- 



