48 SECTIONAL ADDRESSES. 



the first may be too narrow a view and lead progress into a cul-de-sac ; 

 the second may be so broad that it is useless as a guide, and diverges 

 indefinitely from experimental knowledge. When this last case 

 happens, it must be concluded that the knowledge is not yet ripe for 

 theoretical ti'eatment and speculation is premature. The time when 

 speculative theory and observational research may profitably go hand 

 in hand is when the possibilities, or at any rate the probabilities, can 

 be narrowed down by experiment, and the theory can indicate 

 the tests by which the remaining wrong paths may be blocked up one 

 by one. 



The mathematical physicist is in a position of peculiar difficulty. 

 He may work out the behaviour of an ideal model of material with 

 specifically defined properties, obeying mathematically exact laws, and 

 so far his work is unimpeachable. It is no more speculative than 

 the binomial theorem. But when he claims a serious intei'est for 

 his toy, when he suggests that his model is like something going on in 

 Nature, he inevitably begins to speculate. Is the actual body really 

 like the ideal model? May not other unknown conditions intervene? 

 He cannot be sure, but he cannot suppress the comparison; for it is by 

 looking continually to Nature that he is guided in his choice of a sub- 

 ject. A common fault, to which he must often plead guilty, is to use 

 for the comparison data over which the more experienced observer 

 shakes his head; they are too insecure to build extensively upon. Yet 

 even in this, theory may help observation by showing the kind of data 

 which it is especially important to improve. 



I think that the more idle kinds of speculation will be avoided if 

 the investigation is conducted from the right point of view. "When the 

 properties of an ideal model have been worked out by rigorous mathe- 

 matics, all the underlying assumptions being clearly understood, then 

 it becomes possible to say that such and such properties and laws lead 

 precisely to such and such effects. If any other disregarded factors 

 are present, they should now betray themselves when a^ comparison is 

 made with Nature. There is no need for disappointment at the failure 

 of the model to give perfect agreement with observation ; it has served 

 its purpose, for it has distinguished what are the features of the actual 

 phenomena which require new conditions for their explanation. A 

 general preliminary agreement with observation is necessary, otherwise 

 the model is hopeless ; not that it is necessarily wrong so far as it goes, 

 but it has evidently put the less essential properties foremost. We 

 have been pulling at the wrong end of the tangle, which has to be un- 

 ravelled by a different approach. But after a general agreement with 

 obsei-vation is established, and the tangle begins to loosen, we should 

 always make ready for the next knot. I suppose that the applied 

 mathematician whose theory has just passed one still more stringent test 

 by observation ought not to feel satisfaction, but rather disappointment 

 — ' Foiled again ! This time I had hoped to find a discordance which 

 would throw light on the points where my model could be improved. ' 

 Perhaps that is a counsel of perfection; I own that I have never felt 

 very keenly a disappointment of this kind. 



Our model of Nature should not be like a building^a handsome 



