NATURE 703 



important not to multiply hypotheses indefinitely. If we construct a 

 theory based upon multiple hypotheses, and if experiment condemns it, 

 which of the premisses must be changed? It is impossible to tell. 

 Conversely, if the experiment succeeds, must we suppose that it has 

 verified all these hypotheses at once? Can several unknowns be de- 

 termined from a single equation? 



We must also take care to distinguish between the different kinds of 

 hypotheses. First of all, there are those which are quite natural and 

 necessary. It is difficult not to suppose that the influence of very dis- 

 tant bodies is quite negligible, that small movements obey a linear 

 law, and that effect is a continuous function of its cause. I will 

 say as much for the conditions imposed by symmetry. All these 

 hypotheses affirm, so to speak, the common basis of all the theories of 

 mathematical physics. They are the last that should be abandoned. 

 There is a second category of hypotheses which I shall qualify as indif- 

 ferent. In most questions the analyst assumes, at the beginning of 

 his calculations, either that matter is continuous, or the reverse, that 

 it is formed of atoms. In either case, his results would have been the 

 same. On the atomic supposition he has a little more difficulty in 

 obtaining them that is all. If, then, experiment confirms his con- 

 clusions, will he suppose that he has proved, for example, the real 

 existence of atoms? 



In optical theories two vectors are introduced, one of which we 

 consider as a velocity and the other as a vortex. This again is an 

 indifferent hypothesis, since we should have arrived at the same con- 

 clusions by assuming the former to be a vortex and the latter to be 

 a velocity. The success of the experiment cannot prove, therefore, that 

 the first vector is really a velocity. It only proves one thing namely, 

 that it is a vector ; and that is the only hypothesis that has really been 

 introduced into the premisses. To give it the concrete appearance 

 that the fallibility of our minds demands, it was necessary to consider 

 it either as a velocity or as a vortex. In the same way, it was neces- 

 sary to represent it by an a; or a y. but the result will not prove that 

 we were right or wrong in regarding it as a velocity; nor will it 

 prove we are right or wrong in calling it x and not y. 



These indifferent hypotheses are never dangerous provided their 

 characters are not misunderstood. They may be useful, either as arti- 

 fices for calculation, or to assist our understanding by concrete image-s, 

 to fix the ideas, as we say. They need not therefore be rejected. The 

 hypotheses of the third category are real generalizations. They must 

 be confirmed or invalidated by experiment. Whether verified or con- 

 demned, they will always be fruitful; but, for the reasons I rjave 

 given, they will only be so if they are not too numerous. 



Origin of Mathematical Physics. Let us go further and study 

 more closely the conditions which have assisted the development of 



