THE VALUE OF SCIENCE 399 



equally the intuitive idea of the side of the non-Euclidean triangle. 

 Why should I have the right to apply the name of straight to the first 

 of these ideas and not to the second ? Wherein does this syllable form 

 an integrant part of this intuitive idea? Evidently when we say that 

 the Euclidean straight is a true straight and that the non-Euclidean 

 straight is not a true straight, we simply mean that the first intuitive 

 idea corresponds to a more noteworthy object than the second. But 

 how do we decide that this object is more noteworthy? This question 

 I have investigated in ' Science and Hypothesis.' 



It is here that we saw experience come in. If the Euclidean straight 

 is more noteworthy than the non-Euclidean straight, it is so chiefly 

 because it differs little from certain noteworthy natural objects from 

 which the non-Euclidean straight differs greatly. But, it will be said, 

 the definition of the non-Euclidean straight is artificial; if we for a 

 moment adopt it, we shall see that two circles of different radius both 

 receive the name of non-Euclidean straights, while of two circles of 

 the same radius one can satisfy the definition without the other being 

 able to satisfy it, and then if we transport one of these so-called 

 straights without deforming it, it will cease to be a straight. But by 

 what right do we consider as equal these two figures which the Euclidean 

 geometers call two circles with the same radius? It is because by 

 transporting one of them without deforming it we can make it coincide 

 with the other. And why do we say this transportation is effected 

 without deformation? It is impossible to give a good reason for it. 

 Among all the motions conceivable, there are some of which the 

 Euclidean geometers say that they are not accompanied by deforma- 

 tion ; but there are others of which the non-Euclidean geometers would 

 say that they are not accompanied by deformation. In the first, called 

 Euclidean motions, the Euclidean straights remain Euclidean straights, 

 and the non-Euclidean straights do not remain non-Euclidean straights ; 

 in the motions of the second sort, or non-Euclidean motions, the non- 

 Euclidean straights remain non-Euclidean straights and the Euclidean 

 straights do not remain Euclidean straights. It has, therefore, not 

 been demonstrated that it was unreasonable to call straights the sides 

 of non-Euclidean triangles; it has only been shown that that would 

 be unreasonable if one continued to call the Euclidean motions motions 

 without deformation; but it has at the same time been shown that it 

 would be just as unreasonable to call straights the sides of Euclidean 

 triangles if the non-Euclidean motions were called motions without 

 deformation. 



Now when we say that the Euclidean motions are the true motions 

 without deformation, what do we mean? We simply mean that they 

 are more noteworthy than the others. And why are they more note- 



