662 SCIENCE AND HYPOTHESIS 



a solid subjected to unequal dilatations in exact conformity with 

 the law of temperature assumed above. To use an abbreviation, we 

 shall call such a movement a non-Euclidean displacement. 



If a sentient being be in the neighborhood of such a displacement 

 of the object, his impressions will be modified; but by moving in a 

 suitable manner, he may reconstruct them. For this purpose, all that 

 is required is that the aggregate of the sentient being and the object, 

 considered as forming a single body, shall experience one of those 

 special displacements which I have just called non-Euclidean. This 

 is possible if we suppose that the limbs of these beings dilate accord- 

 ing to the same laws as the other bodies of the world they inhabit. 



Although from the point of view of our ordinary geometry there is 

 a deformation of the bodies in this displacement, and although their 

 different parts are no longer in the same relative position, neverthe- 

 less we shall see that the impressions of the sentient being remain the 

 same as before; in fact, though the mutual distances of the different 

 parts have varied, yet the parts which at first were in contact are 

 still in contact. It follows that tactile impressions will be unchanged. 

 On the other hand, from the hypothesis as to refraction and the curva- 

 ture of the rays of light, visual impressions will also be unchanged. 

 These imaginary beings will therefore be led to classify the phenomena 

 they observe, and to distinguish among them the " changes of posi- 

 tion," which may be corrected by a voluntary correlative movement, 

 just as we do. 



If they construct a geometry, it will not be like ours, which is the 

 study of the movements of our invariable solids; it will be the study 

 of the changes of position which they will have thus distinguished, and 

 will be " non-Euclidean displacements," and this will be non-Euclidean 

 geometry. So that beings like ourselves, educated in such a world, 

 will not have the same geometry as ours. 



The World of Four Dimensions. Just as we have pictured to our- 

 selves a non-Euclidean world, so we may picture a world of four di- 

 mensions. 



The sense of light, even with one eye, together with the muscular 

 sensations relative to the movements of the eyeball, will suffice to 

 enable us to conceive of space of three dimensions. The images of 

 external objects are painted on the retina, which is a plane of two 

 dimensions; these are perspectives. But as eye and objects are mov- 

 able, we see in succession different perspectives of the same body 

 taken from different points of view. We find at the same time that 

 the transition from one perspective to another is often accompanied 

 by muscular sensations. If the transition from the perspective A 

 to the perspective B, and that of the perspective A' to the perspec- 

 tive B' are accompanied by the same muscular sensations, we con- 

 nect them as we do other operations of the same nature. Then when 



