Suppose that, by an external change [alpha], we pass from the totality of impressions _A_ to the totality _B_, then that this change [alpha] is corrected by a correlative voluntary movement [beta], so that we are brought back to the totality _A_. Suppose now that another external change [alpha]' makes us pass anew from the totality _A_ to the totality _B_. Experience teaches us that this change [alpha]' is, like [alpha], susceptible of being corrected by a correlative voluntary movement [beta]' and that this movement [beta]' corresponds to the same muscular sensations as the movement [beta] which corrected [alpha]. This fact is usually enunciated by saying that _space is homogeneous and isotropic_. It may also be said that a movement which has once been produced may be repeated a second and a third time, and so on, without its properties varying. In the first chapter, where we discussed the nature of mathematical reasoning, we saw the importance which must be attributed to the possibility of repeating indefinitely the same operation. It is from this repetition that mathematical reasoning gets its power; it is, therefore, thanks to the law of homogeneity, that it has a hold on the geometric facts. For completeness, to the law of homogeneity should be added a multitude of other analogous laws, into the details of which I do not wish to enter, but which mathematicians sum up in a word by saying that displacements form 'a group.' THE NON-EUCLIDEAN WORLD.--If geometric space were a frame imposed on _each_ of our representations, considered individually, it would be impossible to represent to ourselves an image stripped of this frame, and we could change nothing of our geometry. But this is not the case; geometry is only the résumé of the laws according to which these images succeed each other. Nothing then prevents us from imagining a series of representations, similar in all points to our ordinary representations, but succeeding one another according to laws different from those to which we are accustomed. We can conceive then that beings who received their education in an environment where these laws were thus upset might have a geometry very different from ours. Suppose, for example, a world enclosed in a great sphere and subject to the following laws: The temperature is not uniform; it is greatest at the center, and diminishes in proportion to the distance from the center, to sink to absolute zero when the sphere is reached in which this world is enclosed. To specify still more precisely the law in accordance with which this temperature varies: Let _R_ be the radius of the limiting sphere; let _r_ be the distance of the point considered from the center of this sphere. The absolute temperature shall be proportional to _R_^{2} - _r_^{2}. I shall further suppose that, in this world, all bodies have the same coefficient of dilatation, so that the length of any rule is proportional to its absolute temperature.