The Foundations of Science: Science and Hypothesis, The Value of Science, Science and Method. Henri Poincare

are attributed by us to external objects; these are external changes;

      Others, opposite in character and attributed by us to the movements of our own body, are internal changes;

      2º We notice that certain changes of each of these categories may be corrected by a correlative change of the other category;

      3º We distinguish among external changes those which have thus a correlative in the other category; these we call displacements; and just so among the internal changes, we distinguish those which have a correlative in the first category.

      Thus are defined, thanks to this reciprocity, a particular class of phenomena which we call displacements.

      The laws of these phenomena constitute the object of geometry.

      Law of Homogeneity.—The first of these laws is the law of homogeneity.

      Suppose that, by an external change α, we pass from the totality of impressions A to the totality B, then that this change α is corrected by a correlative voluntary movement β, so that we are brought back to the totality A.

      Suppose now that another external change α´ makes us pass anew from the totality A to the totality B.

      Experience teaches us that this change α´ is, like α, susceptible of being corrected by a correlative voluntary movement β´ and that this movement β´ corresponds to the same muscular sensations as the movement β which corrected α.

      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 R² − r².

      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.

      Finally, I shall suppose that a body transported from one point to another of different temperature is put immediately into thermal equilibrium with its new environment.

      Nothing in these hypotheses is contradictory or unimaginable.

      A movable object will then become smaller and smaller in proportion as it approaches the limit-sphere.

      Note first that, though this world is limited from the point of view of our ordinary geometry, it will appear infinite to its inhabitants.

      In fact, when these try to approach the limit-sphere, they cool off and become smaller and smaller. Therefore the steps they take are also smaller and smaller, so that they can never reach the limiting sphere.

      If, for us, geometry is only the study of the laws according to which rigid solids move, for these imaginary beings it will be the study of the laws of motion of solids distorted by the differences of temperature just spoken of.

      No doubt, in our world, natural solids likewise undergo variations of form and volume due to warming or cooling. But we neglect these variations in laying the foundations of geometry, because, besides their being very slight, they are irregular and consequently seem to us accidental.

      In our hypothetical world, this would no longer be the case, and these variations would follow regular and very simple laws.

      Moreover, the various solid pieces of which the bodies of its inhabitants would be composed would undergo the same variations of form and volume.

      I will make still another hypothesis; I will suppose light traverses media diversely refractive and such that the index of refraction is inversely proportional to R² − r². It is easy to see that, under these conditions, the rays of light would not be rectilinear, but circular.

      To justify what precedes, it remains for me to show that certain changes in the position of external objects can be corrected by correlative movements of the sentient beings inhabiting this imaginary world, and that in such a way as to restore the primitive aggregate of impressions experienced by these sentient beings.

      Suppose in fact that an object is displaced, undergoing deformation, not as a rigid solid, but as a solid subjected to unequal dilatations in exact conformity to the law of temperature above supposed. Permit me for brevity to call such a movement a non-Euclidean displacement.

      If a sentient being happens to be in the neighborhood, his impressions will be modified by the displacement of the object, but he can reestablish them by moving in a suitable manner. It suffices if finally the aggregate of the object and the sentient being, considered as forming a single body, has undergone one of those particular displacements I have just called non-Euclidean. This is possible if it be supposed that the limbs of these beings dilate according to the same law 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 their various parts are no longer in the same relative position, nevertheless we shall see that the impressions of the sentient being have once more become the same.

      In fact, though the mutual distances of the various parts may have varied, yet the parts originally in contact are again in contact. Therefore the tactile impressions have not changed.

      On the other hand, taking into account the hypothesis made above in regard to the refraction and the curvature of the rays of light, the visual impressions will also have remained the same.

      These imaginary beings will therefore like ourselves be led to classify the phenomena they witness and to distinguish among them the 'changes of position' susceptible of correction by a correlative voluntary movement.

      If they construct a geometry, it will not be, as ours is, the study of the movements of our rigid solids; it will be the study of the changes of position which they will thus have distinguished and which are none other than the 'non-Euclidean displacements'; it will be non-Euclidean geometry.

      Thus beings like ourselves, educated in such a world, would not have the same geometry as ours.

      The