As a further illustration of the elusiveness of absolute shape and size, Poincaré asks us to conceive of a hollow spherical volume placed anywhere in space, and to assume that the temperature in the sphere decreases progressively from the centre, becoming absolute zero at the surface. He assumes that this hollow sphere is peopled by imaginary beings whose bodies expand and contract with the temperature and that all material bodies in the sphere behave in a similar manner. If we should supplement these suppositions by assuming that the refractive index of the medium in the sphere’s interior varies in a certain definite way, the rays of light in this hypothetical world would describe circles. This closed universe would of course appear infinite to its inhabitants, since as they proceeded from the centre to the surface their bodies would grow smaller, their steps shorter, so that it would be impossible for them to reach its boundary however long they walked. The geometricians of this imaginary world would feel justified in proceeding exactly as we have done ourselves. They would define as remaining congruent when displaced, hence as rigid, those bodies which appeared to them to remain the same wherever they carried them. Owing to the paths devised for the light rays and to the sameness in the reduction of the sizes of all objects as the centre was left behind, the expanding and contracting bodies of this universe would present all the characteristics of rigidity. On conducting measurements with their rigid rods the hypothetical beings would obtain non-Euclidean results, their entire world would appear to them as non-Euclidean, and non-Euclidean geometry would be as inevitable to them as Euclidean geometry is to the average layman. Some Kant among the hypothetical beings would surely arise and explain that non-Euclidean space was the a priori form of pure sensibility, transcending reason and experience. Then eventually some great mathematician would come along, sweep all those cobwebs aside, and prove that there existed other perfectly consistent types of geometries and that the ingrained preference of his fellow citizens for non-Euclidean geometry was due to the dictates of common experience and constituted by no means the a priori form of pure sensibility.