Now, is there any standard to which we could appeal, to say which of the two is right in this argument? There is no standard. We should say that, with a change of position, the configuration and shape of his objects altered. He would say that the configuration and shape of our objects altered in what we called merely a change of position. Hence distance independent of position is inconceivable, or practically distance is solely a property of matter. There is no principle to which either party in this controversy could appeal. There is nothing to connect the definition of distance with our ideas rather than with his, except the behaviour of an actual piece of matter. For the study of the processes which go on in our world the definition of distance given by taking the sum of the squares is of paramount importance to us. But as a question of pure space without making any unnecessary assumptions the shear world is just as possible and just as interesting as our world. It was the geometry of such conceivable worlds that Lobatchewsky and Bolyai studied. This kind of geometry has evidently nothing to do directly with four-dimensional space. But a connection arises in this way. It is evident that, instead of taking a simple shear as I have done, and defining it as that change of the arrangement of the particles of a solid which they will undergo without offering any resistance due to their mutual action, I might take a complex motion, composed of a shear and a rotation together, or some other kind of deformation. Let us suppose such an alteration picked out and defined as the one which means simple rotation, then the type, according to which all bodies will alter by this rotation, is fixed. Looking at the movements of this kind, we should say that the objects were altering their shape as well as rotating. But to the inhabitants of that world they would seem to be unaltered, and our figures in their motions would seem to them to alter. In such a world the features of geometry are different. We have seen one such difference in the case of our illustration of the world of shear, where the square on the hypothenuse was equal to the difference, not the sum, of the squares on the sides. In our illustration we have the same laws of parallel lines as in our ordinary rotation world, but in general the laws of parallel lines are different. In one of these worlds of a different constitution of matter through one point there can be two parallels to a given line, in another of them there can be none, that is, although a line be drawn parallel to another it will meet it after a time.