The fourth dimension
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54 THE FOURTH DIMENSION
shear world, there would evidently be a difference between his views of distance and ours.
We should say that his line AB increased in length in turning to ac. He would say that our line ar (fig. 23) decreased in length in turning to ac. He would think that what we called an equal line was in reality a shorter one.
We should say that a rod turning round would have its extremities in the positions we call at equal distances. So would he—but the positions would be different. He could, like us, appeal to the properties of matter. His rod to him alters as little as ours does to us.
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 Secu 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 ques-
, tion 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.