TEE FOURTH DIMENSION 383 



other than our common, every day three-space. Fortunately a com- 

 parison with lower dimensional geometries furnishes so many analogies 

 that the subject can be very fully explained in a non-mathematical way. 

 Only let me say just here that the geometry of the fourth dimension is 

 a perfectly logical system of theorems and proofs entirely independent 

 of these analogies. 



We, the dwellers in 3-space, can best realize the reasonableness of 

 conceiving of a fourth or higher dimensional space by considering as 

 best we may what would take place in lower-dimensional space did such 

 exist. 



Consider a pipe of indefinite length with a bore of diameter as small 

 as you please, and suppose that there dwell within this pipe " worms " 

 of such diameter that they just fill the pipe. We can not conceive of 

 anything with no breadth or thickness, but let us consider for sake of 

 the illustration that this one-dimensional animal (which for brevity 

 I shall call a unodim) has only length. Of course these unodims may 

 vary in length according to age or family traits, perhaps. Now it is 

 evident that a unodim can never turn around. He may move forward 

 or backward, but one unodim can never pass another. If he possesses 

 an eye in front or behind he can see a neighboring unodim as a mere 

 point. His world is a very limited one. 



Again, we might imagine a two-dimensional animal, taking hold 



Fig. 1. 



of a unodim, turning him around in his (two-dimensional) space and 

 putting him back with his " tail " where his " head " was before. Evi- 

 dently the unodim would be ignorant of the cause of his reversion, for 

 he has no knowledge of a two-dimensional 

 space, and the two-dimensional animal is 

 invisible to him. In other words, if AB and 

 A'B' in the figure are equal in length but 

 running in opposite directions, it is impos- 

 sible to put A'B' in the place of AB, that is, 

 A' where A is and B' where B is. To accom- 

 plish this, it would be necessary to take A'B' 

 into 2-space and turn it around. While this 

 would be an impossible feat for a unodim, a fi G . 2. 



two-space animal could readily do it. 



Now this one-dimensional space may not be " straight " (that is, of 

 zero curvature) ; but it may be the space that we should get by bending 

 the pipe around in the form of a circle, as in Fig. 2. In such a case, 

 as his body would be constantly bent in the same direction and by the 

 same degree, we may suppose that the unodim is totally unconscious of 



