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THE POPULAR SCIENCE MONTHLY 



Fig. 10. 



AB^A'B', AC = A'C, BC = B'C, AB^A'B', BB^B'B', CB 

 = CB'. It is evidently impossible in 3-space to put "b" in the posi- 

 tion " a," or vice, versa. It would be possible to make " b " coincide with 

 the image of "a" in a mirror. In fact it is obvious that "6" is the 

 image of " a " as seen in a mirror. 



Eeaders of that classic nonsense book 

 by Lewis Carroll (Eev. C. L. Dodgson), 

 "Alice Behind the Looking-G-lass," will 

 be interested in the fact that Mr. Dodg- 

 son, himself a mathematician of no mean 

 note, is poking fun at the fourth-dimen- 

 sion students. 



Now, while it is impossible for a tridim 

 to make " b " take the position " a," there 

 would be no difficulty in a fourth-dimensional animal interchanging 

 " a " and " b." In other words, to make " a " and " b " coincide, one 

 must be taken up into 4-space, turned over and put down on the other. 

 It is easy now to see that, while there is no proof of the material 

 existence of 4-space or space of any dimension higher than three, and 

 while we can not even say that there is any likelihood that such exists, 

 yet the conception of hyperspace is a perfectly real and logical concep- 

 tion; moreover, it is by no means an idle question or a useless idea. 

 Assuming hyperspace, mathematicians have built up a perfectly con- 

 sistent geometry which throws much light upon problems of 3-space. 



We have seen that by many analogies it is a simple matter to con- 

 ceive of hyperspace. Let us next observe how algebra invites us to 

 consider the possible existence of higher space. 



The solution of two simultaneous equations in two variables x, y, 

 gives us a point in a plane. The solution of three simultaneous equa- 

 tions in three variables x, y, z, gives us a point in 3-space. The solu- 

 tion of four simultaneous equations in four variables, x, y, z, w, which 

 is easily performed, gives what ? Is there a geometrical equivalent here ? 

 Can the values of x, y, z, w be represented graphically? The answer 

 to both questions is No, at least not in our space. Four-space is neces- 

 sary if we are to give a geometrical representation to the solution of four 

 simultaneous equations, such as: 



dyX + T>iV + o ± g + d t w = e ± , 

 a 2 x + ~b 2 y + c^z -f- d 2 w = e 2 , 

 « 3 £ + & 3 2/ + On? + 3, % w = e 3 , 

 a t x + b 4 y -f- c^s + d^w = e t , 



Again, 



represent a point in 1-space. [Incidentally it would also denote a line 

 in 2-space and a plane in 3-space.] 



