*THE POPULAR SCIENCE MONTHLY*

. It is evidently impossible in -space to put "" in the position "" or *vice versa*. It would be possible to make "" coincide with the image of "" in a mirror. In fact it is obvious that "" is the image of "" as seen in a mirror.

Fig. 10. Readers of that classic nonsense book by Lewis Carroll (Rev. C. L. Dodgson), "Alice Behind the Looking-Glass," will be interested in the fact that Mr. Dodgson, himself a mathematician of no mean note, is poking fun at the fourth-dimension students.

Now, while it is impossible for a tridim to make "" take the position "," there would be no difficulty in a fourth-dimensional animal interchanging "" and "." In other words, to make "" and "" coincide, one must be taken up into -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 -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 conception; moreover, it is by no means an idle question or a useless idea. Assuming hyperspace, mathematicians have built up a perfectly consistent geometry which throws much light upon problems of -space.

We have seen that by many analogies it is a simple matter to *conceive* 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 , gives us a point in a plane. The solution of three simultaneous equations in three variables , gives us a point in -space. The solution of four simultaneous equations in four variables, , which is easily performed, gives what? Is there a geometrical equivalent here? Can the values of be represented graphically? The answer to both questions is *No,* at least not in our space. Four-space is necessary if we are to give a geometrical representation to the solution of four simultaneous equations, such as:

Again,

represent a point in -space. [Incidentally it would also denote a line in -space and a plane in -space.]