*THE POPULAR SCIENCE MONTHLY*

*AB = A'B', AC = A'C', BC = B'C', AD = A'D', BD = B'D', CD = CD'.* It is evidently impossible in 3-space to put "*b*" in the position "*a*" or vice versa. It would be possible to make "*h*" coincide with the image of "*a*" in a mirror. In fact it is obvious that "*b*" is the image of "*a*" as seen in a mirror.

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 "*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 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 3-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 *x, y,* gives us a point in a plane. The solution of three simultaneous equations in three variables *x, y, z,* gives us a point in 3-space. The solution 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 necessary if we are to give a geometrical representation to the solution of four simultaneous equations, such as:

*a*_{1}*x* + *b*_{1}*y* + *c*_{1}*z* + *d*_{1}*w* = *e*_{1},

*a*_{2}*x* + *b*_{2}*y* + *c*_{2}*z* + *d*_{2}*w* = *e*_{2},

*a*_{3}*x* + *b*_{3}*y* + *c*_{3}*z* + *d*_{3}*w* = *e*_{3},

*a*_{4}*x* + *b*_{4}*y* + *c*_{4}*z* + *d*_{4}*w* = *e*_{4},

Again,

*x* = *a*

represent a point in 1-space. [Incidentally it would also denote a line in 2-space and a plane in 3-space.]