. This can be turned into its symmetrical form , the lower half of (a), by opening it out straight and bending it over the other way so that it is turned inside out. This process takes place entirely in the plane and can be performed by a two-dimensional being. The polygon may also be changed into its symmetrical form (b), Fig. 12, by being turned over, in -space, but in this process it is not turned inside out at all. On the other hand, if it is sufficiently flexible, it may be turned inside out by twisting each part upon itself through 180 degrees, and in this process it is not changed into its symmetrical form.
When mathematicians began to talk of higher space, the spiritualists seized upon the idea as affording a habitation for their spirits. These men, naturally wanting a home for their spirits, were rather too eager to believe in the actual existence of the fourth dimension. It is astonishing with what avidity the advocates of spirit rappings and occult demonstrations appropriated the fourth dimension for the abiding place of their unearthly beings. This was, of course, unwarranted as are perhaps most of the claims of such people. While somewhat interesting, it is too trivial to claim our serious attention.
In conclusion, we have no material evidence of a fourth dimension. Our knowledge of the phenomena of -space is empirical. Our experience tells us nothing of -space, if it exists. But the conception, not being dependent upon experience or experiment, is not unreasonable. As a working hypothesis it is not without decided value, as it throws light upon many propositions of our (-space) geometry.
The existence of -space might explain certain phenomena in physics and chemistry; for instance, rotation in hyperspace would explain the changes of a body producing a right-handed polarization of light into one giving a left-handed.
A few months ago an article appeared in the Scientific American by E. L. DuPuy setting forth the use of four dimensions in representing certain chemical compounds graphically. He took as an example a "special steel" consisting of iron, carbon, silicon-manganese and nickel-vanadium.
In this short sketch of what is meant by the fourth dimension, it must be borne in mind that the mathematical investigation of the geometry of the fourth dimension has been omitted altogether. It is hardly necessary to add that all arguments for the existence of a fourth dimension apply equally well for the existence of 5, 6, or dimensional space. The geometry of -space, where is any number, is just as logical as that of -space.
[In this paper the author claims no originality, except to some extent in the mode of presentation and in the manner of introducing the illustrations; but he has not knowingly made use of any ideas that
