392 THE POPULAR SCIENCE MONTHLY 



ABCDEF. This can be turned into its symmetrical form A' BCD' E' F', 

 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 (&), Fig. 12, 

 by being turned over, in 3-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 astonish- 

 ing with what avidity the advocates of spirit rappings and occult demon- 

 strations 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 3-space is empirical. Our experi- 

 ence tells us nothing of 4-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 (3-space) geometry. 



The existence of 4-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 geom- 

 etry of the fourth dimension has been omitted altogether. It is hardly 

 necessary to add that all arguments for the existence of a fourth dimen- 

 sion apply equally well for the existence of 5, 6, or n dimensional space. 

 The geometry of w-space, where n is any number, is just as logical as 

 that of 4-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 



