CONCERNING DIMENSIONS GENERALEY, AND THE 

 EFFECTS OF THE FOURTH DIMENSION 



IN PARTICUEAR. 



r.v A. L. AXXISON. 



The word dimension is tlic ,L;encral term used to 

 indicate extension in space, whilst spare itself is 

 nothin.i; more than the abstract idea of infinite ex- 

 tension \n all known directions. 



There are said to be three dimensions, namel\-, 

 those of length, breadth and thickness, and all our 

 conceptions of shape and size are formed relatively 

 to one or more of these three, which seem quite 

 adequate for the expression of our ideas. The exist- 

 ence of others must, therefore, seem extremeh' 

 problematical, and if at times a guess is hazarded as 

 to the fourth dimension, its nature is assumed to be 

 quite different from that of the other three : so much 

 so, indeed, that in some instances it is held to consist 

 of time itself. 



Now it is one purpose of this article to show that 

 from certain facts which we shall proceed to 

 examine, the existence of an infinite number of 

 dimensions may justifiably be deduced. It will be 

 shown that these dimensions are of the same nature 

 as the three with which we are familiar, and that our 

 ignorance of their existence is due to some personal 

 limitation whereby we are unable to form any mental 

 image of the shape and size of a bodv, beyond that 

 small portion of it which is bounded b\- a surface 

 extending onl\- in the first three dimensions, the 

 others, therefore, seeming to be nothing more than 

 purely abstract ideas derived from mathematical 

 expressions. 



Notwithstanding this, however, we shall also be 

 able to show that some of the effects of the fourth 

 dimension are of such a nature as to be appreciable 

 in spite of our limitations ; and the latter part of this 

 article relates to the nature of these effects antl under 

 what conditions they can be produced. 



The basis of our argument depends on the 

 relation between algebraical equations of two and 

 three variables and geometrical figures of two and 

 three dimensions. It is known, for instance, that the 

 locus of a point moving according to the equation 

 x'^+y'^^r^> is the circumference of a circle, or in 

 other words the equation is the " law of the circle "' : 

 and similarly x'^+y'^+z^ = r'^ is that of the sphere; 

 but there are an infinite number of equations of this 

 type, each containing one more variable than the 

 preceding one ; and arguing by analogy from the 

 first two, the next one x^+y^+z^+u''^=r'' is the law 

 of a four-dimensioned figure : x'" + \--|-/'" + u'- + v" = r', 



of a five-dimensioned one, and generall}" x" + >'H 



(>; terms in all)=r" of an >j dimensioned one, where 

 rj may be anv number between 2 and c^. 



The fact that we are unable to form an\' mental 



image of such figures cannot be ascribed to the 

 equations themselves, which obviously contain no 

 reason either wh\' they should or should not be 

 capable of graphical representation. Prima ftjcic, 

 if some equations can be so treated, the remainder 

 should equall}- admit of such treatment, and our 

 inability to accomplish this must obviously be due 

 to the absence of an\- mental picture that will satisfx' 

 the requirements of the equations. 



Now the pictures that we can conceive consist of 

 nothing more than the figures formed b\- enclosing a 

 portion of space, either by a line or lines (plane 

 figures) or hv a surface or surfaces (solid figures) ; 

 the line and the surface are the materials with which 

 we construct these figures. We cannot, however, 

 construct a four-dimensioned figure of any kind, 

 although we have the necessary material, namely, 

 what for want of a better name must be designated 

 the volume. 



The inabilit\' of our imagination in this respect 

 must be due to some limitation in our powers of 

 perception, for it is from the impressions of external 

 objects, as seen by the e\-es, that we form our 

 mental images of figures, regular or irregular. 



Now it will be shown that our eyes cannot 

 appreciate an\- figure with more than three dimen- 

 sions. It might be argued from this, that it does not 

 follow that there are such figures to be seen, but 

 from an examination which we shall make of the 

 undoubted fact that all visible bodies b\' which we 

 are surrounded are at least three-dimensioned, the 

 existence of these more complex figures can be 

 inferred. 



By the term "body" employed above, is meant 

 something that has an independent existence and 

 is, therefore, an object as distinguished from an 

 attribute. 



All bodies, as we shall see, are defined by their 

 apparent size and shape and are, therefore, figures of 

 some kind, regular or irregular ; but on the other 

 hand all figures are not bodies, for every two- 

 dimensioned figure is onh' found as a bounding 

 surface of a three-dimensioned one, and by itself 

 is quite as abstract as colour, taste, and all other 

 attributes. 



So far as we know, however, the three-dimen- 

 sioned bodies appear to be self-contained and 

 independent of any higher dimensioned body; they 

 do not seem to exist merely as the sections of 

 four-dimensioned figures. 



It will be shown that this can be accounted for 

 by the limitation of our powers of observation, and 



ns 



