THE FOURTH DIMENSION AND ITS BEARING ON 

 THE CAUSE OF UNIVERSAL GRAVITATION. 



By A. G. BLAKE, F.R.A.S. 



Our ideas of the dimensions of a body are very 

 largely derived from the circumstances in which 

 these dimensions may undergo variation. Thus we 

 speak of a piece of paper as being of two dimensions 

 because of the great difficulty of changing its thick- 

 ness compared with the difficulty of changing its 

 length or breadth. 



In " flatland " — a hypothetical region where 

 motion confined to two dimensions only is possible — 

 it is quite conceivable — nay, it is a necessary assump- 

 tion if we are to allow the possibility of concrete 

 bodies in it — that bodies should have a certain 

 thickness in a third dimension which would be 

 invariable in individual bodies, but not necessarily 

 uniform among different bodies. Thus the sum 

 total thickness of bodies in " flatland " would be 

 fixed and invariable. To the inhabitants, who would 

 be incapable of realising thickness, this would result 

 in the conservation of some physical attribute 

 peculiar to bodies of two-dimensional space. 



In seeking evidence of a fourth dimension, then, 

 we must draw our inferences from the conservation 

 of some physical attribute peculiar to three- 

 dimensional space. The most obvious — indeed, the 

 only one — is the Conservation of Mass. We cannot, 

 however, infer that mass is the three-dimensional 

 perception of a four-dimensional thickness ; for the 

 mass of a body is directly alterable by changing its 

 three known dimensions by simply cutting or 

 breaking the body. If we change only three 

 dimensions of a four-dimensional body the fourth 

 must remain unchanged. Suppose (L), (B), (T) to be 

 the units of length, breadth, and thickness, (M) the 

 unit of mass and (F) the unit of " fourth dimension." 



Then, since mass varies directly as volume, 



(M) = m (L), (B), (T), when m is some constant. 



But since the fourth-dimensional unit is a constant 

 for any one body, and (F) is the unit, we may put 



Therefore (F) = 



(M) = (F), (L), (B), (T). 

 (M) 



(L), (B), (T) 



volume 



But 



mass 

 volume 



is what we call density. 



Accordingly, in our three-dimensional universe 

 every body has a thickness in a fourth dimension, 

 which is variable in different bodies, but invariable in 

 the same body, and that fourth-dimensional thick- 

 nest is the body's density. 



That this fits in perfectly with analogies drawn 



from two-space is easily shown. Thus in " flatland " 

 we may consider a two-dimensional body with 

 a small thickness in the third dimension. A 

 " flatlander " would cut down its length and its 

 breadth, but would be powerless to alter its thickness, 

 so that its volume would vary as its area. Extend- 

 ing this to three-dimensional space, we may cut down 

 a body in three dimensions — length, breadth, and 

 thickness — but we cannot alter what we may call its 

 fourth or extent in a fourth dimension, so that its 

 mass varies as its volume. In fact, mass in four- 

 space corresponds to volume in three-space and 

 area in two-space. The volume of a three- 

 dimensional body is infinity times as great as the 

 volume of a two-dimensional body. The mass of a 

 four-dimensional body is infinity times as great as 

 the mass of a three-dimensional body (i.e., one 

 whose density is nil). 



Though we cannot directly change the extent of 

 a body in its fourth dimension we can do so indirectly 

 by taking advantage of the principle of the conserva- 

 tion of mass and compressing the body in three 

 dimensions. The two-dimensional equivalent to 

 this is that in two-space, though it is impossible 

 directly to alter the third dimension ; yet by 

 compressing it in two dimensions the third will be 

 increased, while the volume will remain constant. 

 For in two-space the chief physical principle would 

 be the conservation of volume, though under what 

 aspect volume would present itself to " flatlanders " 

 we can never tell. 



Having shown how a body's density ma)' be our per- 

 ception of its thickness in a fourth dimension, I shall 

 endeavour to explain the cause of Universal Gravita- 

 tion — why any two bodies in space will attract each 

 other, and why the force of attraction will vary 

 directly as the mass and inversely as the square of 

 the distance separating them. 



For the purposes of my theory I assume that 

 matter is surrounded on all sides by an ether of vast 

 extent in every direction — in all four dimensions. I 

 shall show hereafter that the very great density 

 implied in the last clause is not incompatible with 

 absence of gravitation power. In this ether an 

 immense number of waves is being propagated in 

 every possible direction ; the waves themselves may- 

 be small, but their number renders them capable of 

 exerting a finite and constant pressure at every 

 point. Consider a body poised freely in space. For 

 simplicity let it be a homogeneous sphere. This 

 body will cast a penumbral shadow (if I may call it 

 so when light-waves are not involved) out into space 

 in every direction. As an example, I will take the 



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