HoGBEN. — Transcendental Geometry. 613 



In choosing our arbitrary axes, let us suppose we begin by 

 fixing the position of X. In any given plane through X 

 there is only one axis, Y, at right angles to X. By making 

 the plane revolve about X, we shall make it coincide in 

 succession with all the planes that can be drawn through X, 

 and O Y will coincide in succession with all the straight lines 

 that can be drawn perpendicular to X. Let one of these be 

 chosen for the axis Y; let it be Fj. But Z is also at 

 right angles to OX; therefore it is one of the possible Y's ; 

 so is V. But there is only one series of possible Y's ; 

 therefore Z and V must both be in the same series. Now, 

 the particular Y which is taken as Z, must be at right 

 angles to O Fi ; so must the particular O Y taken as V. 

 But in the series of Y's there is only one straight line which 

 is perpendicular to Fj. Therefore Z must be that line ; so 

 also must V. Hence Z and V must be identical : the 

 imaginary being's space is identical with ours, and he would be 

 cognizant of no points, or of no properties of space, of which we 

 were not also cognizant. 



I am aware that this argument is only the reproduction in 

 mathematical form of the argument from common sense ; but 

 the only ground, I think, on which it can be overthrown is, that 

 the fourth dimension is not comparable with the other three — 

 length, breadth, and height, to which we refer our notions of the 

 extension of bodies ; that is, it is not a dimension of space at all, 

 in our sense of the term. Not being a dimension of space, it 

 cannot aid us in finding any points in space other than those 

 known to us by our three dimensions. 



It has been said, I think, by the authors of " The Unseen 

 Universe," that though space may be of three dimensions with 

 us, yet at some great distance it may have a higher number of 

 dimensions. But space, as space, must be homogenous ; to 

 assert anything else is, as Stallo has shown, to confound space 

 with the matter or with the structures which are in it. To 

 explain the use of the word " structure" here, I proceed to 

 distinguish between two of the meanings attached to the word 

 " space." So far, there has been no danger of ambiguity. But 

 we cannot go further without distinguishing between what is 

 sometimes called structure- space, and absolute space. 



Consider the piece of chalk I hold in my hand : it occupies 

 space ; outside it there is space not occupied by the chalk. The 

 space occupied by the chalk, the form of which we identify with 

 the form of the piece of chalk, is what is called structure-space. 

 Other bodies besides the piece of chalk in question are said to 

 occupy space. It is possible, indeed, that no space is empty ; 

 but the very fact of our being able to think of it as empty or not 

 empty shows that we have formed a concept of space apart from 

 the structures which are in it. This is absolute space, or the 



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