XXX11 



section of two quadrics having one common generator; but the equa- 

 tion of the quadrics taken together represent not the cubic curve alone 

 but also the common generator. And the like for other cases. 



Cayley's purpose in his ' Memoir on Abstract Geometry,' already 

 referred to, was the exposition of some of the elementary principles 

 of the subject. The paper is a remarkable instance of his power of 

 presentation of abstract ideas, and of his clear precision of statement. 

 Moreover, he makes it an explanatory paper ; and, in view of the 

 prevailing estimate of him as an analyst, it is worthy of notice that 

 the paper does not contain a single equation, and contains only a few 

 symbols. It is unnecessary to summarize its contents ; the furthest 

 stage reached is the establishment of the notion that underlies the 

 principle of duality in geometry. 



But though the necessity for hyperdimensional geometry can thus 

 be met so far as it arises in connection with analysis, it is a different 

 matter when the geometry is to be regarded as the generalisation of 

 the geometries of two-dimensional space and of three-dimensional 

 space. Cayley's reply to his own question as to the meaning to be 

 attached to hyperdimensional space is* that 



" It may be at once admitted that we cannot conceive of a 

 fourth dimension of space ; that space as we conceive of it, and 

 the physical space of our experience, are alike three-dimensional ; 

 but we can, I think, conceive of space as being two- or even one- 

 dimensional ; we can imagine rational beings living in a one- 

 dimensional space (a line) or in a two-dimensional space (a 

 surface), and conceiving of space accordingly, and to whom, 

 therefore, a two-dimensional space, or (as the case may be) a 

 three-dimensional space would be as inconceivable as a four- 

 dimensional space is to us." 



By not a few people the first clause in this passage has been 

 neglected and the later clauses have not always been read rightly ; 

 and his further remark, " I need hardly say that the first step is the 

 difficulty, and that granting a fourth dimension we may assume as 

 many more dimensions as we please," has left some readers rather 

 puzzled as to whether Cayley had not, after all, some mysterious 

 incommunicable conception of a fourth dimension. His position is 

 stated in the first clause of the former passage : his conclusion is that 

 hypergeometry is, and is only, a branch of mathematics. 



Before passing from the consideration of his larger contributions 

 to hypergeometry, it is proper to mention his introduction of the six 

 co-ordinates of a line. These are six quantities connected by a homo- 

 geneous equation af+lg + ch = 0; and as only their ratios are used, 

 they are thus equivalent to only four independent magnitudes, suffi- 

 * ' Brit. Ass. Keport,' 1883, President's Address, p. 9. 



