XXX111 



cient for the unique specification of a right line. They were first 

 established, and primarily used by him, in connection with his new 

 analytical representation of curves in space ;* and he often recurred 

 to the subject, devoting in particular one paperf to the calculus of 

 the six co-ordinates and to a discussion of Sylvester's involution of 

 six lines. It should, however, be stated that these co-ordinates pre- 

 sented themselves independently to Pliicker ; the development of 

 Pliicker's theory as set forth in his memoirj ' On a New Geometry 

 of Space,' and in his book ' Neue Geometric des Raumes,' is entirely 

 different from that obtained by Cayley, and it ought to be regarded as 

 a separate creation. And it need hardly be remarked that while the 

 introduction of a line, as an entity represented by a set of co-ordinates, 

 leads to a new geometry of space, it is also clear that line-geometry 

 can be regarded as a geometry of four dimensions. 



Another notion, entirely due to Cayley in its first form,, is that of 

 the Absolute ; it was first introduced in, his ' Sixth Memoir on Quan- 

 ticSj'l) which was devoted chiefly to his investigations on the generalised 

 theory of metrical geometry. 



It is a known property that the angle between two lines AB, AC, 

 when multiplied by 2\/ 1, is equal to the logarithm of the cross-ratio 

 of the pencil made up of the lines AB, AC, and (conjugate ima- 

 ginary) lines joining A to the circular points at infinity; and the 

 measure of the angle between two lines can thus be replaced by the 

 consideration of a projective property of an extended system of lines. 

 Other examples of similar changes could easily be quoted. The 

 purpose of Cayley's theory was to replace metrical properties of a 

 figure or figures by projective properties of an extended system 

 corn-posed of the given, figure or figures and of an added figure. 



But it is not solely owing to the generalisation of distance that the 

 memoir is famous. It has revolutionised the theory of the so-called 

 non-Euclidian geometry; aud it has important bearings on the logical 

 and philosophical analysis of the axioms of space-intuition. The 

 independence and the importance of the ideas, originated by Cayley 

 in this memoir, have never been questioned ; but, as is often (and 

 naturally) the case with the discoverer of a fertile subject, Cayley 

 himself did not explain or foresee the full range of application of his 

 new ideas. He did not recognise, at the time when his memoir was 

 first published, the beautiful identification of his generalised theory 

 of metrical geometry with the non-Euclidian geometry of Lobatchew- 

 sky and Bolyai. This fundamental step was taken by Klein in his 



' 0. M. P.,' vol. 4, Nos. 284, 294. 

 t ' C. M. P.,' vol. 7, No. 43.-.. 

 J ' Phil. Trans.,' 1865, pp. 725791. 

 Leipzig, Teubner, 1868. 



|| 'C. M. P.,' vol. 2, No. 158; ' Phil. Trans.' (1859), pp. 619'.). 

 VOL. LVI1I. d 



