XXXVI 
BIOGRAPHICAL NOTICE OF ARTHUR CAYLEY. 
Another notion, entirely due to Cayley in its first form, is that of the Absolute; 
it was first introduced in his Sixth Memoir on Quantics,* 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, AG, when mul 
tiplied by 2 V — 1, is equal to the logarithm of the cross-ratio of the pencil made up 
of the lines AB, AC and (conjugate imaginary) 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 composed of a 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; 
and 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 Lobatchewsky and 
Bolyai. This fundamental step was taken by Klein in his admirable memoir-f*, Ueber 
die sogenannte Nicht-Euklidische Geometrie, which contains a considerable simplification 
in statement of Cayley’s original point of view, and contributes one of the most im 
portant results of the whole theory. The work of the two mathematicians now being 
an organic whole, there is no advantage—at least here—in attempting to subdivide 
the subject for the purpose of specifying the exact share of each in its construction. 
The scope of the Cayley-Klein ideas may briefly be gathered from the following 
sketch. Let A x and A 2 be two points, often called a point-pair; they are to be 
either both real or, if not both real, then conjugate imaginarles so far as their co 
ordinates are concerned. Let P, Q, R be three other points on the line A x A 2 ; and 
let the symbol (PQ) denote 
2y log 
AjP . A 2 Q 
A 1 Q . A 2 P 
or 2i<y log 
AjP . A 2 Q 
AjQ .A 2 P’ 
according as A x and A 2 are 
is manifest that 
a real point-pair, or an imaginary point-pair. 
(PQ) + (QR) = (PR), 
Then it 
so that the functions (PQ), (QR), (PR) satisfy the fundamental property of the dis 
tances between P and Q, Q and R, and P and R. Consequently (PQ) may be taken 
as a generalised conception of the distance between the points P and Q. 
* C. M. P. vol. ii. No. 158; Phil. Trans. (1859), pp. 61—90. 
t Math. Ann. voi. iv. (1871), pp. 573—625.