BIOGRAPHICAL NOTICE OF ARTHUR CAYLEY. 
XXXV11 
hat of the Absolute; 
was devoted chiefly 
AB, AG, when mul- 
the pencil made up 
30 the circular points 
;an thus be replaced 
tem of lines. Other 
se of Cayley’s theory 
tive properties of an 
ided figure. 
that the memoir is 
-Euclidian geometry; 
nalysis of the axioms 
ideas, originated by 
often (and naturally) 
did not explain or 
not recognise, at the 
ion of his generalised 
>f Lobatchewsky and 
able memoir*|*, Ueber 
iderable simplification 
one of the most im- 
ematicians now being 
mpting to subdivide 
its construction. 
d from the following 
fir; they are to be 
i so far as their co- 
the line A 1 A 2 ; and 
point-pair. Then it 
property of the dis- 
(.PQ) may be taken 
id Q. 
Now let a conic be described in a plane, either imaginary, say, of the form 
x 1 + 4- z* = 0 or real, say, of the form x* + y 2 — z 2 = 0. Choosing the latter case, let 
attention be confined to points lying within the conic, so that every straight line 
through a point cuts the conic in a real point-pair. Take two points, P and Q; 
and let the line joining them cut the conic in two points, Ai and A 2 . Then (PQ), 
as defined above (the constant 7 being the same for all such lines), is the generalised 
distance between P and Q. This conic, which has been arbitrarily assumed, and 
upon which the generalised conception of distance depends, is termed by Cayley the 
Absolute. 
Cayley, however, avoided the unsatisfactory procedure of using one conception of 
distance to define a more general conception. As he himself explains more fully,* 
he regarded the co-ordinates of points as some quantities which define the relative 
properties of points, considered without any reference to the idea of distance but 
conceived as ordered elements of a manifold. Thus if a lt ß u <y 1 and a 2 , /3 2 , 7 2 be the 
co-ordinates of the point-pair Aj and A 2 , the co-ordinates of the points P and Q on 
the line AjA 2 can be taken as AjCq + AjOq, \ß t + X 2 ß 2 , X^ + X 2 y 2 and + /¿ 2 a 2 , 
M'ißi + fhßs, P 171 + /¿272 respectively. The function (PQ) can then be defined as 
27 log PA or 2 * 7 log aAT’ 
the generalised idea of distance thus finds its definition without any antecedent use 
of the conception in its ordinary form. Cayley’s view is summed up in his sentenced:— 
“ the theory in effect is, that the metrical properties of a figure are not the pro 
perties of the figure considered per se apart from everything else, but its properties 
when considered in connexion with another figure, viz. the conic termed the absolute.” 
The metrical formulae obtained when the absolute is real are identical with those 
of Lobatchewsky’s and Bolyai’s “hyperbolic” geometry: when the absolute is imaginary 
the formulae are identical with those of Riemann’s “ elliptic ” geometry; the limiting 
case between the two being that of ordinary Euclidian (“parabolic”) geometry. 
Cayley’s memoir leads inevitably to the question, as to how far projective geometry 
can be defined in terms of space perception without the introduction of distance. This 
has been discussed by von StaudtJ (in 1847, previous to Cayley’s memoir), by Klein § and 
by Lindemann ||. The memoir thus points to a division of our space intuitions into two 
distinct parts: one, the more fundamental as not involving the idea of distance, the 
other, the more artificial as adding the idea of distance to the former. The considera 
tion of the relation of these ideas to the philosophical account of space has not yet 
been brought to its ultimate issue. * * * § 
* See the note which he added, G. M. P. vol. n. p. 604, to the Sixth Memoir; it contains some interesting 
historical and critical remarks. 
+ Loc. cit. § 230. 
+ Geometrie der Lage; also in his later Beiträge zur Geometrie der Lage, 1857. 
§ Math. Ann. vol. vi. (1873), pp. 112—145. 
i| Vorlesungen über Geometrie (Clebsch-Lindemann), vol. n. part i.; the third section is devoted to the 
subject.