172 
APOLLONIUS OF PERGA 
A number of other ratios are expressed in terms of the 
straight lines terminating at A, A', H, H', M, M' as follows' 
(VII. 14-20). 
In the ellipse A A' 2 : PP' 2 ~ DD' 2 = A'H: 2 CM, 
and in the hyperbola or ellipse (if p be the parameter of the 
ordinates to PP') 
AA' 2 :p 2 = A'H.MH': MH 2 , 
A A' 2 : {PP' ± pf = A'H. MU': {MH+ MH') 2 , 
A A' 2 : PP'. p = A'H: MH, 
and A A' 2 : (PP' 2 + p 2 ) = A'H . MH':{MH' 2 + MH 2 ). 
Apollonius is now in a position, by means of all these 
relations, resting on the use of the auxiliary points H, H', M, 
to compare different functions of any conjugate diameters 
with the same functions of the axes, and to show how the 
former vary (by way of increase or diminution) as P moves 
away from A. The following is a list of the functions com 
pared, where for brevity I shall use a, h to represent AA', BB' \ 
a', h' to represent PP', DD'; and p, p' to represent the para 
meters of the ordinates to AA', PP' respectively. 
In a hyperbola, according as a > or < h, a' > or < h', and the 
ratio a':, h' decreases or increases as P moves from A on 
either side; also, if a — h, a = h' (VII. 21-3); in an ellipse 
a: h > a':h', and the latter ratio diminishes as P moves from 
A to B (YII. 24). 
In a hyperbola or ellipse a + b < a' + h', and of+ h' in the 
hyperbola increases continually as P moves farther from A, 
but in the ellipse increases till a', h' take the position of the 
equal conjugate diameters when it is a maximum (VII, 
25, 26). 
In a hyperbola in which a, h are unequal, or in an ellipse, 
a~b>a'~b', and a'~b' diminishes as P moves away from A, 
in the hyperbola continually, and in the ellipse till a', h' are 
the equal conjugate diameters (VII. 27). 
ah < a'b', and a'b' increases as P moves away from A, in the 
hyperbola continually, and in the ellipse till a', b' coincide with 
the equal conjugate diameters (VII. 28). 
VII. 31 is the important proposition that, if PP', DD' are