THE CONICS, BOOKS IY-Y 
159 
TR' are 
need with 
tersecting 
alar cases, 
wo conics 
s common 
opositions 
e brought 
particular 
ms affect- 
i in bppo- 
3 (IY. 35); 
not meet 
); a conic 
icave side 
41, 42, 45, 
opositions 
vo conics 
bola) can 
26, 47, 48, 
that two 
nore than 
52, 53, 57) 
ich touch 
rer point, 
ntact. A 
Ants than 
= PF. A 
ve double 
hyperbola 
refore the 
mpossible. 
an ellipse 
le contact 
(IY. 33); 
,ve double 
ntact will 
it is the 
most remarkable of the extant Books, It deals with normals 
to conics regarded as maximum and minimum straight lines 
drawn from particular points to the curve. Included in it are 
a series of propositions which, though worked out by the 
purest geometrical methods, actually lead immediately to the 
determination of the evolute of each of the three conics ; that 
is to say, the Cartesian equations to the evolutes can be easily 
deduced from the results obtained by Apollonius. There can 
be no doubt that the Book is almost wholly original, and it is 
a veritable geometrical tour de force. 
Apollonius in this Book considers various points and classes 
of points with reference to the maximum or minimum straight 
lines which it is possible to draw from them to the CQnics, 
i. e. as the feet of normals to the curve. He begins naturally 
with points on the axis, and he takes first the point E where 
AE measured along the axis from the vertex A is p, p being 
the principal parameter. The first three propositions prove 
generally and for certain particular cases that, if in an ellipse 
or a hyperbola AM be drawn at right angles to A A' and equal 
to \p, and if CM meet the ordinate PN of any point P of the 
curve in H, then PN 2 = 2 (quadrilateral MANH); this is a 
lemma used in the proofs of later propositions, Y. 5, 6, &c. 
Next, in V, 4, 5, 6, he proves that, if AE = \p, then AE is the 
minimum straight line from E to the curve, and if P be any 
other point on it, PE increases as P moves farther away from 
A on either side ; he proves in fact that, if PN be the ordinate 
from P, 
(1) in the case of the parabola PE 2 = AE 2 + AN 2 , 
(2) in the case of the hyperbola or ellipse 
PE 2 = APT- + AN 2 • . ■ f; ■, 
-0.-0. 
where of course p = BB' 2 /AA', and therefore {AA' + p) / A A' 
is equivalent to what we call e 2 , the square of the eccentricity. 
It is also proved that EA' is the maximum straight line from 
E to the curve. It is next proved that, if 0 be any point on 
the axis between A and E, 0 A is the minimum straight line 
from 0 to the curve and, if P is any other point on the curve, 
OP increases as P moves farther from A (Y, 7).