THE CONICS 
133 
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who wrote 
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es described 
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idea of the 
of the sub- 
ality. The 
itroduction, 
:s of conics, 
Dispositions 
i; the term 
i in exactly 
great work, 
devoted to 
of the sub- 
ntent of the 
i first three 
; bearing on 
and in the 
two conics 
may intersect, touch, or both) the part which is claimed 
as new is the extension to the intersections of the parabola, 
ellipse, and circle with the double-branch hyperbola, and of 
two double-branch hyperbolas with one another, of the in 
vestigations which had theretofore only taken account of the 
single-branch hyperbola. Even in Book Y, the most remark 
able of all, Apollonius does not say that normals as ‘ the shortest 
lines ’ had not been considered before, but only that they had 
been superficially touched upon, doubtless in connexion with 
propositions dealing with the tangent properties. He explains 
that he found it convenient to treat of the tangent properties, 
without any reference to normals, in the first Book in order 
to connect them with the chord properties. It is clear, there 
fore, that in treating normals as maxima and minima, and by 
themselves, without any reference to tangents, as he does in 
Book У, he was making an innovation ; and, in view’ of the 
extent to which the theory of normals as maxima and minima 
is developed by him (in 77 propositions), there is no wonder 
that he should devote a whole Book to the subject. Apart 
from the developments in Books III, IV, V, just mentioned, 
and the numerous new propositions in Book VII with the 
problems thereon which formed the lost Book VIII, Apollonius 
only claims to have treated the whole subject more fully and 
generally than his predecessors. 
Great generality of treatment from the beginning. 
So far from being a braggart and taking undue credit to 
himself for the improvements which he made upon his prede 
cessors, Apollonius is, if anything, too modest in his descrip 
tion of his personal contributions to the theory of conic 
sections. For the ‘ more fully and generally ’ of his first 
preface scarcely conveys an idea of the extreme generality 
with which the wdiole subject is worked out. This character 
istic generality appears at the very outset. , 
Analysis of the Conics. 
Book I. 
Apollonius begins by describing a double oblique circular 
cone in the most general way. Given a circle and any point 
outside the plane of the circle and in general not lying on the