A history of Greek mathematics - Wilbourhall.org

THE CONICS 133
may intersect, touch, or both) the part which is claimed
as new is the extension to the intersections of the parabola,
with the double-branch hyperbola, and of
ellipse, and circle
two double-branch hyperbolas with one another, of the investigations
which had theretofore only taken account of the
single-branch hyperbola. Even in Book V, the most remarkable
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, therefore,
that in treating normals as maxima and minima, and by
themselves, without any reference to tangents, as he does in
Book V, 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
himself for the improvements which he made upon his predecessors,
Apollonius is, if anything, too modest in his description
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 whole subject is worked out.
This characteristic
generality appears at the very outset.
to
Analysis of the
Conies.
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