A history of Greek mathematics Vol.II from Aristarchus to Diophantus by Heath, Thomas Little, Sir, 1921

THE CONICS 133
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 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.
Analysis of the
Book I.
Conies.
to
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
134 APOLLONIUS OF PERGA
straight line through the centre of
the circle perpendicular to
its plane, a straight line passing through the point and produced
indefinitely in both directions is made to move, while
always passing through the fixed point, so as to pass successively
through all the points of the circle ; the straight line
thus describes a double cone which is in general oblique or, as
Apollonius calls it, scalene. Then, before proceeding to the
geometry of a cone, Apollonius gives a number of definitions
which, though of course only required for conies, are stated as
applicable to any curve.
1
In any curve,' says Apollonius, ' I give the name diameter to
any straight line which, drawn from the curve, bisects all the
straight lines drawn in the curve (chords) parallel to any
straight line, and I call the extremity of the straight line
(i.e. the diameter) which is at the curve a vertex of the curve
and each of the parallel straight lines (chords) an ordinate
(lit. drawn ordinate- wise, reray/zej/o)? KaTrj-^Oai) to the
diameter/
He then extends these terms to a pair of curves (the primary
reference being to the double-branch hyperbola), giving the
name transverse diameter to any straight line bisecting all the
chords in both curves which are parallel
to a given straight
line (this gives two vertices where the diameter meets the
curves respectively), and the name erect diameter (6p6ia) to
any straight line which bisects all straight lines drawn
between one curve and the other which are parallel to any
straight line ; the ordinates to any diameter are again the
parallel straight lines bisected by it. Conjugate diameters in
any curve or pair of curves are straight lines each of which
bisects chords parallel to the other. Axes are the particular
diameters which cut at right angles the parallel chords which
they bisect ; and conjugate axes are related in the same way
as conjugate diameters. Here we have practically our modern
definitions, and there is a great advance on Archimedes's
terminology.
The conies obtained in the
oblique cone.
most general way from an
Having described a cone (in general oblique), Apollonius
defines the axis as the straight line drawn from the vertex to