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

It follows that
THE CONICS, BOOK I 147
QM(QH+MN) : QM(QT + MU) = R'M* :R'M . MF'
but, from above, QM(QT+MU) = R'M . MF'\
therefore
which is the desired property.
R'M* = QAI(QH+ MN)
= QM.MK,
;
In the case of the hyperbola, the same property is true for
the opposite branch.
These important propositions show that the ordinate property
of the three conies is of the same form whatever diameter is
taken as the diameter of reference. It is therefore a matter
of indifference to which particular diameter and ordinates the
conic is referred. This is stated by Apollonius in a summary
which follows I. 50.
First appearance of principal axes.
The axes appear for the first time in the propositions next
following (I. 52-8), where Apollonius shows how to construct
each of the
conies, given in each case (1) a diameter, (2) the
length of the corresponding parameter, and (3) the inclination
of the ordinates to the diameter. In each case Apollonius
first assumes the angle between the ordinates and the diameter
to be a right angle ;
then he reduces the case where the angle
is oblique to the case where it is right by his method of transformation
of coordinates; i.e. from the given diameter and
parameter he finds the axis of the conic and the length of the
corresponding parameter, and he then constructs the conic as
in the first case where the ordinates are at right angles to the
diameter. Here then w T e have a case of the proof of existence
by means of construction. The conic is in each case constructed
by finding the cone of which the given conic is a
section. The problem of finding the axis of a parabola and
the centre and the axes of a central conic when the conic (and
not merely the elements, as here) is
given comes later (in II.
44-7), where it is also proved (II. 48) that no central conic
can have more than two axes.
L 2
148 APOLLONIUS OF PERGA
It has been my object, by means of the above detailed
account of Book I, to show not merely what results are
obtained by Apollonius, but the way in which he went to
work ; and it will have been realized how entirely scientific
and general the method is. When the foundation is thus laid,
and the fundamental properties established, Apollonius is able
to develop the rest of the subject on lines more similar to
those followed in our text-books.
of the work can therefore for the
summary of the contents.
Book II begins with a section
My description of the rest
most part be confined to a
devoted to the properties of
the asymptotes. They are constructed in II. 1 in this way.
Beginning, as usual, with any diameter of reference and the
corresponding parameter and inclination
of ordinates, Apollonius
draws at P the vertex (the extremity of the diameter)
a tangent to the hyperbola and sets off along it lengths PL, PL'
on either side of P such that PL 2 =PL' =±p 2 . PP' [ = GD%
where p is the parameter. He then proves that CL, GU produced
will not meet the curve in any finite point and are therefore
asymptotes. II. 2 proves further that no straight line
through G within the angle between the asymptotes can itself
be an asymptote. II. 3 proves that the intercept made by the
asymptotes on the tangent at any point P is bisected at P, and
that the square on each half of the intercept is equal to onefourth
of the figure corresponding to the diameter through
' '
P (i.e. one-fourth of the rectangle contained by the 'erect'
side, the latus rectum or parameter corresponding to the
diameter, and the diameter itself) ; this property is used as a
means of drawing a hyperbola when the asymptotes and one
point on the curve are given (II. 4). II. 5-7 are propositions
about a tangent at the extremity of
a diameter being parallel
to the chords bisected by it. Apollonius returns to the
asymptotes in II. 8, and II. 8-14 give the other ordinary
properties with reference to the asymptotes (II. 9 is a converse
of II. 3), the equality of the intercepts between the
asymptotes and the curve of any chord (II. 8), the equality of
the rectangle contained by the distances between either point
in which the chord meets the curve and the points of intersection
with the asymptotes to the square on the parallel
semi-diameter (II. 10), the latter property with reference to