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

CONIC SECTIONS IN ARCHIMEDES 125
2. If two similar parabolic segments with bases BQ 1}
-
BQ
2
be
be any straight
placed as in the last proposition, and if BR Y
R 2
line through B meeting the segments in R 1}
R 2
respectively,
BQ 1
:BQ 2
= BR 1
:BR 2
.
These propositions are easily deduced from the theorem
proved in the Quadrature of the Parabola, that, if through E,
a point on the tangent at B, a straight line ERO be drawn
parallel to the axis and meeting the curve in R and any chord
BQ through B in 0, then
ER:RO = BO: OQ.
3. On the strength of these propositions Archimedes assumes
the solution of the problem of placing, between two parabolic
segments similar to one another and placed as in the above
propositions, a straight line of a given length and in a direction
parallel to the diameters of either parabola.
Euclid and Archimedes no doubt adhered to the old method
of regarding the three conies as arising from sections of three
kinds of right circular cones (right-angled, obtuse-angled and
acute-angled) by planes drawn in each case at right angles to
a generator of the cone. Yet neither Euclid nor Archimedes
was unaware that the section ' of an acute-angled cone ', or
ellipse, could be otherwise produced.
Euclid actually says in
his Phaenomena that ' if a cone or cylinder (presumably right)
be cut by a plane not parallel to the base, the resulting section
is a section of an acute-angled cone which is similar to
a Ovpeos (shield) '. Archimedes knew that the non-circular
sections even of an oblique circular cone made by planes
cutting all the generators are ellipses ; for he shows us how,
given an ellipse, to draw a cone (in general oblique) of which
it is a section and which has its vertex outside the plane
of the ellipse on any straight line through the centre of the
ellipse in a plane at right angles to the ellipse and passing
through one of its axes, whether the straight line is itself
perpendicular or not perpendicular to the plane of the ellipse
drawing a cone in this case of course means finding the circular
sections of the surface generated by a straight line always
passing through the given vertex and all the several points of
the given ellipse. The method of proof would equally serve
126 APOLLONIUS OF PERGA
for the other two conies, the hyperbola and parabola, and we
can scarcely avoid the inference that Archimedes was equally
aware that the parabola and the hyperbola could be found
otherwise than by the old method.
The first, however, to base the theory of conies on the
production of all three in the most general way from any
kind of circular cone, right or oblique, was Apollonius, to
whose work we now come.
B. APOLLONIUS OF PERGA
Hardly anything is known of the life of Apollonius except
that he was born at Perga, in Pamphylia, that he went
when quite young to Alexandria, where he studied with the
successors of Euclid and remained a long time, and that
he flourished (yeyove) in the reign of Ptolemy Euergetes
(247-222 B.C.). Ptolemaeus Chennus mentions an astronomer
of the same name, who was famous during the reign of
Ptolemy Philopator (222-205 B.C.), and it is clear that our
Apollonius is meant. As Apollonius dedicated the fourth and
following Books of his Conies to King Attalus I (241-197 B.C.)
we have a confirmation of his approximate date. He was
probably born about 262 B.C., or 25 years after Archimedes.
We hear of a visit to Pergamum, where he made the acquaintance
of Eudemus of Pergamum, to whom he dedicated the
first two Books of the Conies in the form in which they have
come down to us ; they were the first two instalments of a
second edition of the work.
The text of the Conies.
The Conies of Apollonius was at once recognized as the
authoritative treatise on the subject, and later writers regularly
cited it when quoting propositions in conies. Pappus
wrote a number of lemmas to it ; Serenus wrote a commentary,
as also, according to Suidas, did Hypatia. Eutocius
(fl. a.d. 500) prepared an edition of the first four Books and
wrote a commentary on them ; it is evident that he had before
him slightly differing versions of the completed work, and he
may also have had the first unrevised edition which had got
into premature circulation, as Apollonius himself complains in
the Preface to Book I.