A history of Greek mathematics - Wilbourhall.org

;
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