CONIC SECTIONS IN ARCHIMEDES 
125 
al to QV 2 
)arallel to 
ict of the 
nd QV be 
:abola can 
l chord, 
i property 
3 the axis 
;d parallel 
in V, and 
the curve 
PH> or 
ies, II. 6). 
oof is not 
ze in the 
uadrature 
ig Bodies, 
piently he 
ons, which 
few others 
deductions 
fer mainly 
ises are in 
parabolic 
■aight line 
int on the 
raight line 
ments and 
r and BQ 3 
2. If two similar parabolic segments with bases BQ 1} BQ 2 be 
placed as in the last proposition, and if BR 1 R 2 be any straight 
line through B meeting the segments in R lt R 2 respectively, 
BQi : BQ 2 = BR 1 : BB 2 . 
Those 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 conics as arising from sections of three 
kinds of right circular cones (right-anglecl, 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 êvpeôs (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