A history of Greek mathematics - Wilbourhall.org

124 CONIC SECTIONS
only :
p is simply the line to which the rectangle equal to QV 2
and of width equal to PFis applied.
2. Parallel chords are bisected by one straight line parallel to
the axis, which passes through the point of contact of the
tangent parallel to the chords.
3. If the tangent at Q meet the diameter PV in T, and QV be
the ordinate to the diameter, PV = PT.
By the aid of this proposition a tangent to the parabola can
be drawn (a) at a point on it, (b) parallel to a given chord.
4. Another proposition assumed is equivalent to the property
of the subnormal, NG = \
rpa .
5. If QQ' be a chord of a parabola perpendicular to the axis
and meeting the axis in M, while QVq another chord parallel
to the tangent at P meets the diameter through P in V, and
RHK is the principal ordinate of any point R on the curve
meeting PV in H and the axis in K, then PV :PH > or
= MK : KA '
for this is proved ' (On Floating Bodies, II. 6).
;
Where it was proved we do not know ; the proof is not
altogether easy. 1
6. All parabolas are similar.
As we have seen, Archimedes had to specialize in the
parabola for the purpose of his treatises on the Quadrature
of the Parabola, Conoids and Spheroids, Floating Bodies,
Book II, and Plane Equilibriums, Book II ; consequently he
had to prove for himself a number of special propositions, which
have already been given in their proper places. A few others
are assumed without proof, doubtless as being. easy deductions
from the propositions which he does prove. They refer mainly
to similar parabolic segments so placed that their bases are in
one straight line and have one common extremity.
1. If any three similar and similarly situated parabolic
segments BQ
X 3
lying along the same straight line
, BQ
2
,
BQ
as bases (BQ 1
< BQ 2
< BQ 3 ), and if E be any point on the
tangent at B to one of the segments, and EO a straight line
through E parallel to the axis of one of the segments and
meeting the segments in R%, R 2
,
R
in 0, then
1
respectively and BQ 3
R,R :
2
R 2
R, = (Q2 BQ
Q3 :
3 ) . (BQ, : Q, Q 2 ).
1
See Apollonius of Perga, ed. Heath, p. liv.