132
CONIC SECTIONS
chords drawn parallel to the tangents respectively, the rect
angles contained by the segments of the chords respectively
are to one another as the squares of the parallel tangents;
the by no means easy proposition that, if in a parabola the
diameter through P bisects the chord QQ' in V, and QD is
drawn perpendicular to PV, then
QV*:QD* = p: 2 J a ,
where p a is the parameter of the principal ordinates and p is
the parameter of the ordinates to the diameter PV.
Conic sections in Archimedes.
But we must equally regard Euclid’s Conics as the source
from which Archimedes took most of the other ordinary
properties of conics which he assumes without proof. Before
summarizing these it will be convenient to refer to Archi
medes’s terminology. We have seen that the axes of an
ellipse are not called axes but diameters, greater and lesser;
the axis of a parabola is likewise its diameter and the other
diameters are ‘lines parallel to the diameter’, although in
a segment of a parabola the diameter bisecting the base is
the ‘ diameter ’ of the segment. The two ‘ diameters ’ (axes)
of an ellipse are conjugate. In the case of the hyperbola the
‘ diameter ’ (axis) is the portion of it within the (single-branch)
hyperbola; the centre is not called the ‘ centre ’, but the point
in which the ‘ nearest lines to the section of an obtuse-angled
cone’ (the asymptotes) meet; the half of the axis {CA) is
‘ the line adjacent to the axis ’ (of the hyperboloid of revolution
obtained by making the hyperbola revolve about its ‘ diameter ’),
and A'A is double of this line. Similarly CP is the line
‘ adjacent to the axis ’ of a segment of the hyperboloid, and
P'P double of this line. It is clear that Archimedes did not
yet treat the two branches of a hyperbola as forming one
curve; this was reserved for Apollonius.
The main properties of conics assumed by Archimedes in
addition to those above mentioned may be summarized thus.
Central Conics.
1. The property of the ordinates to any diameter PP',
QV*-.PV.P'V = Q'V' 2 :PV'.P'V'.
