A history of Greek mathematics - Wilbourhall.org

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140 APOLLONIUS OF PERGA
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If in a hyperbola, an ellipse, or the circumference of a circle '
sometimes, however, the double-branch hyperbola and the
ellipse come in one proposition, e.g. in I. 30: 'If in an ellipse
or the opposites (i. e. the double hyperbola) a straight line be
drawn through the centre meeting the curve on both sides of
the centre, it will be bisected at the centre.' The property of
conjugate diameters in an ellipse is proved in relation to
the original diameter of reference and its conjugate in I. 15,
where it is shown that, if DD' is the diameter conjugate to
PP' (i.e. the diameter drawn ordinate- wise to PP'), just as
PP' bisects all chords parallel to DD', so DD' bisects all chords
parallel to PP'
;
also, if DL' be drawn at right angles to DD'
and such that DL' . DD' = PP' 2 (or DL' is a third proportional
to DD', PP'), then the ellipse has the same property in relation
to DD' as diameter and DL' as parameter that it
has in
relation to PP' as diameter and PL as the corresponding para-
PP' = DD' 2 , or PL is
meter. Incidentally it appears that PL .
a third proportional to PP', DD', as indeed is obvious from the
property of the curve QV 2 : PV. PV'= PL : PP' = DD' 2 : PP' 2 .
The next proposition, I. 16, introduces the secondary diameter
of the double-branch hyperbola (i.e. the diameter conjugate to
the transverse diameter of reference), which does not meet the
curve; this diameter is defined as that straight line drawn
through the centre parallel to the ordinates of the transverse
diameter which is bisected at the centre and is of length equal
to the mean proportional between the sides of the figure ' ',
i.e. the transverse diameter PP' and the corresponding parameter
PL. The centre is defined as the middle point of the
diameter of reference, and it is proved that all other diameters
are bisected at it (I. 30).
Props. 17-19, 22-9, 31-40 are propositions leading up to
and containing the tangent properties. On lines exactly like
those of Eucl. III. 1 6 for the circle, Apollonius proves that, if
a straight line is drawn through the vertex (i. e. the extremity
of the diameter of reference) parallel to the ordinates to the
diameter, it will fall
outside the conic, and no other straight
line can fall between the said straight line and the conic
therefore the said straight line touches the conic (1.17, 32).
Props. I. 33, 35 contain the property of the tangent at any
point on the parabola, and Props. I. 34, 36 the property of