140 
APOLLONIUS OF PERGA 
‘ 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 BD' is the diameter conjugate to 
PP' (i.e, the diameter drawn ordinate-wise to PP'), just as 
PP' bisects all chords parallel to D //, so DB' bisects all chords 
parallel to PP'; also, if DL' be drawn at right angles to DB' 
and such that BL'. BD' = PP' 2 (or BL' is a third proportional 
to DB', PP'), then the ellipse has the same property in rela 
tion to DB' as diameter and DL' as parameter that it has in 
relation to PP' as diameter and PL as the corresponding para 
meter. Incidentally it appears that PL. PP' = BD' 2 , or PX is 
a third proportional to PP', BD', as indeed is obvious from the 
property of the curve QV 2 : PV. PV'= PL ; PP' = BD' 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 para 
meter PX, 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. 16 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