Desargues' Brouillon Project and the Conics of ... - J.P. Hogendijk

6 Jan I? Hogendijk
The concepts of diameter and ordinate play a central role in the
theory of Apollonius. We call a chord of a curve a straight segment
joining two points on the curve. A diameter of a curve is a straight
line which bisects all chords of a curve having a certain fixed
direction (Figure 2). The halves of the chords are called the ordinates
corresponding to the diameter. The constant angle o between a
diameter and its ordinates is called the angle ofordinates. If w is a
right angle, the diameter is called an axis of the curve. Because
Apollonius wants to use similar concepts for the two opposite
branches, he defines a diameter ofa pair ofcurues as a straight line
which bisects all segments with an endpoint on each curve and
having a given direction, and so on.
For the circle one can prove on the basis of Euclid’s Elements
II1:3 that every line through the centre is a diameter in the sense of
Apollonius, and that the ordinates are perpendicular to the diameter
to which they correspond. Thus o is always a right angle, and every
diameter is an axis.
Apollonius proves as follows that every conic section r’ which
is not a circle has one diameter d (Conics 1:7, Figure 1): Let the
plane of r’ meet the base plane, that is the plane of r, at line u.
Drop perpendicular MN from the centre M of the circle onto u. Let
plane AMN meet the plane of r’ in line d. Now consider in r‘ a
chord P’Q’ parallel to u, and suppose that P’Q’ meets d at R‘. Draw
AP‘, AQ’, AR’ and extend them, if necessary, to meet the base plane
at P, Q and R. Then PQ is a chord of the circle r, and PQ
Figure 2.