An introduction to mathematics, by A. N. Whitehead.

CONIC SECTIONS 135
of PM21 to AM is constant for the parabola
of fig. 17; and he bases most of his work
on this fact. We are evidently advancing
towards the desired uniform definition which
does not go out of the plane; but have not
yet quite attained to uniformity.
In the diagrams 16 and 18, two points, S
and S', will be seen marked, and in diagram 17
one points, S. These are thefoci of the curves,
and are points of the greatest importance.
Apollonius knew that for an ellipse the sum
of SP and S'P (i.e. SP+S'P) is constant, as
P moves on the curve and is equal to AA'.
Similarly for a hyperbola the difference S'P -
SP is constant, and equal to AA' when P is
on one branch, and the difference, SP' -S'P',
is constant and equal to AA' when P' is on
the other branch. But no corresponding
point seemed to exist for the parabola.
Finally 500 years later the last great Greek
geometer, Pappus of Alexandria, discovered
the final secret which completed this line of
thought. In the diagrams 16 and 18 will be
seen two lines, XN and X'N', and in diagram
17 the single line, XN. These are the directrices of the curves, two each for the ellipse
and the hyperbola, and one for the parabola.
Each directrix corresponds to its nearer focus.
The characteristic property of a focus, S, and
its corresponding directrix, XN, for any one
of the three types of curve, is that the ratio