A history of Greek mathematics - Wilbourhall.org

THE CONICS, BOOKS V, VI 167
tively with the conies give the points at which the normals
passing through are normals.
Pappus criticizes the use of the rectangular hyperbola in
the case of the parabola as an unnecessary resort to a solid
'
locus '
; the meaning evidently is that the same points of
intersection can be got by means of a certain circle taking
the place of the rectangular hyperbola. We can, in fact, from
2
the equation (1) above combined with y = px, obtain the
circle
(x 2 + y
2)
- (x 1
+ ^p)x-iy 1 y = 0.
The Book concludes with other propositions about maxima
and minima. In particular V. 68-71 compare the lengths of
tangents TQ, TQ f , where Q is nearer to the axis than Q\
V. 72, 74 compare the lengths of two normals from a point
from which only two can be drawn and the lengths of other
straight lines from to the curve ; V. 75-7 compare the
lengths of three normals to an ellipse drawn from a point
below the major axis, in relation to the lengths of other
straight lines from to the curve.
Book VI is of much less interest. The first part (VI. 1-27)
relates to equal (i.e. congruent) or similar conies and segments
of conies ; it is naturally preceded by some definitions including
those of ' equal ' and ' similar ' as applied to conies and
segments of conies. Conies are said to be similar if, the same
number of ordinates being drawn to the axis at proportional
distances from the vertices, all
proportional to the corresponding abscissae.
the ordinates are respectively
The definition of
similar segments is the same with diameter substituted for
axis, and with the additional condition that the angles
between the base and diameter in each are equal. Two
parabolas are equal if
the ordinates to a diameter in each are
inclined to the respective diameters at equal angles and the
corresponding parameters are equal ; two ellipses or hyperbolas
are equal if the ordinates to a diameter in each are
equally inclined to the respective diameters and the diameters
as well as the corresponding parameters are equal (VI. 1. 2).
Hyperbolas or ellipses are similar when the 'figure' on a
diameter of one is similar (instead of equal) to the ' figure ' on
a diameter of the other, and the ordinates to the diameters in