THE CONICS, BOOKS V, YI 
167 
lively with the conics give the points at which the normals 
passing through 0 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 
the equation (1) above combined with y 2 = px, obtain the 
circle 
(x 2 + y 2 )-{x 1 + ip)x-±y 1 y = 0. 
The Book concludes with other propositions about maxima 
and minima. In particular Y. 68-71 compare the lengths of 
tangents TQ, TQ', where Q is nearer to the axis than Q'. 
V. 72, 74 compare the lengths of two normals from a point 
0 from which only two can be drawn and the lengths of other 
straight lines from 0 to the curve; Y. 75-7 compare the 
lengths of three normals to an ellipse drawn from a point 
0 below the major axis, in relation to the lengths of other 
straight lines from 0 to the curve. 
Book VI is of much less interest. The first part (VI. 1-27) 
relates to equal (i. e, congruent) or similar conics and segments 
of conics; it is naturally preceded by some definitions includ 
ing those of ‘ equal ’ and ‘ similar ’ as applied to conics and 
segments of conics. Conics are said to be similar if, the same 
number of ordinates being drawn to the axis at proportional 
distances from the vertices, all the ordinates are respectively 
proportional to the corresponding abscissae. 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 hyper 
bolas 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