334 
HERON OF ALEXANDRIA 
The method is the same mutatis mutandis as that used in 
II. 6 for the frustum of a pyramid with any triangle for base, 
and it is applied in II. 9 to the case of a frustum of a pyramid 
with a square base, the formula for which is 
[{$(«+ a')} 2 + £{*(«-“') } 2 ]^. 
where a, a' are the sides of the larger and smaller bases 
respectively, and h the height; the expression is of course 
easily reduced to § h(a 2 + aa' + a' 2 ). 
(y) Frustum of cone, sphere, and segment thereof. 
A. frustum of a cone is next measured in two ways, (1) by 
comparison with the corresponding frrfetum of the circum 
scribing pyramid with square base, (2) directly as the 
difference between two cones (chaps. 9, 10). The volume of 
the frustum of the cone is to that of the frustum of the 
circumscribing pyramid as the area of the base of the cone to 
that of the base of the pyramid; i.e. the volume of the frus 
tum of the cone is | tt, or times the above expression for 
the frustum of the pyramid with a 2 , a/ 2 as bases, and it 
reduces to f^irh (a 2 + aa' + a' 2 ), where a, a' are the diameters 
of the two bases. For the sphere (chap. 11) Heron uses 
Archimedes’s proposition that the circumscribing cylinder is 
1^ times the sphere, whence the volume of the sphere 
— •§ .d.^d 2 or IxfZ 3 ; for a segment of a sphere (chap. 12) he 
likewise uses Archimedes’s result {On the Sphere and Cylinder, 
II. 4). 
(8) Anchor-ring or tore. 
The anchor-ring or tore is next measured (chap. 13) by 
means of a proposition which Heron quotes from Dionyso-- 
dorus, and which is to the effect that, if a be the radius of either 
circular section of the tore through the axis of revolution, and 
c the distance of its centre from that axis, 
, ttci 2 : etc = (volume of tore): ttc 2 . 2a 
[whence volume of tore = 2 n 2 ca 2 \. In the particular case 
taken a = 6, c = 14, and Heron obtains, from the proportion 
113^:84 = F;7392, I r = 9956f. But he shows that he is 
aware that the volume is the product of the area of the