MEASUREMENT OF SOLIDS 
335 
describing circle and the length of the path of its centre. 
For, he says, since 14 is a radius (of the path of the. centre), 
28 is its diameter and 88 its circumference. ‘If then the tore 
be straightened out and made into a cylinder, it will have 88 
for its length, and the diameter of the base of the cylinder is 
12; so that the solid content of the cylinder is, as we have 
seen, 9956$ ’ (= 88 144b 
(e) The two special solids of Archimedes’s * Method 
Chaps. 14, 15 give the measurement of the two remarkable 
solids of Archimedes’s Method, following Archimedes’s results. 
if) The Jive regular solids. 
In chaps. 16-18 Heron measures the content of the five 
regular solids after the cube. He has of course in each case 
to find the perpendicular from the centre of the circumscrib 
ing sphere on any face. Let p be this perpendicular, a the 
edge of the solid, r the radius of the circle circumscribing any 
face. Then (1) for the tetrahedron 
(2) In the case of the octahedron, which is the sum of two 
equal pyramids on a square base, the content is one-third 
of that base multiplied by the diagonal of the figure, 
i.e.|.a 2 . V2a or fV2.a 3 4 ; in the case taken a = 7, and 
Heron takes 10 as an approximation to V{2.7 2 ) or V98, the 
result being ^.10.49 or 163^. (3) In the case of the icosa 
hedron Heron merely says that 
p: a = 93:127 (the real value of the ratio is \ 
V 6 
(4) In the case of the dodecahedron, Heron says that 
7 
p: a = 9:8 (the true value is \ 
put equal to f, Heron’s ratio is readily obtained). 
Book II ends with an allusion to the method attributed, to 
Archimedes for measuring the contents of irregular bodies by 
immersing them in water and measuring the amount of fluid 
displaced.