A history of Greek mathematics - Wilbourhall.org

MEASUREMENT OF SOLIDS 335
describing circle and the length of the path of its centre.
For, he says, since 1 4 is a radius (of the path of the centre),
28 is its diameter and 88 its circumference.
c
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, '
9956f (= 88 . iJ
.
144K
(e) The tivo 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.
(() 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 circumscribing
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
a 2 = 3r 2 ,
p = 2 a 2 — \a 2 = §a 2 .
(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. J .a2 , a/2 a or J a/2, a3 ;
in the case taken a = 7, and
Heron takes 10 as an approximation to \/(2 . 7 2 ) or a/98, the
result being J. 10.49 or 163|. (3) In the case of the icosahedron
Heron merely says that
p
:
a = 93 : 127 (the real value of the ratio is \ / 7 + 3 n/ 5 \ .
(4) In the case of the dodecahedron, Heron says that
~ n ali
• , /25 + 11 \/5 . .„ / .
p:a = 9 : 8 (the true value is -
J
/ -> and, if v5 is
put equal to J, 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.