THE SAND-RECKONER 
85 
Draw from E 
0, and from 
With centre 
isent the path 
iis circle meet 
ng CO in M. 
the side of a 
e circle AOB, 
lar polygon of 
d in the circle 
eter); 
in all respects, 
in) > CH + OK, 
at of the earth; 
■ CO. 
VQ > OF. 
'i. 
JO^’ 
Hence AB is greater than the side of a regular polygon of 
812 sides, and a fortiori greater than the side of a regular 
polygon of 1,000 sides, inscribed in the circle AOB. 
The perimeter of the chiliagon, as of any regular polygon 
with more sides than six, inscribed in the circle AOB is greater 
than 3 times the diameter of the sun’s orbit, but is less than 
1,000 times the diameter of the sun, and a fortiori less than 
30,000 times the diameter of the earth; 
therefore (diameter of sun’s orbit) < 10,000 (diam. of earth) 
< 10,000,000,000 stades. 
But (diam. of earth): (diam. of sun’s orbit) 
= (diam. of sun’s orbit): (diam. of universe); 
therefore the universe, or the sphere of the fixed stars, is less 
than 10,000 3 times the sphere in which the sun’s orbit is a 
great circle. 
Archimedes takes a quantity of sand not greater than 
a poppy-seed and assumes that it contains not more than 10,000 
grains; the diameter of a poppy-^eed he takes to be not less 
than Jg-th of a finger-breadth; thus a sphere of diameter 
1 finger-breadth is not greater than 64,000 poppy-seeds and 
therefore contains not more than 640,000,000 grains of sand 
(‘ 6 units of second order + 40,000,000 units of first order ’) 
and a fortiori not more than 1,000,000,000 (‘10 units of 
second order of numbers ’). Gradually increasing the diameter 
of the sphere by multiplying it each time by 100 (making the 
sphere 1,000,000 times larger each time) and substituting for 
10,000 finger-breadths a stadium (< 10,000 finger-breadths), 
he finds the number of grains of sand in a sphere of diameter 
10,000,000,000 stadia to be less than ‘ 1,000 units of seventh 
order of numbers’ or 10 51 , and the number in a sphere 10,000 3 
times this size to be less than ‘ 10,000,000 units of the eighth 
order of numbers ’ or 10 03 . 
The Quadrature of the Parabola. 
In the preface, addressed to Dositheus after the death of 
Conon, Archimedes claims originality for the solution of the 
problem of finding the area of a segment of a parabola cut off 
by any chord, which he says he first discovered by means of 
mechanics and then confirmed by means of geometry, using 
the lemma that, if there are two unequal areas (or magnitudes