Since the diameter of a poppy-seed is not less than 1⁄40th of a dactylus, and spheres are to one another in the triplicate ratio of their diameters, a sphere of diameter 1 dactylus is not greater than 64,000 poppy-seeds, and, therefore, contains not more than 64,000 × 10,000 grains of sand, and a fortiori not more than 1,000,000,000, or 109 grains of sand. Archimedes multiplies the diameter of the sphere continually by 100, and states the corresponding number of grains of sand. A sphere of diameter 10,000 dactyli and a fortiori of one stadium contains less than 1021 grains; and proceeding in this way to spheres of diameter 100 stadia, 10,000 stadia and so on, he arrives at the number of grains of sand in a sphere of diameter 10,000,000,000 stadia, which is the size of the so-called universe; the corresponding number of grains of sand is 1051. The diameter of the real universe being 10,000 times that of the so-called universe, the final number of grains of sand in the real universe is found to be 1063, which in Archimedes’s terminology is a myriad-myriad units of the eighth order of numbers.
CHAPTER VI.
MECHANICS.
It is said that Archytas was