“number of the sand,” to go beyond the eighth order of the first period. The orders of the first period begin respectively with 1, 108, 1016, 1024, ... (108)99,999,999; and we can express all the numbers required in powers of 10.
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