ordinary numerals all numbers up to a myriad myriads (100,000,000). Let us, says Archimedes, call all these numbers numbers of the first order. Let the second order of numbers begin with 100,000,000, and end with 100,000,000². Let 100,000,000² be the first number of the third order, and let this extend to 100,000,000³; and so on, to the myriad-myriadth order, beginning with 100,000,00099,999,999 and ending with 100,000,000100,000,000, which for brevity we will call P.

Let all the numbers of all the orders up to P form the first period, and let the first order of the second period begin with P and end with 100,000,000 P; let the second order begin with this, the third order with 100,000,000² P, and so on up to the 100,000,000th order of the second period, ending with 1,000,000,000100,000,000 P or P². The first order of the third period begins with P², and the orders proceed as before.

Continuing the series of periods and orders of each period, we finally arrive at the 100,000,000th period ending with P100,000,000. The prodigious extent of this scheme is seen when it is considered that the last number of the first period would now be represented by 1 followed by 800,000,000 ciphers, while the last number of the 100,000,000th period would require 100,000,000 times as many ciphers, i.e. 80,000 million million ciphers.

As a matter of fact, Archimedes does not need, in order to express the “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 ×