On these assumptions, and seeing that the perimeter of a regular chiliagon (as of any other regular polygon of more than six sides) inscribed in a circle is more than 3 times the length of the diameter of the circle, it easily follows that, while the diameter of the earth is less than 1,000,000 stadia, the diameter of the so-called “universe” is less than 10,000 times the diameter of the earth, and therefore less than 10,000,000,000 stadia.

Lastly, Archimedes assumes that a quantity of sand not greater than a poppy-seed contains not more than 10,000 grains, and that the diameter of a poppy-seed is not less than 1⁄40th of a dactylus (while a stadium is less than 10,000 dactyli).

Archimedes is now ready to work out his calculation, but for the inadequacy of the alphabetic system of numerals to express such large numbers as are required. He, therefore, develops his remarkable terminology for expressing large numbers.

The Greek has names for all numbers up to a myriad (10,000); there was, therefore, no difficulty in expressing with the 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