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 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 × 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 the first to treat mechanics in a systematic way by the aid of mathematical principles; but no trace survives of any such work by him. In practical mechanics he is said to have constructed a mechanical dove which would fly, and also a rattle to amuse children and “keep them from breaking things about the house” (so says Aristotle, adding “for it is impossible for children to keep still”).