Archimedes / by Sir Thomas Heath.

  • Thomas Little Heath
Date:
1920
Catalogue details

Licence: In copyright

Credit: Archimedes / by Sir Thomas Heath. Source: Wellcome Collection.

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    ioo.ooo.ooo100’000’000 F 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 mo,ooo,oooth period ending with pioo,ooo,ooo 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 I followed by 800,000,000 ciphers, while the last number of the 100,000,oooth 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 I, IO8, ioiC, io24, . . . (i o8)99’999’999 ; and we can express all the numbers re¬ quired in powers of io. Since the diameter of a poppy-seed is not less than voth of a dactylus, and spheres are to one another in the triplicate ratio of their diameters, a sphere of diameter i dactylus is not greater than 64,000 poppy-seeds, and, therefore, contains not more than 64,000 x 10,000 grains of sand, and a fortiori not more than 1,000,000,000, or 1 o9 grains of sand. Archimedes multiplies the diameter of the sphere continually by 100, and states the corre¬ sponding number of grains of sand. A sphere of diame¬ ter 10,000 dactyli and a fortiori of one stadium contains less than io21 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 ol 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.