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 100,000,000^2. Let 100,000,000^2 be the first number of the _third order_, and let this extend to 100,000,000^3; and so on, to the _myriad-myriadth_ order, beginning with 100,000,000^(99,999,999) and ending with 100,000,000^(100,000,000), 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 100,000,000^2 P, and so on up to the _100,000,000th order_ of the _second period_, ending with 1,000,000,000^(100,000,000) P or P^2. The _first order_ of the _third period_ begins with P^2, 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 P^(100,000,000). 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, 10^8, 10^16, 10^24, ... (10^8)^(99,999,999); and we can express all the numbers required in powers of 10.