276 M. Biofs Abstract of Mr Napier's 



up to tens of thousands of units, which they termed myriads, 

 and indicated by the capital letter M, surmounted by that letter 

 of the alphabet which expressed the number of myriads intended. 

 This being arranged, Archimedes supposes an indefinite progres- 

 sion of numbers, commencing by simple unity, successively mul- 

 tiplied by ten, so that when written out, in the notation now in 

 use, the progression would be, 



1; 10; 100; 1000; 10000; 100000; &c. 



But as in this method of notation we would soon find ourselves 

 embarrassed by the multiplicity of zeros following the unit, we 

 shall abridge the expression by the aid of that ingenious artifice 

 contrived by Descartes, and which consists in only writing out 

 the common factor 10, qualified by a numerical index of more 

 or less value, which marks (becomes the expone7it) how often 

 the common root 10 is, in the particular term, to be found 

 multiplied byitself. Then writing out, according to this method, 

 the successive terms of the progression, and noting beneath each 

 term the rank which it occupies after the first term, we shall 

 obtain the following lines : — 



1; 101; 102; 10»; W; 10^ 10^ W; 10^; &c. indefinitely. 

 0123 45678 



It is evident, upon mere inspection, that the number in the 

 lower line, which expresses the rank of each term, is equal to the 

 exponent which marks how often the common root 10 \% factor 

 in that particular term. This was not evident to the eye, at a 

 glance, in the literal notation employed by Archimedes; and, 

 moreover, it was not possible for him to express, as we do, the 

 character of indefinite extension which he proposed to give to 

 such a progression. What then was his expedient ? In the 

 first place, he considers apart the nine first terms from 1 to 10^ ; 

 but he can write out, and even name these terms ; for the highest 

 of them, that is to say ten thousand times ten thousand, is only 

 equal to a myriad of myriads. Placing, then, these eight first 

 terms by themselves, he calls them numbers of the first order. 

 In the next place, with the ninth term 10^, he composes a new 

 unity, which he calls of the second order, and he arranges these 

 new units like the former, in numbers progressively multiplied 



