166 Memoir of John Napier, 



Then, what did he do? He considered first, the nine first 

 terms from 1 to 10 3 separately. Now, he can write them, 

 and even name them, since the last term 10 8 or ten times 

 ten thousand only was equal to a myriad of myriads ; 

 separating, then, the first 8 terms, he called them numbers of 

 the first order ; then, with the ninth term 10 8 he forms a 

 new unit, which he terms of the second order; and he 

 arranges these new units, like the preceding, in numbers 

 progressively decupled the one by the other, until he 

 arrives at the 8th term, which is 10 15 ; so that the follow- 

 ing 10 16 is found to be a myriad of myriads of numbers of 

 the second order. Working with this term 10 16 as with 

 10 s he forms a new unit, which he terms of the third order ; 

 and, continuing to form successive numbers of units, each 

 of which commences at the myriad of myriads of the preced- 

 ing numbers, it is evident that we may proceed in the series 

 as far as we please, and even express all these terms orally, 

 for it is sufficient to consider them all placed consecutively, 

 and to separate them by orders or octades, as in the follow- 

 ing lines, 



1st Order, 



1; 10* ; 10«; 1Q3; 10 4. 105; 1Q6; 10'. 



2d Order, 

 10 8 ; 10 9 ; 10 10 ; 10 11 ; 10 12 ; 10 13 ; 10 1 *; 10 15 . 



3d Order, 

 10 16 ; 16 1 ' ; 10 18 ; 10 19 ; 10 20 ; 10 2 1 ; 10 22 ; 10 23 . 



4th Order, 

 10 24 ;10 25 ;10 26 ;10 27 ; 10 28 ; 10 29 ; 10 30 ; 10 3 * ; 10 32 ,&c. 

 Thus, any term, however far distant from the first, may 

 be completely defined and named, by stating the order or 

 octade to which it belongs, with its place in the octade ; 

 and besides, this method of characterizing it will be in- 

 finitely more simple, than if we wished to write it in an 

 explicit manner ; for example, in proceeding from the 

 dimensions of a small grain of sand, and raising it from 

 multiple to multiple by means of his series, until we 

 conceive it to fill a sphere of an equal diameter to that of 

 the stars, Archimedes proves that the total number of these 

 grains, will be less than a thousand myriads of 8th numbers ; 

 now, according to this table, it is easy to see that the simple 

 units of this 8th order have for their expression the 



