Memoirs of John Napier of Merchiston. 277 



by ten, till he arrives at the eighth term of this new order, which 

 is 10^^ ; so that the next term 10^^, is in like manner found to 

 be a myriad of myriads of numbers of the second order ; and 

 thus, by continuing to compose successive orders of units, the 

 first term of which is always the myriad of myriads in the pre- 

 ceding order, it is evident that he could extend the series as far 

 as he chose, and even express evefy term orally ; for, in order 

 to do so, no more was necessary than to conceive them all placed 

 consecutively after each other, and then to separate them into 

 orders, or ociades, as in the following lines : — 



102^ 10^^ 10«; W^\ 1028; 10«^ W% 10^^ 



Thus any term, at whatever distance from the first, may be 

 perfectly defined, and named, by announcing the order or octade 

 to which it belongs, in addition to its place in that octade itself; 

 and, moreover, that mode of characterising will be infinitely 

 more simple than if one attempted to write it out in an explicit 

 manner ; for, to take the example, in commencing with the 

 dimensions of a little grain of sand, and rising, from multiplica- 

 tion to multiplication, by means of this series, so as to conceive 

 a sphere, composed of these grains, equal to the sphere of the 

 whole galaxy, Archimedes proves that the sum-total of these little 

 grains will be less than a thousand myriads of numbers of the 

 eighth order ; now, from the table given above, it is easy to see 

 that the unit's term of that eighth order will be expressed by the 

 number 10 multiplied by itself 5Q times; and as a thousand 

 myriads of units make a thousand times ten thousand, or, 10 

 seven times factor, we find that the number announced by Archi- 



VOL. XX. NO. XL. APRIL 1886. T 



