38 DISSERTATION SECOND. |>art i. 



found from mere inspection, if the progression were far 

 enough continued. If, for instance, the third term of the 

 progression were to be multiplied by the seventh, the pro- 

 duct must be the tenth, and if the twelfth were to be divid- 

 ed by the fourth, the quotient must be the eighth ; so that 

 the multiplication and division of such terms was reduced 

 to the addition and subtraction of the numbers which indi- 

 cated their places in the progression. 



This observation, or one very similar to it, was made by 

 Archimedes, and was employed by that great geometer to 

 convey an idea of a number too vast to be correctly express- 

 ed by the arithmetical notation of the Greeks. Thus far, 

 however, there was no difficulty, and the discovery mi^ht 

 certainly have been made by men much inferiour either to 

 Napier or Archimedes. What remained to be done, what 

 Archimedes did not attempt, and what Napier completely 

 performed, involved two great difficulties. It is plain, that 

 the resource of the geometrical progression was sufficient, 

 when the given numbers were terms of that progression ; but 

 if they were not, it did not seem that any advantage could 

 be derived from it. Napier, however, perceived, and it 

 was by no means obvious, that all numbers whatsoever 

 might be inserted in the progression, and have their places 

 assigned in it. After conceiving the possibility of this, the 

 next difficulty was, to discover the principle, and to execute 

 the arithmetical process, by which these places were to be 

 ascertained. It is in these two points that the peculiar merit 

 of his invention consists ; and at a period when the nature 

 of series, and when every other resource of which be could 

 avail himself were so little known, his success argues a depth 

 and originality of thought which, I am persuaded, have 

 rarely been surpassed. 



The way in which he satisfied himself that all numbers 

 might be intercalated between the terms of the given pro- 



