THE GENESIS OF LOGARITHMS 167 



to be effected in this manner, that o should become the logarithm 

 of unity, and 10,000,000,000 that of the whole sine ; which I 

 could not but admit was by far the most convenient of all. 

 So, rejecting those which I had already prepared, I commenced, 

 under his encouraging counsel, to ponder seriously about the 

 calculation of these tables." 



The notably high characters of both Napier and Briggs, 

 the strong friendship which existed between the two writers 

 and the unqualified admiration and veneration which Briggs 

 ever shows of his master, justify us in taking these words 

 at their plain meaning ; it may be added that a more exhaustive 

 study of the evidence afforded serves to confirm the views 

 stated above. 



It is not necessary here to go into any great detail con- 

 cerning the manner of computation of logarithms to the base 10, 

 as a clear account of some of the methods used is easily ac- 

 cessible in the article " Logarithms " in the Encyclopedia 

 Britannica. Thus, for example, in computing the logarithm 

 of 5, given log 1 and log 10, the geometric mean of 1 (A) 

 and 10 (B) is taken, giving C= V A B. Then, finding D = </ B C, 

 E— \/ C D, etc., we finally arrive at a mean which may be 

 made to approach as closely as we please to the value 

 5*00000. . . . And to every geometric mean there corresponds 

 a logarithm obtained by continually taking arithmetic means 

 of the logarithms in like manner, finally giving 



log 5'oooooo = '6989700. 



A second method is outlined by Napier in the Appendix 

 to the Constructio. In his own words — 



"... the Logarithm of any given number is the number 

 of places or figures which are contained in the result obtained 

 by raising the given number to the 10,000,000,000th power. 



"Also if the index of the power be the Logarithm of 10 

 the number of places, less one, in the power or multiple, will 

 be the Logarithm of the root. 



" Suppose it is asked what number is the Logarithm of 2. 

 I reply, the number of places in the result obtained by 

 multiplying together 10,000,000,000 of the number 2. 



" But, you will say, the number obtained by multiplying 

 together 10,000,000,000 of the number 2 is innumerable. I 

 reply, still the number of places in it, which I seek, is 

 numerable. 



