napier's logarithms : the development of his theory. 137 



prepared, I commenced, under his encouraging counsel, to ponder 

 seriously about the calculation of these tables." 



Napier also mentions his discovery of the new system in 

 the dedication of his Rabdologia (1617) in a passage quoted 

 in my previous paper. 1 



It will be seen from Briggs' own words, that the modifi- 

 cation which he suggested to Napier was to keep the 

 logarithm of the radius as zero, but to take the logarithm 

 of one-tenth of the radius as 10,000,000,000. His reference 

 to the Canon is sufficient to show that he does not look 

 upon the radius as unity. In the construction of the Table 

 of Logarithms, after Napier's death, he takes it as 10 10 , and 

 it is for this reason that the characteristics 9, 8, etc., are 

 to be found in the logarithms of the sines, etc. 



Using the notation bl r x for the logarithm of x in the 

 system suggested by Briggs when the radius is 10 r , we have 



bl r ci — bl r b = blr c — bl r d, 

 when a : b = c : d. 



Also blrW = 0, and bhlO"- 1 = 10 10 . 



In this system we have 



blr(uv) = bl r U -f bl r V — U r l, 

 bl r (u/v) = bl r U - blrV + bl r l. 



Also MiolO 10 = 10 WiolO - 9 bZ l0 l = 

 bZiolO 9 = 9 WiolO - 8 Wiol = 10 10 . 

 Thus bholO = 9 x 10 10 and W, 1 = 10 x 10 10 . 

 The advantage of the new system consists in the fact 

 that the logarithms of numbers with the same figures in 

 the same order could be read off from each other, since we 

 have bl r (10 m a) = bl,a - m X 10 10 . 



§ 6. The change upon which Napier had resolved, previous 

 to Briggs' visit, was a much more important one. He 

 "conceived that the change ought to be affected in this 



1 See also Macdonald's English translation of the Constructio, p. 88. 



