482 Prof. H. S. Carslaw on Napier's Logarithms : 



is for this reason that the characteristics 9, 8, &c, are to be 

 found in the logarithms of the sines, &c. 



Using the notation h\ r x for the logarithm of x in the system 

 suggested by Briggs when the radius is 10% we have 



blr a — h\ r b = bl r c— hl r d, 



when a:b = c : d. 



Also bl r 10^=0, and bl r 10 r - 1 =10 , °. 



In this system we have 



bl,. (w) = bl r w + bl r v— bl,. 1, 



bl,. (ujv) = h\ r u — b\ r v + bl r 1. 



Also bl 30 10 10 = 10bl 10 10 - 9bl 10 l = 0. 



bl 10 10 9 = 9bl 10 10 - 8bl 10 l = 10™. 



Thus bl 10 10 = 9 x 10 10 and bl 10 l = 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 n V) = bl r a-mXl0 10 . 



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

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

 " conceived that the change ought to be effected in this 

 manner, that should be the logarithm of unity, and 

 10,000,000,000 the logarithm of the whole sine/' And in 

 the Appendix we see that he often passes from logarithms of 

 sines, and drops all reference to the radius. In the new 

 system, logarithms were to be defined by the relations : — 



If a : b = c : d, then 



nla— nl5 = nlc — nld 

 with nl 1 = and nl 10 = 10 10 . 



It need hardly be added that 10 10 was taken for the 

 logarithm of 10 instead of unity, for the same reason that 

 10 7 (or 10 10 ) was taken for the radius in dealing with the 

 trigonometrical ratios. 



Later, Briggs takes the logarithm of 10 as unity, and 

 introduces the notation of decimal fractions in his Tables, a 

 notation employed, probably for the first time, by Napier 

 himself. 



If this account of the growth of the idea of a logarithm in 

 Napier's work is correct *, it seems unfortunate that the 



* See also Gibson's paper in the 'Napier Tercentenary Memorial 

 Volume,' pp. 111-137. 



