134 



H. S. CARSLAW. 



In this paper I follow his notation, and log e x is used in its 

 modern sense for the logarithm of x to the base e, the 

 system commonly called hyperbolic logarithms. 



The fundamental theorem, referred to above, can now be 

 stated as follows: — 



If a : b = c : d, then nl r a - nl r b = ril r c - nhd (1) 



Also we are given that nl r 10' = (2) 



Napier's Canon consists of a Table of Logarithms in which 

 (1) and (2) are satisfied. His definition of the logarithm by 

 means of the velocities of two points moving in two different 

 lines leads to the formula 



nl r x = 10 r loge (— ). 



\ X ' 



But, of course, neither this, nor the fact that his function 

 nl r x has -1 for its differential coefficient, when # = 10% 

 could be known in his time. 



The Third Stage. 

 § 4. Since uv :u = v : 1, 

 we have nl, {uv) — nl r u = nl r v — nl r l. 



Thus nl r (uv) = nlrit + nl r v — nl r l, 

 and it must be remembered that nl r l is not zero. 



When r = 7, nl r l = 161180896*38 (Of. Construct™, Sec- 

 tion 53). x 



Similarly nl r (u/v) = nl v u - nl r v + nlrl. 



Thus multiplication and division are changed into addition 

 and subtraction. But the logarithms of numbers with the 

 same figures in the same order cannot be read off from one 

 another, since, in this system, 



nlr (10 m a) = nl r a - m (nl r l - nl r 10), 



1 The error in Napier's Second Table affects the accuracy of his Canon 

 and this number should be 161180956 - 51. The alteration can be made 

 from the corrected result given by Macdonald in his English translation 

 of the Constructio pp. 94-5, for it is not difficult to show that nZ 7 1 = 7 nl 7 10°. 



