napier's logarithms : the development of his theory. 139 



1619), and it is stated that they contain the first table of 

 logarithms to the base e. 1 Attention is also usually called 

 to the fact that, while logarithms to the base e are fre- 

 quently spoken of as Napierean logarithms, they are quite 

 different from the logarithms of Napier's Canon; and it is 

 pointed out that the place of the number e in the theory of 

 logarithms and the possibility of defining logarithms as 

 exponents were discoveries of a much later day. These 

 two statements, at first sight, seem inconsistent. A word 

 or two regarding Speidell's system will make the matter 

 clearer, and will also confirm the view I have taken above 

 as to Napier's final conception of the logarithm. 



Speidell's New Logaritlimes, like Napier's Canon, refer 

 to the trigonometrical ratios. Using Glaisher's notation 

 shx for Speidell's logarithm of x when the radius is 10', we 

 have sl r x = 10 r+1 — ril Y x. 



It follows that 



Sly (UV) = Sl r U + Sl r V — Sl r l, 

 Slr(liv) — Sl r U — Sl r V + SlA, 



and sl,l is not zero. 



The sole advantages of this system was that it avoided 

 the use of negative quantities in calculation with logarithms. 

 Such quantities were then outside the range of the "vulgar 

 and common arithmetic." 



Since nhx = 10 r log e (—) , 



\ x J 



we have six = 10 7+1 + 10 J> log e f-^rX 

 Thus sh Sin,* = 10 r+1 + 10 r log r 



10 1 

 = 10 r (10 + loge sin x). 



1 In Glaisher's paper already referred to, he published the interesting- 

 discovery that an Appendix (1618) to Wright's English translation of the 

 Descriptio contains a table of hyperbolic logarithms by an anonymous 

 author, whom he identifies with Oughtred. 



