484 Prof. H. S. Carslaw oa Napier s Logarithms : 

 Since nl r x = 10 r 1 og e I — ) , 



we have , . - . . . _ , / # \ 



sl^^lO^ + lOMog.^j. 



Thus , ~. ^/v ., -.^ i /Sin r #\ 



sl r Sm r ^ ^lO^i + lC log, (-j^r), 



= 10 r (10+log e sin.£). 



In a sense SpeidelPs ' New Logarithmes ' may be said to 

 be hyperbolic logarithms, but the sense is the same as that 

 in which the logarithms of Napier's Canon are sometimes 

 said to be logarithms to the base e~ l . But this is a misuse 

 of the term *. Still Speidell's logarithms of sines, from the 

 accident that the sine is now used in a different sense, have 

 actually the same figures as our hyperbolic logarithms of 

 sines. 



In the 'New Logarithmes ' (1619) he takes the radius 

 as 10 5 , so that these tables give 



sl 5 Sin 5 # = 10 5 (10 + log e sin x) . 



§ 8. But subsequently Speidell did publish a table of 

 hyperbolic logarithms of numbers, which gives the values 

 of 10 6 log e .a? for the numbers 1 to 1000. This table probably 

 appeared either separately, or attached to an impression of 

 the ' New Logarithmes,' in 1622 or 1623. In this system 

 he takes 



sl r # = nl r 1— uArW. 

 It follows that 



sly (uv) = sl r u +- sl r v, 



sl r (ujv) = sl r u — sl r v ; 

 nl r x = 10 r log e ( — j , 



we ha ve sl r x = 10 r log e so. 



But it is clear that in both Speidell's systems of logarithms 

 the connexion with hyperbolic logarithms is accidental, and 

 the same is true of the logarithms discovered by Glaisher, 

 to which reference is made at the beginning of this section. 



Like Napier and Briggs, Speidell sees that the fundamental 

 property, that the logarithms of proportional numbers have 



* Cf. Glaisher, loc, cit. p. 146, footnote. 



and since 



