140 H. S. CARSLAW. 



In a sense Speid.ell's 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" x . However this is a mis- 

 use of the term. 1 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 6 (10 + loge sin x). 



§ 8. But subsequently Speidell did publish a table of 

 hyperbolic logarithms of numbers, which gives the values 

 of 10 6 loge # for numbers 1 to 1,000. 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 X = nlrl - nl r X. 



It follows that 







Sir (UV) =Sl r U + Sir V, 





Sir (u/v) = Sir U — sl r V\ 



and since 







nlr X = 10 r lOge (— ), 



we have 



Sir X = 10 r lOge X. 



But it is clear that in both SpeidelPs systems of logarithms 

 the connection 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 funda- 

 mental property, that the logarithms of proportional 

 numbers have equal differences, can be taken as the starting 



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



