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



His tables cover the range 10 8 to 10 9 , and for all practical 

 purposes are as satisfactory as Napier's Table of Logarithms 

 of 1614. If Napier had simply used the idea of the cor- 

 respondence between the terms of a geometrical series 

 and the terms of an arithmetical series, his work could 

 not be regarded as so great an advance upon Biirgi's as it 

 really is. 



But it is clear that at the beginning of his labours, which 

 extended over a period of 20 years, Napier's mind was 

 working on the same lines as Biirgi's, and that he used the 

 series 



0, 1, 2, 



10', w(l-jj,), 10'(l-^) 2 



The geometrical series occurs in the Constructio. He 

 employed it in the calculation of his logarithms, but neither 

 then, nor later, are his logarithms the terms of the corre- 

 sponding arithmetical series. His word logarithm (see 

 above, § 1) is evidently a survival of this stage of his work. 



Napier meant his Tables to be used in calculations 

 involving the trigonometrical ratios. In his time, the sine, 

 cosine, &c, were lines — or, more exactly, the measures of 

 lines — in a circle of given radius. Napier took the radius 

 as 10 7 . It may be that Biirgi chose 10 8 in his Tables for a 

 similar reason. With our notation, Napier's numbers would 

 correspond to 7-figure Tables of Natural Sines, &c. If 

 greater accuracy were required, the radius was taken as 10 10 , 

 and sometimes even a higher power of 10 was used. These 

 sines, &c, following Glaisher *, we shall refer to as line- 

 sines, &c. 



The Second Stage. 



§ 3. Napier opened out entirely fresh ground when he 

 passed to his kinematical definition of the logarithm of a 

 sine or number. By this definition he associated with the 

 sine, as it continually diminished from 10 7 for 90° to zero 

 for 0°, a number which he called its logarithm; and the 

 logarithm continually increased from for the sine of 90 to 

 infinity for the sine of 0°. 



The fundamental proposition in Napier's theory in the 



* The Quarterly Journal of Pure and Applied Mathematics, vol. xlvi. 

 p. 125 (1915). To this paper I am indebted, not only for a most con- 

 venient notation for the different systems of logarithms, but also for 

 an account of Speidell's work, ''hitherto. inaccessible to me. 



