132 H. S. CARSLAW. 



The Swiss Biirgi in his Arithmetische and Geometrische 

 Progress Tabulen, 1 constructed some time between 1603 

 and 1611, but first published in 1620, used the series 

 10 x 0, 10 x 1 , 10 x 2 ,...10 x n, 



10°, 10° (i +4) 10' ( 1+^)1 ... 10° (i+^V. 



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 corres- 

 pondence 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 twenty years, Napier's mind was 

 working on the same lines as Biirgi's, and that at this stage 

 he used the series 



0, 1 ,2 



ioMo^i-^), io'(i-^:...... 



in a similar way. This 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 corresponding arithmetical series. His 

 word logarithm, (See §1), is evidently a survival of the 

 first stage of his work. 



Napier meant his tables to be used in calculations involv- 

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

 cosine, etc., were lines— or, more exactly, the measures of 

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



1 A facsimile of the title page of Biirgi's work and of one of the pages 

 of the Tables will be found in the Napier Tercentenary Memorial Volume 

 (Plates XII and XIII). Comparison with the references in Cantor's 

 Geschichte der Mathematik, Tropfke's Geschichte der Elementar-Mathematik^ 

 and BraunniuhTs Geschichte der Trigonometric will show that in none of 

 these works is the title quoted correctly. 



