the Development of his Iheory. 477 



Construct™. In the third, referred to in the Appendix to 

 the Construction he has reached the idea of a logarithm as 

 defined by the property : — 



The logarithms of proportional numbers have equal differences, 

 with the additional condition that the values of the logarithms 

 of two numbers are given. 



In the second and third stages he has obtained, what we 

 would now call, a function of the independent variable — the 

 number — , but the function of the third stage is more general 

 than that of the second, which it includes as a special case. 



If this view is correct, the statement that " Napier's theory 

 rests on the establishment of a one-one correspondence 

 between the terms of a geometric series and the terms of an 

 arithmetic series " * should not be taken too literally. 

 Further, the custom of employing the term " Napier's 

 logarithms " to describe only the logarithms of his Canon is 

 unfortunate. It will be seen in the course of this paper that 

 logarithms to the base 10 — as we now know them — are 

 Napier's logarithms just as much as the logarithms of his 

 Canon. 



The First Stage. 



§ 2. The idea that multiplication and division could be 

 reduced to addition and subtraction by the correlation of a 

 geometrical series and an arithmetical series was not a new 

 one. Aristotle was familiar with it, and since his time many 

 mathematicians had returned to it. If we take the series 



1, 2, 3, 4, 5, 6, 7, 8, 15. ... 



2, 4, 8, 16, 32, 64, 128, 256, 32768, ... 



the product of 128 and 256 in the geometrical series can be 

 read off as 32768, which corresponds to 15, the sum of 7 

 and 8, in the arithmetical series. 



The Swiss Biirgi in his Arithmetische und Geometrische 

 Progress Tabulen f, constructed some time between 1603 

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



10x0, 10x1, 10x2, 10 xn,... 



108 ' w { 1+ w\ H 1+ M> w ( 1+ mf-- 



* Cajori, 'The American Mathematical Monthly,' vol. xxiii. p. 71 

 (1916). 



t A facsimile of the title-page of Biirgi's work, and of one of the 

 pages of his 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 Braunmuhl's Geschichte der Trigonometric will show 

 that in none of these is the title quoted correctly. 



