napier's logarithms : the development of his theory. 131 



The logarithms of proportional numbers have equal 

 differences, with the additional condition that the loga- 

 rithms 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 correspond- 

 ence between the terms of a geometric series and the terms 

 of an arithmetic series" 1 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 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. 



1 Cajori, Napier's Logarithmic Concept : A Reply. American Mathe- 

 matical Monthly, Vol, 23, p. 71, (1915). 



