170 SCIENCE PROGRESS 



Ball, A Short History of Mathematics. 

 De Morgan, Trigonometry and Double Algebra. 



Glaisher, Articles in the Philosophical Magazine for Oct. and Dec. 1872, 

 and May 1873. 



Note B 



The fiftieth term in Napier's Second Table, given as 9,995,001 '222927 is 

 incorrect, the true value being 9,995,001 "224804. 



This, of course, introduces a corresponding error into the logarithms attached 

 to the Radical Table, inasmuch as we have seen that the logarithm of the first 

 proportional in the Radical Table is obtained from the logarithm of the last 

 proportional in the Second Table by means of a theorem which involves the 

 difference of the proportionals ; and, one of the proportionals being in error, 

 the logarithm will also be incorrect. The magnitude of the error introduced 

 may be shown by noting that the logarithm of the last term in the Radical Table 

 is given as 6,934,250 8, its true value being 6,934,253*4 — an error of rather less 

 than one in 2J millions. 



The mistake, unnoticed by Hutton, seems to have been first pointed out by 

 Biot in 1835 and later in 1865 by Sang. 



Note C 



The derivation of the word logarithm is not without interest. Even when 

 we know that logarithm = Xoyav dpifffios = " the number of the ratios," the 

 modern mode of deriving logarithms as powers, and of computing logarithms 

 by means of series, is apt to render the meaning underlying the phrase 

 "number of the ratios" somewhat obscure. But the originators of the word 

 looked at the subject of logarithms from the point of view of compounded 

 ratios. Suppose, then, that the ratio of 10 to 1 is compounded of, say, a 

 million small ratios or ratiunculas, each of which is, of course, the millionth 

 root of ten. Then the ratio of 2 to 1 is compounded of 301,030 of these 

 small ratios, so that the logarithm of 2 is given by the number of the ratios 

 or ratiunculae which is contained in the ratio of 2 to 1. Hence the word 

 logarithm. 



It may be noted that whilst Napier uses the word Logarithmus in the 

 Descriptio, published in 16 14, he uses, in the text of the posthumously published 

 Construction the phrase Numerus artificialis, or simply Artificialis, as opposed 

 to Numerus naturalis for the ordinary numbers. The term Logarithmus, how- 

 ever, is used in the title-page, headings and Appendix to the Constructio. 



