THE GENESIS OF LOGARITHMS 169 



But, before concluding, one widespread error calls for notice. 

 In many text-books on trigonometry we find the statement 

 that, to avoid the inconvenience of printing negative charac- 

 teristics, the number 10 is always added to the logarithms of 

 sines, cosines, etc., thus giving the so-called logarithmic sines, 

 etc. This is by no means an exact statement of the facts. The 

 trigonometrical tables most in use at the beginning of the 

 seventeenth century were constructed, as previously explained, to 

 the radius io 10 . It follows, therefore, that the logarithm of radius 

 (the sine of 90°) is 10, with corresponding numbers having as 

 characteristics 9, 8, . . . etc., for the remaining sines. The 

 numbers, therefore, given as " tabular logarithms " in a modern 

 book of tables are actual logarithms, the manner of printing 

 them having never been altered ; the modern conception of 

 the trigonometrical functions as ratios gives us, however, 

 unity as the sine of 90 ; consequently, the tabular logarithms 

 as printed are, in ever}' case, too great by ten. It seems a 

 pity that the account of such an interesting remnant of seven- 

 teenth-century usage should be obscured by the usual " expla- 

 nation " of trigonometrical text-books. 



's' 



APPENDIX 

 Note A 



The following short list of authorities consulted may be useful to those who 

 wish to pursue the subject further : 



Napier, Mirifici Canonis Logarithmorum Descriptio. First edition, 1614. 

 A reprint is contained in vol. vi. of Baron Francis Maseres' compilation en- 

 titled Scriptores Logarithmici. English translations made by Edward Wright 

 (1616) and Herschell Filipowski (1857). 



Napier, Mirifici Canonis Logarithmorum Constructio. First Edition, 1619. 

 English translation, together with an exhaustive bibliography of Napier's works* 

 made by W. R. Macdonald (1889). 



Hutton's Mathematical Tables. A full and for the most part accurate 

 historical introduction is prefixed to the earlier editions of the above Tables. 

 This introduction is reprinted in Hutton's Mathematical Tracts, vol. i. (1812). 



Napier, Mark, Memoirs of John Napier of Merchiston (1834). 



Encyclopaedia Britannica. Tenth edition. Articles Logarithms, Mathematical 

 Tables, Napier, etc. 



Montucla, Histoire des Mathematiques ; completed and published by 

 Lalande (1 799-1 802). 



Fink, Geschichte der Elementar-Mathematik. English translation by Beman 

 and Smith (1903). 



Cajori, A History of Mathematics. 



