napier's logarithms : the development of his theory. 135 



and nhl - tiJ 7 10 = 23025842*34 (Of. Constructio, Section 

 53). * 



It is obvious that if a system of logarithms could be 

 devised in which the logarithm of unity is zero and the 

 logarithmof 10 is unity, the calculations would be immensely 

 simplified, and the table curtailed; because one of the chief 

 defects of Napier's Canon, as well as of Biirgi's Tables, was 

 that, if the numbers did not come within the range covered 

 by it, more or less awkward calculations were needed to 

 overcome this difficulty. 



Napier's Canon was first printed in the Descriptio (1614). 

 After his death in 1617 the Constructio was published by 

 the care of his son. It had been written several years 

 before the Descriptio, To this work was added an Appen- 

 dix, by the hand of Napier himself, "On the Construction 

 of another and better kind of Logarithms, namely one in 

 which the Logarithm of unity is 0." This Appendix begins 

 with the words: — 



"Among the various improvements of Logarithms, the 

 more important is that which adopts a cypher as the 

 Logarithm of unity, and 10,000,000,000 as the Logarithm 

 of either one-tenth of unity or ten times unity. Then, 

 these being once fixed, the Logarithms of all other numbers 

 necessarily follow.''' 



It is clear from Napier's words that, when he wrote the 

 Appendix, not only did he see the advantage of such a 

 system, but he was in a position to draw up a Table of 

 Logarithms in which these conditions would be satisfied. 

 Indeed he gives three distinct methods of finding these 

 logarithms. The kinematical definition of the logarithm 

 was superseded, and the correspondence between the terms 



1 This number should be 23025S50 - 93, since it is easy to show that 

 nl n \ - wi.10 =nZ 7 10 6 , and Macdonald gives the corrected logarithmof 

 10 6 (loc. cit., pp. 94-5). 



