480 Prof. H. S. Oarslaw on Napier's Logarithms : 



Thus multiplication and division are changed into addition 

 and subtraction. But the logarithms of numbers with the 

 same figures in the same order cannot be read off from one 

 another, since in this system, 



nl r (10 m a) = nl r a — m (nl r 1 — nl r 10), 



and nl 7 l—nl 7 10 = 23025842-34 (cf. Construct™, Section 53). 



It is obvious that if a system of logarithms could be 

 devised in which the logarithm of unity is zero and the 

 logarithm of 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 Appendix, by 

 the hand of Napier himself, " On the Construction of another 

 and better Mnd of Logaritinns, 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 w T ords 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 superseded, and the correspondence between the terms of a 

 geometrical series and the terms of an arithmetical series 

 was left far behind. This is the third and final stage of his 

 work. 



Briggs and Napier. 



§ 5. In the change from the logarithms of the Canon to 

 this " better kind of logarithms " Briggs was associated with 

 Napier ; but, chiefly because of the unsatisfactory account of 

 the matter given by Hutton in his ' History of Logarithms' *, 



* Hut ton's l Tracts on Mathematical and Philosophical Subjects/ 

 vol. i. Tract 20. 



