Baron of Merchiston. 173 



Besides the merit of the invention, the tahles of Napier 



are a prodigy of patient labour. When we think of the 



time and labour required to calculate these tables, we are 



led to be anxious lest any chance should have prevented 



him from realizing his idea, and that it should have died 



with him. It has been said, and Delambre has repeated 



the statement, that the last cyphers of his numbers were 



incorrect. This is true, but a more useful fact would have 



been, to know if the error arose from the method, or from 



some fault in the calculation. I have done this, and have 



ascertained that it was in reality produced by a fault of the 



latter kind, a very small error in the last term of the second 



progression which he forms for preparing the calculation of 



his table. Now, all the following steps are deduced from 



this which produces the small error remarked. I have 



corrected the error, and with his method, but abridging the 



operations by our more rapid processes of developement, 



have calculated the logarithm of 5000000, which is the 



last of the table of Napier, upon which, consequently, all the 



errors accumulate; I have found its value 6931471808942, 



while, by the modern series, it ought to be 6931471805599. 



Thus, the difference begins at the tenth figure. I have 



calculated, likewise, the hyperbolic logarithm from 10 



according to the corrected numbers of Napier; I have found 



for its value 2,30258 50940 346, while, by our actual tables, 



it is 2,30258 50929 940; the real difference then takes 



place at the ninth decimal. If Napier had possessed at his 



disposal a village teacher to calculate, by subtraction, a 



geometrical progression more slow still than that of which 



he has made use, the tables of Briggs with 14 decimals 



would not have been superior to his. 



After this immense invention of logarithms, we can 

 scarcely mention some other works, which proceeded from 

 him. The former was sufficient for his life and for his 

 glory. He discovered some ingenious theorems for con- 

 tracting, in certain cases, the resolution of spherical tri- 

 angles, and these have been called from his name the ana- 

 logies of Napier. But their utility was greater in his time 

 than at present. The advancement of analytical processes 

 lias superseded, in a great measure, the use of particular 

 reductions; and we now understand, that these general 



