10 



J. 



out of I + y is extrafted a root whofe index is i, that root will be i + y ; 



and fince (by lem. 3.) 14-^-1:1+;' — i : : n : m, it follows that 



i+e -I : i+y — I : : n : m. But it was proved above (art. 6.) that 

 the logarithms of the ratios of i to i+f and of i to i+y, are as n and m; 



therefore thefe logarithms are as i-^e -i and i+y -i ; that is, if out of 

 two numbers both greater than i be extrafted the fame root, the excefles 

 of thefe roots above i will be as the logarithms of the ratios that i has : 



to thefe numbers : and therefore if one of thefe excefles i +e -i (or 

 any multiple of it) be made the logarithm of the ratio of i to i +e, the 



other excefs i-fy — i (or the fame multiple of it) mufl: be made the lo- 

 garithm of the ratio of i to 1+7. 



8. Nov/ according to Neper's firfl: plan, as publiflied in the Canon Mirificus, 

 when he refolved any ratio (as of i to i +f) into a fufficient number of rati- 

 unculce, or which is the fame thing, when he had placed between i and i-\-e 

 a fufficient number of mean proportionals, he made the excefs of the firft, 

 or leafl: of them above i , to be the logarithm of one of thefe ratiunculx : 



X 



thus, if I +6 be the firft, or leafl:, of the mean proportionals between i 



and I + f, then he made i+f -i, to be the logarithm of the ratio of 



I to !+<? , or, as it is ufually called, the logarithm of the number i+e ; 



J. 



and then it will follow from the nature of logarithms, that «xi +f -i will 

 be the logarithm of the ratio of i to 1 +e, or of the number i -f ^ ; and hence 



J. 



n 



again (by what was faid in art. 7.) it follows, that nXi-\-y — i mull be 

 the logarithm of the ratio of \ to i-Vyt or of ^^ number i-\-y. 



9. Let 



