the ratio of i to 2 will be refolved into 301029995 of thefe ratiun- 

 cul^; and thefe numbers, looooooooo and 301029995, will be the 

 logarithms of the ratios of i to 10 and of i to 2, as being the num- 

 bers of equal ratiunculse of which thefe ratios arc compounded. 



4. But though thefe numbers be immediately and properly the lo- 

 garithms of thefe ratios, they are not the only ones that can be ufed as 

 fuch ; any two numbers (or indeed any two quantities of the fame kind) 

 that have the fame ratio with them, may be made their logarithms. Thus, 

 if there be any convenience in having i for the logarithm of the ratio of 

 I to ID, and if 0,301029995 be to i, as 301029995 is to looooooooo, 

 then may 1 and 0,301029995 be made the logarithms of thofe ratios. 



5. If I be made the antecedent of any ratio, that ratio may be re- 

 folved into any number of equal ratiuncula;, by extrafting out of the 

 confequent a root, the dominator of whofe index is the number of rati- 

 unculse that is required. Thus, if it be required to refolve the ratio of 



jr. 



I to i+e into n ratiunculie, the ratio of i to i+e will be tlje firft of 

 them ; and it is fufEcient to find one of them, each of the others being 



equal to it. 



6. Let now e and y be any two numbers, of which e is the greater, 

 and between i and i + e let a feries of mean proportionals be placed, 

 whofe number is n — i, the ratio of i to i -j-e will be refolved into n 

 ratiunculse ; if of thefe means any number denoted by m ftands between 

 I and I + y, the ratio of i to i + y will be refolved into m ratiunculcc, 

 each equal to one of the former ; and from what was faid above (art. 3.) 

 the logarithms of the ratio's of i to i + e and of i to i+y will be as n 

 and m. 



7. The firfl: of thefe mean proportionals is 1 -{-e , and if this quantity 

 be involved to a power whofe index is m, that power will be equal to 

 1 -hy, (or fo near to it, that it may be ufed for it without any error,) 

 1 i i 



■ ■ ' n • I ™ ft "■ PI 



that is i + e = 1 +y, and therefore 1 + e = i+y. Suppofe now th^ 

 Vol. IX. ( B ) out 



