VOL. XIX. 3 PHILOSOPHICAL TRANSACTIONS. 21 



Again, if the logarithm of a decreasing ratio be sought, the infinite root of 

 1 

 1 — o, or 1 — « K IS 1 — -o — — -o^ — -— o^_-_o* q^ — 



•^ 9*, &c. whence the decrement of the first of our infinite number of propor- 

 tionals will be - into 9 + -r ? ? + t 9^ + 4 9* + i 9* + t 9^ &c. which 



therefore will be as the logarithm of the ratio of unity to 1 — q. But if m be 

 put 10000 &c. then the said logarithm will be q-\-±.qq-\-:^q^-\-^q^-\. 

 i 9= + i q\ &c. 



Hence, the terms of any ratio being a and b, q becomes — ^, or the differ- 

 ence divided by the lesser term, when it is an increasing ratio ; or ~ when 



it is decreasing, or as b to a. Whence the logarithm of the same ratio may 

 be doubly expressed ; for putting x for the difference of the terms a and b, it 

 will be either 



I . , X X- . Xi . X* x^ , x^ - 



-m '"^° 6 + ^ + 361 + n^ + 56-5 + 64^ ' &C. Or 



1 . , X J^ j^ X3 X* xi X* 5 



- mto - — ^-^ + — _ -_ -J. _ _ _ J &c. 



But if the ratio of a to i be supposed divided into two parts, viz. into the 

 ratio of a to the arithmetical mean between the terms, and the ratio of the 

 said arithmetical mean to the other term b, then will the sum of the loffarithms 

 of those two rationes be the logarithm of the ratio of a to ^ ; and substituting 

 -L z instead oi ^ a -\- -^ b, the said arithmetical mean, the logarithms of those 

 rationes will be by the foregoing rule. 



be the logarithm of the ratio of a to b, whose difference is x and sum z. And 

 this series converges twice as swift as the former, and therefore is more proper 

 for the practice of making of logarithms: which it performs with that expedition, 

 that where x the difference is but a 100th part of the sum, the first step — suf- 

 fices to seven places of the logarithm, and the second step to twelve ; but if 

 Briggs's first 20 chiliads of logarithms be supposed made, as he has very care- 

 fully computed them, to 14 places, the first step alone is capable to give the 

 logarithm of any intermediate number true to all the places of those tables. 



