VOL. XIX.J PHILOSOPHICAL TRANSACTIONS. 25 



that in the inscribed polygon of 512 sides, in the 18th figure was a O instead 

 of 9, which being rectified, and the subsequent work corrected therefrom, the 

 result agreed to a unit with our number. x\nd this I propose, not to cavil at an 

 easy mistake in managing, of so vast numbers, especially by a hand that has so 

 well deserved of the mathematical sciences, but to show the exact coincidence 

 of two so very differing methods to make logarithms, which might otherwise 

 have been questioned. 



From the logarithm given, to find what ratio it expresses, is a problem that 

 has not been so much considered as the former, but which is solved with the 

 like ease, and demonstrated by a like process, from the same general theorem 

 of Mr. Newton : for as the logarithm of the ratio of 1 to 1 + 9 was proved to 



be J + q\"' — 1 , and that of the ratio of 1 to I — 9 to be 1 — 1 — ^j"" *• so 

 the logarithm, which we will from henceforth call l, being given, 1 -f- l will 



1 



be equal to I + ^|'"in the one case; and 1 — l will be equal to l — r^f in 



the other: consequently l + lV" will be equal to i -\- q, and I — l)"* to 

 1 — q; that is, according to Mr. Newton's said rule, I -\- nih -\- Vm" l^ 

 + im^L^ + ^'-7n' L-* + -pl-m'L' &c. will be = 1 + y, and I — m l 

 + -A-Tn'-L- — -^m^ v^ + -p-m* 1.* — -j^-g-m'* l^ &c. will be equal to 1 — q, m 

 being any infinite index whatever; which is a full and general proposition, from 

 the logarithm given, to find the number, be the species of logarithm what it 

 will. But if Napier's logarithm be g ven, the multiplication by m is saved, 

 which multiplication is indeed no other than the reducing the other species to 

 his, and the series will be more simple, viz. 1 + L + 4-i-L-f- -^l^ + -Vl* 

 + ^t.l'- &c. or 1 - L + iL L - 4-L^ + VtL^ — twL' &c. This series, 

 especially in great numbers, converges so slowly, that it were to be wished it 

 could be contracted. 



If one term of the ratio, whereof l is the logarithm, be given, the other 

 term will be had easily by the same rule : for if l were Napier's logarithm of the 

 ratio of a the less, to h the greater term, h would be the product of a into 

 1 + L + iLL + -iLLL &c. = a + a l + -la ll + ^a l' &c. But if h 

 were given, a would be = i- — ^l + i^LL — -^^l^ &c. Whence, by the 

 help of the chiliads, the number appertaining to any logarithm will be exactly 

 had to the utmost extent of the tables. If you seek the nearest next logarithm, 

 whether greater or less, and call its number a if less, or b if greater than 

 the given l, and the difference thereof from the said nearest logarithm you call 

 /; it will follow that the number answering to the logarithm l, will be eitlier a 

 Into 1 + / -}- ^// -L ^lll -j- _•_/* -j- _4_/5 &c. or else h into i — / -j- j// 

 VOL. IV. E 



