LOGARITHMS. 



arrive at any conftant or invariable fluxion. If the log;\ritlims 

 of Uvo quantities be always to each other in any invariable 

 ratio, the fluxions of thofc quantities fhall be in a ratio that is 

 compoundedof a ratio of the quantities themfelves, and of 

 the invariable ratio of tliuir logarithms. 



Let o^be greater than oa; ail: up : : oa -.op; and let oa, 



q a d e f g h k p x 



»d, lie, ef, fg, &c. be in continued proportion : then by 

 adding together ad, ^ d e, ye/, \fg, &c. we approxi- 

 mate continually to the value «f A P, the logarithm of 

 Ob. And we approximate continually to the logarithm 

 or d, by fumming up the differences betwixt a d and 

 i de, 3 ef, and ^fg, \ g h and ^ h k, &c. See Mackurin's 

 Fluxions, art. 171, 172. From what has been faid, it 

 follows, that \i ao : d : : p : X, then the logarithm 

 of ox will be equal to the fum of the logarithms ol p 

 and d : that is, to the fum o{ a d + ^d e + j f / + \fg 

 + \gb+ yji,Sic.mdad+ide + ief+ifg + \gl, 



A = (r- i)-iir~iy + i{r-iy ~i,c. 

 A' = (r - 1)^ - (r- !)■ + &c. 

 A" = &c. 



where A, A', A", A"', &c. are condant but unknown 

 quantities. And now, in order to determine the law by 

 which they are connetled with each other, let x be increafed 

 by any indeterminate quantity z ; then r'*''' — i + A 



(.r + z) +A' {x + z)- + A" (x + z)' A' "- 



(x + !!i)";or, expanding the powers of * + z, and Hopping 

 at the firll two terms, we have 



r' + '= 1 + A{x + z) 



+ A' (.r= 4- 2 .r z + &c.) 



+ A" {x^ + 3 X- z + &c.) 



+ A"— Ma-" + « *■'"' ~ + Sec.) 



+ A<"' (.v'' + ' + (« + i) x" + &.C.) 



Again 



(i + A« + A':r' -I- A"«^ + &c.) X 

 ( I + A s f A' a'' + A" z' + &c.) 



•^ \h i, &c. and a d ■ 



■\dc^ 



h k, &c. which fum is 1 ad -{■ ^ ef -\- \ g h, &c 



Let a q =: ad ; then the logarithm of x will meafure 

 the ratio of odto q. But od andoy have half their 

 fum equal to a, and half their difference equal to a^, which 

 are the two firll terms of the geometric progrelfion a, a il, 

 de, ef, fg, g h, hi, &c. Hence, if oiz = i, and a d = 

 X, de, ef, fg, die. will be rcipcdtively, x\ x^, x^, &c. and 

 the ratio of i -f- x to i — ,v will be equal to that of d to 

 q. But the logarithm of this ratio is 2 a r/ + §■ f /" -|- ig ^ 



■+ , &c. therefore the logarithm of '- = 2 x * + J .v' 



ef — ~fg -)- ^glj — i the aftual multiplication of which gives 



+ T •■''^ + T -^ ? + '^c. agreeably to what has been fliewn 

 by Dr. Halley and others. 



Having thus given an idea of the forms under which lo- 

 garithms were confidered, and the methods by which 

 they were computed by fome of the early writers on this 

 fubjeft, it will be proper now to bellow a few columns, to 

 explain the more modern way of invefligating the principles 

 and of computing thefe very ufeful numbers ; in doing 

 which, however, the limits of our article will neccflarily 

 confme our obfervation to only the inoft popular and ufeful 

 formulae. 



We have already defined a logarithm to be the index of a 

 certain number called the radix, which beinir raifed to the 

 power denoted by that index or logarithm, will produce the 

 given number. If, therefore, r = N, then .v is the loga- 

 rithm of N, and r is the radix of the fy (lem. Now, lirft, in or- 

 der to find an analytical expreffion for N in terms of .v and r ; 

 r' mull be converted into a ieries, for which purpofe it may 

 be put under the form 



r' = (I + (r -,))'= 1 + ^.(^-1) + 'LI^HA 



{>■ - I)' + 



■{■■<:- l) {x - a) 



{r - i)' + &c. 



= I + X 



I + A « -t- A' .r* + A" *' + &c. 

 by writing 



»■■ + »= I .f A (« + a) + A' x' + A"x' . . . Af ' 



A\vs + A', Ai-'-s . 



&c. 



whence, by comparing the correfponding terms in the two 

 expa.dions, we have 



2 A' = A'-, or A' =^ ; A" = A A' = — ; 



A' 

 and therefore A" = 



in the fame way A 



I . 3.3 ...(« + i) 



And coufequentlv, 



A^ \} 



r ' = N = i + A .V H X- ^ .r' + &c, 



1.3 1.2.^:; 



which is the analytical expreflion for any number in terms 

 of the radix r and its logarithm x ; but the reverfe of this, 

 by which the logarithm is exprefled in terms of its number 

 and radix, is the formula which is more particularly appli- 

 cable in the prefenl enquiry. This may be found as 

 follows. 



In the preceding article we found 



r' =: N = I + A .f -I x^ + —^_ •_ .-«5 + &c. 



J . 2 1.2.3 



here A = (r — i ) — -i (r — 1 )' + i (r - 



where A = (r — i ) - i (r — 1 )' + i (r — l)^ - &c. ; 

 and if now we maice 



B = (N- I) - HN- i)'' + HN- I)'- &c. 

 we fhall have on the fame principles 



N' ^ I + B a + 

 But 



B^ 



A^ A"" 



M» = r"^ =: 1 + A .-c z + .v" z' H A-' 1:^ + 



1.2 ' • - • 3 



&x. ; whence, by comparing the co-eiScients of a in bot" 



ferics, we have 



3 A., 



