LOGARITHMS. 893 
and r, by reverfion of feries, will be found — i‘7>3i8iS, c ___Lv • (a _i)— 
as before. And if, on the contrary, the radix r be af- Lo *»• 1 ^ ^ a ' &c._ 
fumed = 10, the value of the feries , ,a-ix 
f + 5 ? 3 + i ? 3 + '* ? 4 + f ? 5 &c * or its ^ ual L °s- a ~m x '~ M“J 2 + H'v) 3 + h“J 4&c ‘ 
J i<T>irKr>«(T 
5 &c. 
Log. +t(^ 7 ; +y(^) 7&c » 
will become = 2*30258509, as before 5 and the common a +b be put =5, and a^l—d, thefe general for- 
logarithm of (*+tf a +*P 8 +fc 4 -H* 6 mul* may be eafily converted into the following. 
1 —p 2*30258509 
&c.) or the common logarithm of a 
2*30258509 
« 1 ± d* d*_ 
■^°S* £ - ,rt X 'Z> lb 2 ' 2 b 3 4 6 4 <t^ 5 C ’ 
e 1 d , d 2 d 3 d* d* 
Lo S'j=» X! i + ^ + r^ + 5 I + 7 ? &c ‘ 
, « »«.£ + iL + ii + iL+ 
«*'1 + 3 * 3 + 5 j5 7 i? 
9 39 
&c. 
1 
Or the latter formula, for the logarithm of or 
its equal c, may be more concifely derived from the firft, From which laft exprefiions, if^ or its equal a > *■»*■ b be 
as follows; The logarithm of i+p has been fliown put — .1, we (hall have, y P 1 p - L > 
# _i A 2i.*3_.u4j..,s &c , r , nature of logarithms, 
to be = - f44^s and lf be fub ‘ 
q ——4? 4 *T'52 
ftituted in the place of + p, the logarithm of 1— p 
Log.fl=log.(a 1 ) +“ X + &c * 
1 1 
will 
become = —^ whence Log.a_log.(a-i)+ m X2^-1)^4-0*-?)' 
&c 
&c 
$—2? 2 + i? 3 —$J 4 —i2 5 
III t * 
the logarithm of = log. i — log. (i — f) = o — I t og.«=log.(«-a)+-X •— 3 ^- I ^+ 5 ^ I ^+ 7 ^ I y 
* 1*2 1,3 *.4 U 5 J,,. .j.ujiifSj.wj.nst-,, And, from the addition and fubtraaion of thefe feries,. 
it * 3 & c _J fr _ + ?-P + 3/ 1 + T + =,/ y_c\ p evera i others may be derived ; but, in the aftual compu- 
1 ?—£2 2 -B? 3 —i? 4 +i? 5 &o 1 — i? 2 +M 3 — ! 2 4 + i‘? 5 & c >' tation of logarithms, they will be found to poffeis little 
a —1 ,a —i» —1\ /a —1, . or no advantage above thofe here given. The fame gene- 
-b 21 -J 2 +3('* ‘ ) ^ c • ral formula may be derived from the original logarithmic 
or log. a — . a A - A,,- ,, \ 3 ,, (4 T ~ equation = « in a different way, thus: Let r=i+ ?t 
(r—i)—i( r —1) + 3 (^— 0 3 —?(r—i ) 4 &c. th » en y , ==J -^). = 1 +(? _i}*+i ? 3 -ij 4 & c.) x+Uq-lf 
where the denominator is the fame as in the firft formula, 1 , _ , , , 1 
f being here = r- 1. +—(q-W+k 3 ~i 
le I pi h =”^"l^ithm“r,°+] l »Tb^ {f-U’+h’-H*= or if r b e put = 
—X:(P —!#*+!?’ &C. or the logarithm of a by y_Z7’ we have =‘-0+« + s »’+il &c 3 
772 * j 
—X: (a— i)-J(a—1)»—*(«— i) 4 +i(«—i ) 5 ^+ 2 (?+^ 2 +^ 3 +i 2 4&c *) 2 ^ 2 —— (?+l?“ + S«*+l 2 * 
774 I 
&c. And the logarithm of —-— will be denoted by— &c.) 3 * 3 +^ + (?+2? +s? "\"i 1 ^' c *) x &c a * 
/* 1 i a 2 1 m I 1*4 .,*s e *1 , v. r 1. And by denoting i? 4 &c. in the firft cafe, 
X:C*+l/> 2 +3^ 3 +^ 4 +^ 5 &c. or the logarithm of« by Qr it$ eq ^ al in the lat£eii cafe3 b y m > 
l X :—+b( —V+if—V+/—> + ! T—V&c. thefe expreffions will become 
* a ^ \ a J \ -m*** + &c. = «i 
1 1 ** 1 2.3 a.3.4. 
-A-« 3 X 3 +.-&c. ~~i 
2.3 a. 3 ' 4 - a 
And, (ince the fum of the logarithms of any two num¬ 
bers is equal to the logarithm of their produft, the loga- 
14 .p 2 and i—mx + i« 2 x 2 
nthm of —— will become =—X(/’+i/ ) 3 +i/ , 5 +f/ >7 & c ') 
or the logarithm of a ” vvbich are , the two antidogarithmic feries of Halley, from 
•whence, by reveriion ot feries, may be found the loga- 
_ 2 a — 1 | ^ ritlim of any number, a, as before. ' 
m a+i \a+i) \a+i) + \a+rj ' C ‘ NeU) Met j lod l y T . Manning, Efq. Phil. Tranf. 1 $06. 
If'til f 6nera ! formula, that converges fafter If there already exifted (fays the author) as full and 
T . 4 1 , oimei. extenfive logarithmic tables as will ever be wanted, and 
u ■ , Jr ,', m 0 an y number may, therefore, be exhi- 0 f whofe accuracy we were abfolutely certain, and if the 
- 1 f a y» or ^cording to any fyitem of logarithms, evidence for that accuracy could remain unimpaired 
in the following forms: throughout all ages, then any new method of computing 
Log. (i+/>) = — x ; *_ 1*2 J.ii53 „JL*4 J_4 >5 Arc logarithms would be totally fuperfluousfo'far as concerns 
m r nr sP tP ~rip • the formation of tables, and could only be valuable indi- 
j j rectly, inafmuch as it might (how fome curious and new 
Log. -- =— X •P J r%P i +i/!' 3 4 -l-^ 4 4 -^i 5 &c. views of mathematical truth. But this kind of evidence 
1 P m 5 is not in the nature of human affairs. Whatever is re- 
1 -p/> 2 corded is no otherwife believed than on the evidence of 
^*°S* ~^Tp — — X : / 4 * I /* 3 + a /* 5 +^P 7 + &c. teftimcny j and luch evidence weakens by the lapfe of 
tirne^ 
