LOGARITHMS. 
C8? 
and the meafures of thofe two ratios deftroy each other 
when added together; fo that, when the one is confidered 
as pofitive, the other is to be confidered as negative. By 
fuppofing the logarithms of quantities greater than oa 
(which is fuppofed to reprefent 'unity) to be pofitive, and 
the logarithms of quantities lefs than it to be negative, 
the fame rules ferve for the operations by logarithms, 
whether the quantities be greater or lefs than ao. When 
op increafes proportionally, the motion of p is perpetually 
accelerated ; for the fpaces ac, cd, de, Sec. that are defcribed 
by it in any equal times that continually fucceed after 
each other, perpetually increafe in the fame proportion as 
the lines oa, oc, od. See. When the point p moves from a 
towards o, and op decreafes proportionally, the motion of 
p is perpetually retarded; for the fpaces defcribed by it 
in any equal times that continually fucceed after each 
other, decreafe in this cafe in the fame proportion as op 
decreafes. If the velocity of the pointy be always as the 
diftance op, then will this line increafe or decreafe in the 
manner fuppofed by lord Napier; and the velocity of the 
point/<, being the fluxion of the line op, will always vary 
in the fame ratio as this quantity itfelf. 
The fluxion of any quantity'is to the fluxion of its lo¬ 
garithm as the quantity itfelf is to unity. Hence the 
, X X 
fluxion of the logarithm of x will be -. For x : i ;: x : - 
x x 
~ the fluxion of the logarithm required. 
When op increafes proportionally, the increments gene¬ 
rated in any equal times are accurately in the fame ratio 
as the velocities of p, or the fluxions of op, at the begin¬ 
ning, end, or at any fimilar terms, of thofe times. When 
op' increafes or decreafes proportionally, the , fluxions of 
this line, in all the higher orders, increafe or decreafe in 
the fame proportion as the line itfelf increafes or decreafes; 
fo that one rule ferves for comparing together thofe of any 
kind at different terms of time; and in this cafe we never 
arrive at any conftant or invariable fluxion. If the loga¬ 
rithms of two quantities be always to each other in any 
invariable ratio, the fluxions of thofe quantities (hall be 
in a ratio that is compounded of a ratio of the quantities 
themfelves and of the invariable ratio of their logarithms. 
Let ep be greater than oa ; ad : ap :: oa : Op ; and let oa, 
oq ad tfgkkp x 
A-P 
ad, de, ef,fg, &c. be in continued proportion: then by 
adding together ad, %de, %ef, %fg, See. we approximate 
continually to the value of AP, the logarithm of op. And 
we approximate continually to the logarithm of od, by 
fumming up the differences betwixt ad, \de, \ef, and \fg, 
$g/i, and fhk, See. See Maclaurin’s Fluxions, art. 171, 172. 
From what has been faid, it follows, that if ao : od :: op : ox, 
then the logarithm of ox will be equal to the fum of the 
logarithms of op and od : that is, to the fum of adJr\de-\- 
Sec. and ad+ \de + i«/+ ifg+^g/i + 
\hk. See. and ad — %de+$ef — zfg+^gh — \hk. Sec. which 
fum is lad+^ef+^gh, Sec. 
Let aq — ad\ then the logarithm of ox will meafure the 
ratio of od to oq. But od and cq have half their fum equal 
to oa, and half their difference equal to ad, which are the 
two firft terms of the geometric progreflion oa, ad, de, cf, 
fg, gk, hk. Sec. Hence, if oa= 1, and adzezx, de, Sec. 
will be refpeftively, x 2 , x 3 , x 4 , Sec. and the ratio of 1 -f-x 
to 1 — x will be equal to that of od to oq. But the loga¬ 
rithm of this ratio is 2<zd4-§r/4--|g/^4-, Sec. therefore the 
j_l_ # --—- 
logarithm of -- ~ 2 Xx-{-fx 3 + -^x 5 + fx-£-p, &c. agree¬ 
ably to what has been fhown by Dr. Halley and others. 
Having thus given an idea of the forms under which 
logarithms 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 explain the more modern 
way of invefligating the principles and of computing thefe 
very ufeful numbers. 
We have already defined a logarithm fo be the index of 
a certain number called the radix, which, being raifed to 
the power denoted by that index or logarithm, will pro¬ 
duce the given number. If, therefore, r* = N, then x is 
the logarithm of N, and r is the radix or the f’yflem. Now, 
firft, in order to find an analytical exprelfion for N in terms 
of x and r\ r x must be converted into a l'eries, for which, 
purpofe it may be put under the form 
: C 1 + ( r 
(r — i) 2 4- 
+ * {0—0 
I .21 
1))*= i +x(r 
(x- I ) (x- 
0 4- 
2 ) 
(r 
-Mr-0 2 -R.(r 
i) 2 — ( r — 0 3 + & c .^ = 
• O- O 
1 . 2 . 
- r) 3 + &c- 
O 3 — &c. \ 
Sec » 
1 4. A 1: + A' x 2 + A" x 3 4- Sec. 
by writing 
A — (r— O — l (r— i) z (r— j)s 
A' = (r — i) 2 — (r— i) 3 -J- Sec. 
A" — Sec. 
where A, A', A", A"', Sec. are conftant but unknown 
quantities. And now, in order to determine the law; by 
which they are connected with each other, let x be increafea 
by any determinate quantity z; then r x ± x — 1 + A 
(i4-z)+A'(x + 2) 2 +A "(x+z) 3 .A(»-« 
(x -f- z ) n ; or, expanding the powers ot x + z, and flopping, 
at the firft two terms, we have 
r x f x — i 4* A (x 4* -0 
4- A' (x 2 4- 2 x z 4~ Sec.} 
4- A" (x 3 4- 3 x 2 2 4- Sec.) 
A ( ”~ 1) (x”4- n x”- 1 z 4- Sec.) 
4- A (n) (x” + * 4- (« 4- 1) x”4- Sec.) 
Again : 
T* •+• x ZZZ 7 X ^ 7^ — 
(1 4- A x 4- A' x 2 4- A" x 3 4- &c.) X 
(1 4- A z + A' z 2 4- A" z 3 4- &c.) 
the aclual multiplication of which gives 
r x + * = J 4- A (x 4- z) 4- A' x 2 4- A" x 3 . . . 
A 2 x z 4- A' . A x 2 z .*. Scco. 
whence, by comparing the correfponding terms in the two 
expanfions, we have 
iA' = A 2 , or A' = — 3 A" = A A' = 
2 > 
A 3 
A3 
and therefore 
in the fame way 
and generally 
And confequently. 
A" == 
A'" = 
2 • 3 
A 4 
A'” -1 ) ~ 
3 • 4 
A” 
A" =■ 
2 • 3 • • . 
An* 
= N = I 4- A x 4- 
A 2 
( s 4 *) 
A 3 
X 3 4- Sec, 
which is the analytical exprefiion for any number in terms 
of the radix r and its logarithm x; but the reverfe of this, 
by which the logarithm is expreffed in terms of its num¬ 
ber and radix, is the formula which is more particularly 
applicable in the prefent enquiry. This may be found’as 
follows. 
In the preceding article we found 
„ A 2 A3 
? .*_N—1 4- A x 4-— x 2 4- —— 1 — 
1 1.2 ‘1.2.3 
where A= (r — 1) — •■§ (r — j) 2 -J- ) ( r — i) 3 — See, 
4- Sec. 
