Ij02 
LOGARITHMS. 
« 
As the numerator of the index denotes the power to 
which the number is to be raifed, and the denominator 
denotes the root of that power to be extracted, the reafon 
of the operation is manifeft; for, by firft multiplying the 
logarithm of the number by the numerator of the index, 
you have the logarithm of the power, and then, dividing 
by the denominator, you have the logarithm of the root 
of that power. 
To find a fourth proportional to three given numbers The 
fourth term of a proportion is found by multiplying the 
fecond and third terms together, and dividing the product 
by the firft ; hence we have the following Rule : Let 
a, b, c, be the firft, fecond, and third, terms; put m~ the 
number of decimal figures in a, n — the number in b and c 
together, and r— i; and let A, B, C, be the refpedtive 
values of a, b, c, confidering the fignificant figures of each 
term as whole numbers; then the logarithm of the fourth 
term 2= ar. co. log. A + log. B + log. C + m — n — 10. 
Ex. i. What is the fourth proportional to 24, 35, and 79 ? 
Here, m 222 o, n = o ; hence, we have to fubtrail 10 from 
the index. 
8-6197888 ar. co. log. of 24 
' 35 
79 
115-2083 the fourth proportional. 
Ex. 2. What is the fourth proportional to 0-073, 042, and 
0-009. Herera=3,«=5, m—n —10=—12; hence, we have 
to lubtraft 12 from the index. 
8-1366771 ar. co. log. of 73 
1-6232493 log. of - 42 
0-9542425 —-- - 9 
"a'7 141689 ——- 0 0517808 the fourth proportional. 
Here, the index of the fum being 10, by fubtrafling 12 
from it, the remainder is—2. 
In trigonometrical operations, when radius is the firjl 
term, its logarithm in the Tables being io-ooooooo, and 
therefore its arithmetical complement o-ooooooo, we have 
only to add the logarithms of the fecond and third terms 
together, and fubtraft 10 from the index. If radius be 
one of the mean terms, we then add 10 to the logarithm 
of the other mean term, and fubtraft the logarithm of 
the firft term from it. In the Tables of natural fines, 
cofines, See. we generally fuppofe radius =1, and there¬ 
fore its logarithm =20; but in the Tables of the logarith¬ 
mic fines, cofines, See. the logarithm of radius is io-ooooooo ; 
if therefore we make any natural number, the fine, cofine. 
See. of an arc, to radius unity, and we want to exprefs 
its logarithm fo as to agree with the logarithm of the fines, 
cofines, Sec. in the Table of thofe quantities, we muft add 
10 to the index of the logarithm, as taken from the Table 
of the logarithms of the natural numbers. And, vice 
verfa, if we want to reduce the logarithm of a fine, cofine, 
See. as found in the Table of thofe quantities, to the lo¬ 
garithm of the natural numbers, in order to determine the 
value of fuch a fine, cofine, See. we muft l'ubtraft 10 from 
the index. 
To find any number of geometrical means between two given 
numbers. —Rule: Take the difference between the loga¬ 
rithms of the two numbers, and divide it by the number 
of means increafed by unity, and add the quotient conti¬ 
nually to the leaf! logarithm, and you get the logarithms 
of the means. 
Ex. Find four geometrical-means between iof and 14.Z. 
The logarithm of iof — log. of — log. 32_ log. 
3 — , ' 5 ° 5 , 5 0 ° — °'477i2i3 — 1-0280287 ; and log. of 
= lo g- 1 1 3 = log. 133 — log. 92222-1238516'— 
°.‘ 954*+ a 5 — 1-1696091. The difference of thefe loga¬ 
rithms is 0-1415804, which divided by 5, the quotient is 
0-0283x61 ; hence. 
1-5440680 log of 
1-8976271-— 
2-0614839 ■ 
1-0280287 
0-0283161 
1-0563448 log. of 11-38531 — rft mean. 
1-0846609-12-15236 — 2d 
1-1129770 •- 12-971x1 — 3 d 
1-1412931-13-84500 — 4 th 
The reafon of this operation is,- that, when the numbers 
are in geometrical progreffion, their logarithms are in 
arithmetical progreffion ; and the number of intervals of 
the terms is always greater by unity than the number of 
means ; and therefore you muft divide the difference of 
the logarithms by a number greater by unity than the 
number by means. 
Given the hypothenufe (h) of a right-angled plane triangle, 
and one leg (l J, to find the other leg (L ), By Euc. B. i. p. 47. 
L h 2 / 2 ~ \Jh -j - l x h — /; hence, log. T. — x 
(log. h -f l -f- log. h — l ). 
Ex. Let h — 48796,21435 j then 
{ + {2=70230 - - log. 4-8465227 
h l —27361 - - log. 4-4371320 
2)9-2836547 
L — 43835-64 - . log. 4-6418273 
For the more general application of logarithms to trio-o- 
nometrical computation, fee the article Trigonometry. 
Hyperbolic Logarithms. 
If from the centre of an equilateral hyperbola you take 
upon one of its afymptotes an abfeiffa which fhall be 
equal to the correfponding ordinate, and reprefent each 
by unity; then, if 1 + x 2= any other abfciifa, the cor¬ 
refponding ordinate = ~~; and (by Vince’s Prin. of 
Flux. Art. 49. Ex. 3.) the area comprehended between 
the ordinates 1 and is the logarithm of 1 + * to 
the modulus unity. Hence thefe logarithms are called 
hyperbolic logarithms ; and a Table thus conftrufted is very 
uleful for finding fluents; for, the modulus being unity, 
the fluent found by thefe logarithms requires no multi¬ 
plication. The general equation exprefling the relation 
between a logarithm x and its natural number b, \s a* = b\ 
and in this iyltem, 2*7182818 5 hence, the equation 
is 2-7182818" —b. If b=z, x — 0-69314708 the hyperbolic 
logarithm of 2. Now the common logarithm of 2 is 
0 3010300; and the ratio of 0-69314708 to 0-3010300 is b 
to 0-43429448 ; therefore 0-43429448 is the modulus of 
the common fyftem; becaufe the logarithms of any num¬ 
ber are to each other as the moduli of the fy Items. (Fluxions, 
Art. 103.) Hence, if we divide the common logarithm 
by 0-43429448, or multiply it by the reciprocal thereof 
2-302585, we get the hyperbolic logarithm. 
A Table of hyperbolic logarithms would not be fo con- 
venient foi pia£lice as that now- in ufe ; for in the latter 
fyftem the index of the logarithm is more readily known, 
it being always lefs by unity than the number of integral 
figures of the given number; on which account, the com¬ 
putations by this fyftem are made more readily and with 
greater certainty. 
Logistical Logarithms. 
Befides the common logarithms here treated of, there are 
others, called logijhcal logarithms, which are the common 
logarithms fubtrafted from 3-5563 the logarithm of 3600, 
the number of feconds in 60 minutes. By this means the 
logarithm of 3600" (2=60') becomes nothing; and the lo¬ 
garithm ol 360 is i-oooo. For numbers greater than 3600 
the logarithms would be negative; but, inftead of putting 
them down fo, a unit is added to the index of the loga¬ 
rithm of 3600, before the iubtraiftion; the logarithms are 
therefore 
