LOGARITHMS. 
0.000000000542813 ; and if the logarithm of the geometrical mean, viz. 
4.301051709302416 be added to the quotient, tlie sum will be 
4.301051709845230 z= the logarithm of 20001. 
Wherefore it is manifest that to have the 
logarithm to 14 places of figures, there is 
no necessity of continuing out the quotient 
beyond 6 places of figures. But if you have 
a mind to have tlie logarithm to 10 places 
of figures only, the two first figures are 
enough. And if the logarithms of the num- 
bers above 20000 are to be found by this 
way, the labour of doing tliem will mostly 
consist in setting down the numbers. This 
series is easily deduced from the considera- 
tion of the hyperbolic spaces aforesaid. The 
first figure of every logarithm towards the 
left hand, which is separated from the rest 
by a point, is called the index of that loga- 
rithm ; because it points out the highest or 
remotest place of tlrat number flora the 
place of unity in the infinite scale of propor- 
tionals towards the left hand ; thus, if the 
index of the logarithm be 1, it shows that 
its highest place towards the left hand is the 
tenth place from unity; and therefore all 
logarithms which have 1 for their index, 
will be found between the tenth and hun- 
dredth place, in the order of numbers. And 
for the same reason all logarithms wliich 
have 2 for their index,will be found between 
the hundredth and thousandth place in the 
order of numbers, &c. Whence universally 
the index or characteristic of any logarithm 
is always less by one than the number of 
figures in whole numbers, which answer to 
the given logarithm; and, in deciniqls, the 
index is negative. 
As all systems of logarithms whatever are 
composed of similar quantities, it will be 
easy to form, from any system of logarithms, 
another system in any given ratio; and 
consequently to reduce one table of loga- 
rithms into another of any given form. For 
as any one logarithm in the given form is to 
jts correspondent logarithm in another form, 
so is any other logarithm in the given form 
to its correspondent logarithm in the re- 
quired form; and hence we may reduce 
the logarithms of Lord Neper into the 
form of Briggs’s, and conti-arywise. For as 
2.302585092, &c. Lord Neper’s logarithm 
of 10, is to 1.0000000000, Mr. Briggs’s 
logarithm of 10; so is any other logarithm 
in Lord Neper’s form to the correspondent 
tabular logarithm in Mr. Briggs’s form : and 
because the two first numbers constantly 
remain the same ; if Lord Neper’s logarithm 
of any one number be divided by 2.302585, 
&:c. ormulliplied by .4342944, &c. the ratio 
of 1.0000, to 2.30258, Ac. as is found 
by dividing 1.00000, &c. by 2.302.58, &c. 
the quotieirt in the former, and the product 
in the latter, will give the correspondent 
logarithm in Briggs’s form, and the con- 
trary. And, after the same manner, the 
ratio of natural logarithms to that of Briggs’s 
will be found = 868588963806. 
The use and applkation of Logarithms. 
It is evident, from what has been said of 
the cqnstruction of logarithms, that addition 
of logarithms must be the same thing as 
multiplication in common arithmetic ; and 
subtraction in logarithms the same as divi- 
sion : therefore, in multiplication by loga- 
rithms, add the logarithms of the multipli- 
cand and multiplier together, their sum is 
the logarithm of the product. 
num. logarithms. 
Example. Multiplicand.. 8.5 0.9294189 
Multiplier 10 1.0000000 
Product 85 1.9294189 
And in division, subtract the logarithm of 
the divisor from the logarithm of the divi- 
dend, the remainder is the logarithm of the 
quotient. 
num. logarithms. 
Example. Dividend.. 9712.8 3.9873444 
Divisor.... 456 2.6589648 
Quotient.. 21.3 1, .328.3796 
Logarithm, to find the complement of a. 
Begin at the left hand, and write down 
what each figure wants of 9, only what the 
last significant figure wants of 10 ; so the 
complement of the logarithm of 456, viz. 
2.6589648, is 7.3410352. 
In the rule of three. Add the logaritlims 
of the second and third terms together, and 
from the sum subtract the logarithm of the 
first, the remainder is the logarithm of the 
fourth. Or, instead of subtracting a loga- 
rithm, add its complement, and the result 
will be the same. 
Logarithms, to raise powers by.- Mul- 
tiply the logarithm of the number given by 
the index of the power required, the pro- 
duct will be the logarithm of the power 
sought. 
Example, Let the cube of 32 be required 
