$96 
LOGARITHMS. 
time, even while the original record remains; and it weakens 
on a twofold account, if the record muft from time to 
time be replaced by copies. Nor is this deftruftion of 
evidence arifing from the uncertainty of the copy’s being 
accurately taken, any where greater than in the cafe of 
copie.d numbers. It is ufeful then to contrive new and 
eafy methods for computing not only new tables, but 
even thofe we already have. It is ufeful to contrive methods 
by which any part of a table may be verified independently 
of the reft; for, by examining parts taken at random, we 
may in forne cates fatisfy ourlelves of its accuracy, as well 
as by-examining the whole. Among the various methods 
of computing logarithms, none, that I know of, poffelfes 
this advantage of forming them with tolerable eafie inde¬ 
pendently of each other by means of a few eafy bafes. 
'This defideratum I trull the following method will fup- 
ply, while at the fame time it is peculiarly eafy of appli¬ 
cation, requiring no divifion, multiplication, or extraction 
of roots, and has its relative advantages highly increafed 
by incregfing the number of decimal places to which the 
computation is carried. The chief part of the working 
confifts in merely fetting down a number under irtfelf re¬ 
moved one or more places to the right, and fublratfing, 
and repeating this operation ; and confequently is very 
little liable to miftake. Moreover, from the commodious 
manner in which the work Hands, it may be revifed with 
extreme rapidity. If may be performed after a few' mi¬ 
nutes inltruftion by any one who is competent to fubtraft. 
It is as eafy for large numbers as for final! ; and on an 
average about 27 iubtraftions will furnifn a logarithm 
TL *4“ I 
accurately to 10 places of decimals. In general, 9X - 
z 
Subtractions will he accurate to 211 places of decimals.” 
The method employed may be thus Hated : Suppofe x 
S.Q be any-number; then 
€C 
T 1 
again : r -; 
a. 
&c. and fo on, fuppofe, to r" r ! 
(3—i 
Again; r' v — -^=z 
r iv = fV 
1 ^- 
(3" f 
•Hence, if we Hop at r T, » 
0 
== r»> Sec. and fo ots. 
(«- 
& c- 
i) s 
Hence h . log. x .= 
-iY 
J!_ r v, 
03—O 2 
5 k. 1. 
'(«— 0 ; 
* 5 (3 
.O— 0* '(3—i 
OL -1 
-J- h. 1 . r~ l 
■ 1 r 
1 — j_ — • 
[ a. s 
• I 
+ ^ ’ «* +ScC -} 
+ k . rw 
In the author’s plan, a. is 10, (3 either 100 or 1000, See. 
and r 7 ‘, fuppofing.it to be the lait remainder, is to be equal 
:to 1 followed by half as many ciphers as the number of 
.decimal places to which it is intended to work. Thus, 
i'uppofe r 71 = i-oooo 3141, then h. 1. — -ocoo 5141 if we 
work only to 8 figures, a and (3 being taken powers of 
ct. (3 
10, it is plain that h. 1. -—, h. l.- 0 -&c. are readily cal- 
a ,— 1 p— i 
x x 
ciliated; and the operations, fuch as x —- or x -are 
* a, 10 
.performed.by.the aid of the decimal notation with the 
greateft-eafe. In our inftance, we have only employed a, 
/3, and made fix fubtraCUon.s with,a* and two with.(3; b.ut 
2 
it is obvious that we might employ «, j?, y, Sec. and make, 
generally, m fubtraclions with a, n with (3, s with y, Sec. 
If the quantities be greater than two, they muft be "re¬ 
duced by divifion; thus, if the A. 1. 17 were required, 
17 = z4 —r = a4 (i-ofizs ) A. l- 17 = 4 h. 1 . 2 4- h. 1 . 
I Q 
(1.0625); and any number between 1 and 2 is eafily found 
by the method above deferibed. 
The principle of this method of computing logarithms 
is very fimple, and the practice is both fafe and eafy. If 
the conftruction of new logarithmic tables were required, 
it would be valuable ; and it is efrimable for the fkilland 
ingenuity with which it has befit invented aud conftructed. 
We fitali conclude this article with Mr. Profelfor Vince’s 
Introduction to his Trigonometry. 
On the Nature and Use of Logarithms. 
Logarithms are a fet of artificial numbers adapted to 
the common or natural numbers, 1, 2, 3, 4, 5, &c. in order 
to facilitate arithmetical calculations. 
Definition. Let a be a conftant quantity, and x be va¬ 
riable, and put a x —b\ then x is the logarithm of b. 
Corollary. Hence, if for x we write o, 1,2, 3, 4, 5, See. 
the correfponding values of b will be a°, a*, a 2 , a 3 , a 4 ,a 5 . 
Sec. If therefore the natural numbers be taken in geo¬ 
metrical progrelfion, their refpeftive logarithms will be in 
arithmetical progrefiion. From this definition of loga¬ 
rithms, we immediately deduce thofe properties which 
render them ufeful in fhortening numerical computa¬ 
tions. 
To multiply numbers together, by logarithms. —Rule. Add 
the logarithms of the numbers together, and the fum is 
the logarithm of their product. 
For, let a x =. b, a’ — c, a x = d ; then, a x l } * z — bed-, 
and x is the logarithm of b, y the logarithm of c, 2 the 
logarithm of d, and x -\-y -J- 2 is the logarithm of b.cd. 
To divide one number by another, by logarithms .—Rule. 
From the logarithm of the dividend fubtraCl the loga¬ 
rithm of the divifor, and the remainder is the logarithm 
of the quotient. 
b a x 
For let a*=. b-, a? = c, then —— a x ~*; and * is the 
c ar 
logarithm of b, y the logarithm of c, and x—y is the lo¬ 
garithm of -• 
c 
To raife a number to any power, by logarithms. —Rule, 
Multiply the logarithm of the root by the index of.th’e 
power, and the product is the logarithm of the power. 
For, let a' — b\ then a rx — b r 5 and x is the logarithm of b 3 
and rx is the logarithm of b r . 
To extratd the root of any number, by logarithms. —Rule, 
Divide the logarithm of the given number, by the num¬ 
ber which exprefies the root to be extracted, and the quo¬ 
tient is the logarithm of the root. For, let a x =.b ; then 
f. L . 
a r —b T ; and x is the logarithm of b, and ~ is the loga¬ 
rithm of b r . 
Thus it appears, that multiplication may be performed 
by the addition of logarithms ; divifion, by the fubtrac- 
tion of logarithms ; the railing of pUWers, by the multi¬ 
plication of a logarithm; and the extracting of roots, by 
the divifion of a logarithm. 
As a may be affumed any number, we may have the 
fame fet of logarithms adapted to different fets of natural 
numbers ; and the fame fet of natural numbers may have 
different fets ofslogarithms. But, as a 0 — 1, in every lyf- 
tem the logarithm of 1 is o. 
The tables of logarithms in common ufe are conftruCted 
upon fuppofition that a — 10. Now io° = 1, ro 1 — to, 
to 2 = 100, io 3 = 1000, io 4 — 10000, Sec. hence, in this 
fyftent, the log. of 1 is o ; the log. of 10 is 1 ; the log of 
100 is 2 ; the log. of 1000 is. 3 ; the log. of 10,000 is 4; 
Sec. and, in general, if 10 x z=.b, x is the logarithm of b. 
Hence, the logarithms of all numbers from 1 to 10 
are 
