152 
but those are not regarded in fixing the decimal 
point. See above. 
The reason for fixing the decimal point as 
directed, may be inferred from the rule followed 
in multiplication. The quotient multiplied by 
the divisor produces the dividend; and there- 
fore the number of decimal places in the divi- 
dend is equal to those in the divisor and quotient 
together. 
Some decimals, . though extended to any length, 
are never complete ; and others, whieh termi- 
nate at last, sometimes consist of so many places, 
that it would be difficult in practice to extend 
them fully. In these cases, we may extend the 
decimal to three, four, or more places, according 
to the nature of the articles, and the degree of 
accuracy required. In this manner we may per- 
form any operation with ease by the common 
rules, and the answers we obtain are sufficiently 
exact for any purpose in business. 
On the Extracting of Roots. 
The root is a number, whose continual mul- 
tiplication into itself produces the power; and is 
denominated the sqtiare, cube, 4th, 5th, root, 
See. according as it is, when raised to the 2d, Sd, 
4th, 5th, Sec. power, equal to that power. Thus 
2 is the square root .of 4, because 2 X 2=4; 
and 4 is tire cube root of 64, because 4 x 4 x 
=: 64; and so on. 
Although there is no number of which we 
cannot find any power exactly, yet there may be 
many numbers of which a precise root can never 
be determined. But, by the help of decimals, we 
can approximate towards the root, to any as- 
signed degree of exactness. 
The roots which approximate are called surd 
roots, and those which are perfectly accurate are 
called rational roots. 
Roots are sometimes denoted, as in algebra, 
by writing the character before the power, 
with the index of the root against it : thus, the 
third root of 70 is expressed f/ 70, and the se- 
cond root of it is 70, the index 2 being al- 
mays omitted when the square or second root is 
designed. 
If the power be expressed by several num- 
bers, with the sign -j- or — between them, a 
line is drawn from the top of the sign over all 
the parts of it; thus, the third root of 28 — 13 
is \/ 28 — 13. 
Sometimes roots are designed like powers, 
with fractional indices ; thus, the square root of 
£ I 
5 is 5 2 , the third root of 19 is 19 T , and the 
akrnrth root of 40 — 12 is 40 — 12*’ & c. 
To EXTRACT THE SQUARE ROOT. 
Rule. (1) Distinguish the given number into 
periods of two figures each, by putting a point 
•over the place of units, another over the place of 
hundreds, and so on. (2) Find a square number 
either equal to, or the next less than, the first 
period ; and put the root of it to tbe'right hand 
of the given number, after the manner of a 
quotient figure in division, and it will be the 
first figure of the root required. (3) Subtract 
the assumed square from the first period, and to 
the remainder bring down the next period for a 
dividend. (4) Place the double of the root al- 
ready found, on the left hand of the dividend, 
for a divisor. (5) Consider what figure must be 
annexed to the divisor, so that, if the result be 
multiplied by-it, the product may be equal to, or 
the next l6ss than, the dividend, and it will be the 
:id figure of-the root. (6) Subtract the product' 
from the dividend, and to the remainder bring 
down the next period, for a new dividend. (?) 
Find a divisor as before, by doubling the figures 
already in the root; and from these find the next 
figures of the root, as in the last article ; and so 
oh through all the periods to the last. 
ARITHMETIC. 
Note. If there nre decimals in the given num- 
ber, it must be pointed both ways from unity, 
and the root be made to consist of as many 
whole numbers and decimals as there are periods 
belonging to each; and when the figures belong- 
ing to the given number are exhausted, the ope- 
ration may be continued at pleasure by adding 
cyphers. 
EXAMPLES. 
Required the square root of 5499025. 
5499025(2345 the root. 
4 
43)149 
129 
464)2090 
1856 
4685)23425 
23425 
0 
Extraction of the Cube Root. 
Rule. (1) Find by trial the nearest rational 
cube to the given number, and call it the as- 
sumed cube. (2) Then, as twice the assumed 
cube added to the given number, is to twice the 
given number added to the supposed cube, so is 
the root of the supposed cube to the root re- 
quired nearly. (3) By taking the cube of the 
root thus found for the supposed cube, and re- 
peating the operation, the root may be had to 
a still greater degree of exactness. 
Ex. 1. What is the cube root of 12484? 
By trial I find the nearest root, less than the 
given number, is 23, the cube of which is 12167. 
Then 12167 xH 12484 = 36818, and 
12484 X 2 12167 = 37135. There- 
fore as 36818 : 37135 ” 23 ) 23.198, 
and 23.198 is the root required nearly. 
2. What is the cube root of 2 ? 
As the nearest rational root is 1, we have 
1 X 2 -f- 2 = 4, and 2 x 2 -f- 1 z= 5. 
Then 4 * 5 ** 1 ) A. = 1.25 = root nearly. 
Again, the cube of 5 = 1*5 therefore 
4 6 4 
W X 2 -f 2 : 2 X 2 + 1.25 :: 5 , or 
To EXTRACT THE ROOTS OF POWERS IN 
General. 
Rule I. Prepare the given number for extrac- 
tion, by pointing off from the units place as the 
root required directs. 
2. Find the first figure of the root by trial, and 
subtract its power from the given number. 
3. To the remainder bring down the first 
figure in the next period, and call it the dividend. 
4. Involve the root to the next inferior power 
to that which is given, and multiply it by the 
number denoting the given power for a divisor. 
5. Find how many times the divisor may be 
had in the dividend, and the quotient will be an- 
other figure of the root. 
6. Involve the whole root to the given power, 
and subtract it from the given number as before. 
7. Bring down the first, figure of the next pe- 
riod to the remainder for a new dividend, to 
which find a new divisor, and so on till the whole 
is finished. 
Ex. What is the cube root of 53157376? 
53157376(376 ? 
27 = 3’ 
3 2 X 3 = 27)261 dividend. 
50653 = 37 5 
3 - 4107)25043 second dividend. 
53157376 
0 • 
The reason of the process in the Extraction of 
Roots may be seen in Algebra, p. 53. 
Thus have we gone over all the principal ! 
rules in cojruhon arithmetic, giving under each 
an example or examples, by the assistance of 
which the reader may invent any number of 
others for his own improvement in the branch 
of science. We have not touched upon simple 
interest, discount, loss and gain, &c. because these 
are but modifications of the Rule of Three; and 
may be done either by the rules there given, or 
by Practice. We doubt not that this article, if 
followed by that under the word Algebra, 
(which though first in order in a dictionary ne- ■ 
cessarily comes last in practice) will be found 
sufficient for almost all purposes in common 
life. 
ARITHMETIC, decimal, that containing j 
the doctrine of decimal fractions. See Arith- * 
METIC. 
Arithmetic of infinites, the doctrine of i 
infinite series. See Series. 
Arithmetic, instrumental, that perform- j 
ed by means of instruments, as the abacus or j 
counting-board, Napier’s bones, &c. 
Arithmetic, literal, the same with spe- i 
cious. See Algebra. 
Arithmetic, logarithmetical, that per- I 
formed by means of logarithms. See Lo- 1 
garithm. 
Arithmetic, logistical, the same with. ] 
sexagesimal. 
Arithmetic, sexagesimal, the doctrine of ; 
sexagesimal fractions. 
AKiTHMETic,specious, the same with alge- ' 
bra. See Algebra. 
ARITHMETICAL complement a loga- j 
rithm, the sum or number which a logarithm j 
wants of 10,000000: thus the arithmetical 
complement of the logarithm 8.154032 is 
1.845968. 
Arithmetical progression. See Alge- i 
bra, p, 54. 
Arithmetical proportion. See Alge- 
bra, p. 54. 
ARITHMOMANCY, a species of divina- 
. tion performed by means of numbers. 
ARLEQUIN, an English trivial name ap- ; 
plied to some birds, insects, shells, &c. re- 
markable for their striking colours. 
ARM, in respect of the magnet. A load- ■ 
stone is said to be armed when it is inclosed, j 
capped, or set in iron or steel, in order to in- 
crease its magnetic virtue. 
Arm, in sea language. A ship is said to 
be armed when fitted out and provided in all 
respects for war. 
ARMADILLO. See Dysapus, 
ARMED, in the sea language. A cross- | 
bar shot is said to he armed when some rope- 
yarn or the like is rolled about the end of the 
iron bar which runs through the shot. 
Armed, in heraldry, is used when the 
horns, feet, beak, or talons, of any beast or 
bird of prey, are of a different colour from 
the rest of their body. He bears a cock or a ] 
falcon armed or, &c. 
ARMENIANS, in church history, a sect 
or division amongst the eastern Christians; 
thus called from Armenia, the country an- 
ciently inhabited by them. There are two 
kinds of Armenians ; the one catholic, and 
subject to the pope, having a patriarch in 
Persia and another in Poland; the other' 
makes a peculiar sect, having two patriarchs 
in Natolia. They are generally accused of 
being monophysites, only allowing of oive na- 
