igo 
hand, and put the root of it as the firfi figure of the root 
required. Subtract the affumed fquare from the firft pe¬ 
riod, and to the remainder bring down tire next period for 
a dividend. Place the double of the root, already found, 
on the left hand of the dividend, for a divifor. Confider 
v.hat figure mull be annexed to the divifor, fo that, if the 
refult be multiplied by it, the product may be equal to, 
or next lefs than, the dividend, and it will be the (econd 
figure of the root. Stibtraft the prod lift from the divi¬ 
dend, and to the remainder bring dow n the next period for 
a new dividend* Find a divifor as before, by doubling the 
figures already in the root, and front thefe find the next 
figure of the root, as explained above ; and io on through 
all the periods to the lafi; and if there be a remainder you 
may bring down cyphers two at a time, and carry the root 
into decimals at pleafure. 
Inftead of doubling the quotient every time for a divi- 
ior, you may always add the lafi quotient figure to the lafi: 
divifor for a new divifor, and proceed as before. To prove 
the work, multiply the root by itfelf, and to the produft 
add tire remainder, and the fum will be the given number. 
Ex. What is the fquare root of 408970-0727789824? 
408970-0727789824(639-50768 the root required. 
36 
ARITHMETIC. 
Ex. 4. What is the fquare root of ? Firfi —Lf r 
then, 
4/11X6 8-124038 
1-354006-1- the anfwer. 
!23 489 
+3 369 
1 269 
+ 9 
12070 
I I 42 I 
12785 
+5 
64907 
63925 
1279007 -9822778 
+7 8953049 
12790146 
+6 
■86972998 
76740876 
127901528 
-1023212224 
1023212224 
Proof 639-50768 X 639-50768=2408970-0727789824- 
Where the operation is large, and the root required to 
be carried to many places in decimals, the work may be 
much abbreviated thus ; when more than half the figures 
of the root are found by the common method, all therefi 
may be as truly found by common Divifion. 
To extraEl the Square Root of a Vulgar FraQion or Mixed 
Number. Firfi prepare all vulgar fractions by reducing 
them to their lowed: terms, both for this and all other 
roots. Then take the root of the numerator, and of the 
denominator, for the refpeftive terms ot the root requi¬ 
red. And this is the belt way, if the denominator is a 
complete power. But, if it is not, multiply the numerator 
and denominator together ; take the root of the produft ; 
this root being made the numerator to the denominator of 
the given fraftion, or made the denominator to the nu¬ 
merator of it, will form the fractional root required. 
That is, J-~ y And this rule will ferve 
y y \J xy 
’whether the root be finite or infinite. Or, reduce the vul¬ 
gar fraftion to a decimal, and extraft its root. 
Mixed numbers may be either reduced to improper 
fractions, and extracted by the firfi and fecond rule; or, 
the vulgar fraction may be reduced to a decimal, then 
joined to the imeger, and the root of the whole extracted. 
Ex. 1. What is the fquare root of Firfi in 
its lowefi terms; then 92=3, and 4/252=5; therelore, 
4 ! ^=-|, the anfwer. 
Ex. 2. What is the fquare root of ||, and of -§-|? -^ n ‘ 
fwer, £ and *. 
Ex. 3. What is the fquare root of 
4/11X15 12-84523 .0. 
Thus,-=•-= * 5 * 
fiS *3 
Application of the Square Root. To find a mean propor¬ 
tional between any two numbers, a and b. y/«i=2 the an- 
fiver. That is, multiply the two numbers together, the 
fquare root cf the produft is the proportional required. 
What is the mean proportional between 7 and 343 ? 
Anfwer, 4/7x343=49- For as 7 : 49 :: 49 : 343. .-.343 
X 7 =49X49=2401. 
A tub of IriIh butter weighed in one fcale 641b. and in 
the other 49th. w-hat was the true weight of the butter, 
Anfwer, 4/64X49=56^. 
To EXTRACT the CUBE ROOT. 
Separate the given number into periods of three figures 
each, thus, Put a point over the units place, and over 
every third figure from thence, to the left hand in whole 
numbers, and to the right, beginning at the place of thou- 
hinds, in decimals. Find the greatefi cube in the firfi pe¬ 
riod, and put its root in the quotient. Subtraft the cube 
thus found from the firfi period, and to the remainder an¬ 
nex the firfi figure of the fecond period, and call this the 
dividend. Divide this dividend by thrice the fquare of 
the root lafi found, and the quotient will be the fecond 
figure of the root fought. Cube the root thus found, and 
fubtraft it from the two firfi periods, and to the remainder 
bring down the firfi figure of the next period for a new 
dividend. Divide this new dividend by the triple fquare 
of the whole root thus found, the quotient figure annex: 
to the root again. Cube the whole root, and fubtraft it 
from as many periods as you have brought down, and pro¬ 
ceed in this manner till the whole work is ended. If 
there be a remainder, annex periods of cyphers, and carry 
the root into decimals, as far as you pleafe. 
Ex. What is the cube root of 146363183? 
146363183(527 the root required 
125 = cube of 5 
5*X3=75) 2 x 3 
146363 =2 two firfi periods 
140668 222 cube of 52 
52”X3=8ii 2) 57551 2= new dividend 
Prnr,f / 146363183 =2: the three periods 
\146363183 2= the cube of 527. 
Find by trials a cube as near the given number as po fti' 
ble, either greater or lefs, and call it the affiimed cube. 
Then, by the Rule of Three, fay. As the film of twice the 
affiimed cube and given number is to the funr of twice the 
given number and affumed cube, fo is the root of the af¬ 
fiimed cube to the root required. By taking the cube of 
the root thus found for the fecond affumed cube, and re¬ 
peating the operation, the root will be found to a fiill 
greater degree of exaftnefs. 
Ex. What is the cube root of 98003449 ? 
Let 5"ool’=: 125000000, the affumed cube ; then 
125000000x2+98003449222348003449, the firfi term, 
And 98003449 X 2+125000000222321006898 fecond term. 
As 348003449 : 321006898 :: 500 : 461, the firfi correft- 
ed root. 
Again, let 46T) 3 2=97972i8i, the fecond affumed cube; 
97972181 x2+98003449=2:293947811, firfi term, and 
98003449x2+97972181=2293979079, fecond term; then 
As 293947811 : 293979079 :: 461 : 461-04903778, the 
root required, and true to the lafi place of decimals. 
The refolvend being pointed into periods, as taught in 
the firfi rule, find a number whofe cube comes the neareft 
to the firfi period of the refolvend, to this annex fo many 
cyphers as there are remaining points over the whole num¬ 
bers in the refolvend, and call this the affumed root. Di¬ 
vide the given refolvend by three times the affumed root, 
1 and 
