ALGEBRA. 
287 
by the fecond (b), and fubtraft this produCt (zab^-b 2 ) from 
the remainder. If there be more terms, proceed as be¬ 
fore, always dividing the remainder by twice the root al¬ 
ready obtained, for a new faftor in the root; and, lor a 
new fubtrahend, multiply this factor by twice the fum of 
the former factors increafed by this faCtor. 
* 2 
Ex. r. To extract the fquarerootof a 2 — ax -\— . 
X 2 X 4 
a 2 —ax-1-(a-' 
4 3 
— ax-\~- 
-ax-f- 
It appears from this example, that a trinomial a~—ax 
4—•, in which four times the produCt of the fird and lad 
4 . 
terms is equal to the fquare of the middle term, is a com¬ 
plete fquare. 
The method of extracting the cube root is difcovered in 
the fame manner. The cube root of a?-\-ia 1 b-\-T > ab 2 -\-b 3 
is a-^b. ^ 
Having arranged the a 3 -\-T,a 2 b-\-T,ab 2 
terms as before, the cube a 3 
root of the fird term a. 3 is 
a, the fird faCtor in the 
root; fubtraft its cube 
from the whole quantity, 
3a 2 ) 
3 a 2 b-^-^ab 2 -\-b 3 
3a 2 b^-^ab 2 -\-b 3 
and divide the fird term of the remainder by 3a 2 , the re- 
fultis^, the fecond faClorin.the root; then fubtraft 3 a 2 b 
-f-3 ab 2 -\-b 3 from the remainder, and the whole cube of 
a-f b has been fubtrafted. If any quantity be left, proceed 
wi th a -\-b as a new a, and divide the lad remainder by 
2-a-\-b] 2 for a third faCtor in the root; and thus any num¬ 
ber of faftors may be obtained. 
Scholium. Thefe rules for the extraction of the roots 
of compound quantities, are but little ufed in algebraical 
or fluxional operations; but it was necedary to give them 
at full length, for the purpofe of invedigating rules for 
the extraction of the fquare and cube roots in numbers. 
The fquare root of 100 is 10, of 10,000 is 100, of 
1,000,000 is 1,000, &c. from which confideration it fol¬ 
lows that the fquare root of a number lefs than 100 mud 
conlid of only one figure, of a number between 100 and 
10,000 of two places of figures, of any number from 10,000 
to 1,000,000 of three places of figures, See. If then a 
point be made over every fecond figure in any number, 
beginning with the units, the number of points will dievv 
the number of figures or places in the fquare root. Thus 
the fquarerootof 4357 confids of two figures, the fquare 
root of 56478, of three figures, &c. Ex. Let the fquare 
root of 4357 be required. 
4357( 6 °+ 6 °r 66 
3600 [the root. 
120+6) 757 
or 126] 756 
1 remainder. 
Having pointed it accord¬ 
ing to the direction, it ap¬ 
pears that the root confids 
of two places of figures; 
let a-j-i be the root, where 
ads the value of the figure 
in the ten’s place, and b of 
that in the unit’s; then is a the neared fquare root of 4300, 
which does not exceed the true root, this appears to be 
60; fubtrail the fquare of 60 (a 2 ) from the given num¬ 
ber, and the remainder is 757; divide this remainder by 
120 (2a), and the quotient is 6 (the value of b), and the 
fubtrahend, or quantity to be taken from the lad remain¬ 
der 757, is 126x6 (2 a-\-b.b) 01*756. 
It is laid that a mud be the greated number wliofe fquare 
does not exceed 4300. It evidently cannot be a greater 
number than this, and if podible let it be fome quantity 
• 2 
( x ) lefs than this; then fince .v is in the ten's place, and 
b in the unit’s, x-\-b is, lefs than a, therefore the fquare 
of x-\-b, whatever be the value of b, mud be lefs than a?, 
and confequently x-\-b lefs than the true root. 
If the root confids of three' places of figures, let a re- 
prefent the hundreds, and b the tens; then, having obtain¬ 
ed a and b as before, let the new value of a be the hundreds 
and tens together, and find a new value of b for the .units ; 
and thus the procefs may be continued if there be more 
places of figures in the root. 
The cyphers being omitted for the fake of expedition, 
the following rule is obtained from the foregoing procefs;' 
Point every fecond figure beginning 
with the unit’s place, dividing by this 
means the whole number into feveral pe¬ 
riods; find the greated number wliofe 
fquare is contained in the firfit period, this 
is the fird figure in the root; fubtraft its 
fquare from the fird period, and to the 
remainder bring down the next period ; divide this quan¬ 
tity, omitting the lad figure, by twice the part of the root 
already obtained, and annex the refult to the root and alfo 
to the divifor; then multiply the divifor as it now dands by 
the part of the root lad obtained, for the fubtrahend. If 
there be more periods to bringdown, the procefs mud be 
repeated. 
In extracting the fquare root of a decimal, the pointing 
mud be made the contrary way, beginning with the place 
of hundredths; or care mud be taken to have an even 
number of decimal places, becaufe, if the root has 1, 2, 
3, 4, &c. decimal places, the fquare mud have 2, 4, 6, 8* 
See. places. ' 
Ex. To extract the fquare root of 64*853. 
64*8530(8*053 &c, 
64 
1605) 
8530 
8025 
16103) 50500 
48309 
2191 See. 
For every pair of cyphers which we fuppofe annexed to 
the decimal, another figure is obtained in the root. 
The cube root of 1,000 is 10, of 1,000,000 is 100, Sec, 
therefore the cube root of a number lefs than 1,000 con- 
lids of one figure, of any number between 1,000 and 
1,000,000 of two places of figures, <i*c. If then a point 
be made over every third figure contained in any number, 
beginning with the units, the number of points will Ihevv 
the number of places in its cube root. 
Let the cube root .of 4.05224 be required. 
405224(70-1-4 
3« 2 —.14700) 62224 the fird remainder. 
58800 — 3 a 2 b 
3360 2= 3 ab 2 
64 = b 3 
6Z224 fubtrahend*. 
By pointing the number according* to the direction, it 
appears, that the root confids of two places.; let a be the 
value of the figure in the ten’s place, and b of that in the 
unit’s. Then a is the greated number wliofe cube is con¬ 
tained in* 405000, or 70 ; fubtraft its cube from the whole 
quantity, and the remainder is 62224; divide this remain¬ 
der by 3a 2 , or 14700, and the quotient 4, 01* b, is the fe¬ 
cond term in the root; then fubtraft the cube of 74 from 
the original number, and, as the remainder is nothing, 74 
is the cube root required. Obferve, that the cyphers may 
be omitted in the operation; and that, as a? was at drlt 
fubtrafted, it, from the fird remainder, 2 a2 b-\-T,ab 2 -{-b 3 \ e 
taken, the whole cube of a-\-b will be taken from the ori¬ 
ginal quantity. 
Ira 
