4. Seek how often the divifor is contained in the dividend, (except the right hand figure) and place the anfwer in the root for the fecond figure of it, and likewife on the right hand of the divifor: Multiply the divifor, with the figure latt annexed, by the figure laft placed in the root, and fubtract the product from the dividend: To the remainder join the next period for a new dividend. 5. Double the figure already found in the root, for a new divifor, (or bring down your last divifor for a new one, doubling the right hand figure of it) and from thefe find the next figure in the root as laft directed; and continue the operation in the fame manner, till you have brought down all the periods.

Note 1. If, when the given power is pointed off as the power, requires, the left hand period fhould be deficient, it must nevertheless ftand as the first period.

Note 2. If there be decimals in the given number, it must be pointed both ways from the place of units; If when there are in-. tegers, the first period in the decimals be deficient, it may be completed by annexing fo many cyphers as the power requires; And the root must be made to confift of fo many whole numbers. and decimals as there are periods belonging to each; and when the periods belonging to the given number are exhausted, the operation may be continued at pleasure by annexing cyphers.

EXAMPLES.

1. Required the fquare root of 30138696025 ?

30138696025(173665 the root.

Note. When more than half the root is found, the remaining figures of it may be found by Divifion, making use of the laft divifor, and bringing down fo many of the next figures of the refolvend, as there were periods to come down, when you began the divifion.

RULES for the SQUARE ROOT of VULGAR FRACTIONS and MIXED NUMBERS.

After reducing the fraction to its loweft terms, for this and all other roots; then,

1. Extract the root of the numerator for a new numerator, and the root of the denominator for a new denominator, which is the best method, provided the denominator be a complete power. But if it be not,

2. Multiply the numerator and denominator together; and the root of this product being made the numerator to the denominator of the given fraction, or made the denominator to the numerator of it, will form the frac tional part required :-Or,

3. Reduce the vulgar fraction to a decimal, and extract its root.

4.

Mixed numbers may either be reduced to improper fractions, and extracted by the first or second

rule;

rule; or the vulgar fraction may be reduced to a deci-mal, then joined to the integer, and the root of the whole extracted.

EXAMPLES.

144 ?

1. What is the fquare root of 120

144

By Rule 1.

1120 1681 16(4 root of the numerator.

10116

1681(41 root of the denominotor. 16

81)81 Therefore the root of the given fraction.

81

By Rule 2.

16× 1681=26896 and ✔✅ 26896=164. Then,

09756+

2

8208

,0095181439=

Anj. 71

2. What is the fquare root of 787 ? 3. What is the fquare root of 424 ?

Anf. 6

Note. In extracting the fquare or cube root of any furd number there is always a remainder or fraction left, when the root is found: To find the value of which, the common method is to annex pairs of cyphers to the refolvend, for the fquare, and ternaries of cyphers to that of the cube, which makes it tedious to difcover the value of the remainder, cfpecially in the cube. Now this trouble may be faved by the following method.

1

In the fquare the quotient is always doubled for a new divifor Therefore when the work is completed, the root doubled is the true divifor, or denominator* to its own fraction; as, if the root be 12, the denominator will 0.3

Thefe denominators give a fmall matter too much in the fquare root, and too little in the cube, yet they will be fuffi cient in common ufé.

will bec2469 to be placed under the remainder, which vulgar fraction, or its equivalent decimal, must be annexed to the quotient, or root, to complete it.

* If to the remainder either of the fquare or cube, cyphers be annexed, and divided by their refpective denominators, the quotient will produce the decimals belonging to the root.

ORULE. Multiply the given numbers together, and extract the fquare root of the product; which root will be the mean proportional fought. r Tic p?Pz1*

What is the mean proportional between 24 and 96?

PROBLEM II. To find the fide of a Square equal in Area bus dege to any given Superficies whatever.

RULE. Find the Area, and the fquare root is the fide of the fquare fought.

EXAMPLES.

1. If the area of a circle be 184,125 what is the side of a square equal in area thereto sa mi quiba pa √ 184,125=13,569+ Anf.

2. If the area of a triangle be 160; what is the fide of a fquare equal in area thereto ?

✓ 160=12,649+ Anf. PROB.

PROB. III. A certain General has an army of 5625 men; pray how many muft he place in rank and file, to form them into a square of a to jaspe ptá

165625=75 Anf.*

PROB. IV. Let 10952 men be fo formed, as that the number in rank may be double the file. ✓ 10952

2

jeur sa-of gorgato! =74 in file, and 74× 2=148 in rank. MOITASI PROB. V. If it be required to place 2016 men fu as that there may be 56 in rank and 36 in file, and to stand 4 feet diftance in rank, and as much in file; how much ground do they ftand on

To anfwer this, or any of the kind, ufe the following proportionAs unity to the diftance : fo is the number in rank lefs by one to a fourth number ;-next do the fame by the file, and multiply the two numbers together, found by the above proportion, and the product will be the anfwer.t

As 14:56—1: 220. And as 1 : 4 :: 36—1: 140. Then 230X14030800 fquare feet, the Anf.

PROB. VI. Suppose I would fet out an orchard of 600 trees, fo that the length fhall be to the breadth, as 3 to 2, and the distance of each tree, one from the other, 7 yards; how many trees must it be in length, and how many in breadth; and how many fquare yards of ground do they stand on?

To

343 If you would have the number of men be double, triple, or quadruple, &c. as many in rank as in file extract the fquare Foot of,,, &c. of the given number of men, and that will be the number of men in file, which double, triple, quadruple, 51&c, and the product will be the number in rank.

+ The above rule will be found useful in planting trees, having the diftance of ground between each given.