THE EXTRACTION OF 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, which points shew the number of figures the root will consist of.

2. Find the greatest square number in the first, or left hand, period, place the root of it at the right hand of the given number, (after the manner of a quotient in division) for the first figure of the root, and the square number, under the period, and subtract it there. from, and to the remainder bring down the next period for a dividend.

3. Place the double of the root, already found, on the left hand of the dividend for a divisor.

4. Seek how often the divisor is contained in the dividend, (except the right hand figure) and place the answer in the root for the second figure of it, and likewise on the right hand of the divisor: Multiply the divisor with the figure last annexed by the figure last placed in the root, and subtract the product from the dividend: To the remainder join the next period for a new dividend.

5. Double the figures already found in the root, for a new divisor, (or, bring down your last divisor for a new one, doubling the right hand figure of it) and from these, find the next figure in the root as last directed, and continue the operation, in the same 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 figure should be deficient, it must nevertheless stand 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 integers, the

* In order to fhew the reason of the rule, it will be proper to premife the following Lemma. The product of any two numbers can have, at moft, but so many places of figures as are in both the factors, and at least but one less.

Demonftration. Take two numbers confisting of any number of places; but let them be the leaft poffible of those places, viz. Unity with cyphers, as 100 and 10: Then their product will be 1 with so many cyphers annexed as are in both the numbers, viz. 1000; but 1000 has one place lefs than 100 and 10 together have: And fince 100 and 10 were taken the least poffible, the product of any other two numbers, of the fame number of places, will be greater than 1000; confequently, the product of any two numbers can have, at least, but one place less than both the factors.

Again, take two numbers, of any number of places, which fhall be the greateft poffible of thofe places, as 99 and 9. Now, 99 x 9 is lefs than 99 X 10; but 99 X 10 (990) contains only fo many places of figures as are in 99 and 9; therefore, 99 X 9, or the product of any other two numbers, confifting of the fame number of places, cannot have more places of figures, than are in both its factors.

Corollary 1. A fquare number cannot have more places of figures than double the places of the root, and at least but one lefs.

Corollary 2. A cube number cannot have more places of figures than triple the places of the root, and at least but two lefs,

the first period in the decimals be deficient, it may be completed by annexing so many cyphers as the power requires: And the root must be made to consist of so 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 pleas ure by annexing cyphers,

EXAMPLES.

1st. Required the square root of 30138696025 ?

30138696025(173605 the root,

1

Ist. Divisor-27)201

189

2d. Divisor-343)1238

1029

3d. Divisor 3466)20969

20796

4th. Divisor-347205) 1736025

1736025

2d. Required the square root of 575-5 ?

575-50(23-98+, the root.

4

43)175

129

469)4650
4221

4788)42900
38304

4596 Remainder.

3d. What is the square root of 10342656?

Ans. 3216.

4th. What is the square root of 964-5192360241? Ans. 31-05671.

5th. What is the square root of 234.09 ?

Ans. 15.3.

6th. What is the square root of ⚫0000316969?

Ans. 00563.

7th. What is the square root of ⚫045369?

Ans. 213.

RULES FOR THE SQUARE ROOT OF VULGAR FRACTIONS AND MIXED NUMBERS.

After reducing the fraction to its lowest terms, for this and all oth er roots; then,

1st. Extract

1st. 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,

2d. 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 fractional part required. Or,

*

3d. Reduce the vulgar fraction to a decimal, and extract its root. 4th. Mixed numbers may either be reduced to improper fractions, and extracted by the first or second rule, or the vulgar fraction may be reduced to a decimal, then joined to the integer, and the root of the whole extracted.

81)81 Therefore, the root of the given fraction.

81

By Rule 2.

16x1681-26896, and ✔ 26896=164. Then,

1681)16(-0095181439+. And 0095181439=09756+.

2d. What is the square root of 2293 ?

8208

3d. What is the square root of 42?

7

Ans. g.

Ans 6.

Note. In extracting the square or cube root of any surd 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 resolvend, for the square, and ternaries of cyphers to that of the cube, which makes it tedious to discover the value of the remainder, especially in the cube, whereas this trouble might be saved if the true denominator could be discovered.

As in division the divisor is always the denominator to its own fraction, so likewise it is in the square and cube, each of their divisors being the denominators to their own particular fractions or numera

tors.

* That is, fuppofe a=7, and b=2, the rule may be thus expreffed:

In

a

this rule will ferve whether the root be finite or infinite,

2

2

√7X2

In the square the quotient is always doubled for a new divisor; therefore, when the work is completed, the root doubled is the true divisor or denominator to its own fraction; as, if the root be 12, the denominator will be 24, 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 square or cube, cyphers be annexed, and divided by their respective denominators, the quotient will produce the decimals belonging to the root.

APPLICATION AND USE OF THE SQUARE ROOT. PROB. I To find a mean proportional between two numbers.

RULE. Multiply the given numbers together, and extract the square root of the product; which root will be the mean proportional sought.

EXAMPLE.

What is the mean proportional between 24 and 96?

96x24-48 Answer.

PROB. II. To find the side of a square equal in area to any given superficies whatever.

RULE. Find the area, and the square root is the side of the square sought.

EXAMPLES.

1st. If the area of a circle be 184-125, What is the side of a square equal in area thereto ?

√184·125=13.569+ Answer. 2d. If the area of a triangle be 160, What is the side of a square equal in area thereto ✔/160=12·649+ Answer. PROB. III. A certain general has an army of 5625 men: pray How many must he place in rank and file, to form them into a square? 5625 75 Answer.† PROB. IV. Let 10952 men be so formed, as that the number in rank may be double the file.

✔10052-74 in file, and 74x2-148 in rank. PROB. V. If it be required to place 2016 men so as that there may be 56 in rank and 36 in file, and to stand 4 feet distance in rank, and as much in file, How much ground do they stand on?

To answer this, or any of the kind, use the following proportion: As unity to the distance :: so is the number in rank less by one: to a fourth number; next, do the same by the file, and multiply the

two

Although thefe denominators give a small matter too much in the fquare root, and too little in the cube, yet they will be fufficient in common use, and are much more expeditious than the operation with cyphers.

If you would have the number of men be double, triple, or quadruple, &c. as many in rank as in file, extract the square root of †,†, 1, &c. of the given number of men, and that will be the number of men in file, which double, triple, quadruple, &c. and the product will be the number in rank,

two numbers together, found by the above proportion, and the product will be the answer.'

As 1 : 4 :: 56 — 1: 220. And, as 1 : 4 :: 36 1:140. Then, 220×140-30800 square feet, the Answer.

PROB. VI. Suppose I would set out an orchard of 600 trees, so that the length shall 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 square yards of ground do they stand on?

To resolve any question' of this nature, say, as the ratio in length is to the ratio in breadth :: so is the number of trees: to a fourth number, whose square root is the number in breadth.

And as the ratio in breadth : is to the ratio in length :: so is the number of trees: to a fourth, whose root is the number in length.

203 × 133 = 26999 square yards, the Answer.

PROB. VII. Admit a leaden pipe

inch diameter will fill a cistern

in 3 hours; I demand the diameter of another pipe which will fill the same cistern in 1 hour.

RULE. As the given time is to the square of the given diameter, so is the required time to the square of the required diameter.

275 and 75x75=5625. Then, as 3h. : ·5625 ::

:

1h. 1.6875 inversely, and 1.6875=1.3 inch nearly, Ans. PROB. VIII. If a pipe whose diameter is is 1.5 inch, fill a cistern in 5 hours, in what time will a pipe whose diameter is 3.5 inches fill the same?

1.5×1.5=2.25; and 3.5×3 5=12:25. Then, as 2-25: 5 :: 12:25 : ·918+ hour, inversely,=55 min. 5 sec. Answer.

PROB. IX. If a pipe 6 inches bore, will be 4 hours in running off a certain quantity of water, In what time will 3 pipes, each four inches bore, be in discharging double the quantity?

6×6=36. 4×4=16, and 16x3=48. Then, as 36: 4h. :: 48: 3h. inversely, and as 1w. : 3h. :: 2w.: 6h. Answer.

PROB. X. Given the diameter of a circle to make another circle, which shall be 2, 3, 4, &c. times greater or less than the given circle. RULE. Square the given diameter, and if the required circle be greater, multiply the square of the diameter by the given proportion, and the root of the product will be the required diameter. But if the required circle be less, divide the square of the diameter by the given proportion, and the root of the quotient will be the diameter required.

There is a circle whose diameter is 4 inches; I demand the diameter of a circle 3 times as large?

4x4-16; and 16x3=48; and ✔ 48=6·928+ inches Answer.

PROB.

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