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144(12 1 22)44 Now the first figure of the 44 root shows the number of pie-- ces there are on a side of Fig. 1,

viz. 10 In order to preserve the square form, the additions must be made on two adjoining sides of the square, as in Fig. 2. Now it is evident that if there were just 20 pieces left, after disposing of 100, there would be just enongh to make a row on two sides of Fig. 1, and if there were 40 pieces left, they would make two rows, on two sides, as represented by the rows a, e, and o, n, Fig. 2. Hence the reason of placing the double of the |o | n | root on the left of the dividend for a divisor. In making the additions a, e, and o, n, you will observe there is a deficiency, A. which is not filled. To fill this deficiency, the rule directs to “except the right hand figure,” and likewise to “place the quotient figure on the right hand of the divisor.” Now the deficiency A. must be limited by the additions a, e, and o, n, consequently the figure, expressing the width of these additions, expresses the root of this deficiency, which, multiplied into itself, gives the superficial contents of the deficiency. Thus Fig.2 shows the disposition of 144 pieces into a square form.

2. Required the square root of 575.5 !

$15.56(23.984, root, 4

43)175 129

469)4650 4221

4788)42900 38304

4596 remainder.

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3. What is the square root of 10342656 7. * - Ans. 3216.

4. What is the square root of 964,5192360241 ? Ans. 31.05671.

5. What is the square root of .0000316969 7 - Ans. 0.0563.

Rules for extracting the Square Root of Vulgar Fractions and JMixed JN'umbers.

1. Reduce the fraction to its lowest terms for this and all other roots.

2. Extract the root of the numerator for a new numerator, and’ the root of the denominator for a new denominator.

3. Or reduce the vulgar fraction to a decimal, and extract its TOOt.

4. Mixed numbers may be reduced to improper fractions, and, the root of the numerator and denominator extracted, or the vulgar

fraction may be reduced to a decimal, and annexed to the whole

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2, What is the square root of ####! Ans. I's.

3. What is the square root of 42+! Ans. 64.

291. What is the rule for extracting the Square Root of Vulgar Fractions and mired numbers ?

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

RULE.—Multiply one of the given numbers by the other, and extract the square root of the product, and the root will be the mean proportional required.

Note. When the first number is as many times greater than the second, as the second is times greater than the third, the second number is called a mean proportional between the other two.

EXAMPLE. 1. What is the mean proportional between 36 and 144? 36×144=5184, and A/ 5184-72 Ans.

Probley 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.

1. If the area of a triangle be 160, what is the side of a square equal

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Problem 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'

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Problem IV. 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 4 inches bore, be in discharging double the quantity?

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* A square is a figure of four equal sides, each pair meeting perpendicularly, or a figure whose length and breadth are equal. As the area, or number of square feet, inches, &c. in a square, is equal to the product of two sides which are equal, the second power is called the square. Hence the rule of PRob LEM II. is evident.

# 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 $ 3, 4, &c. of the given number of men, and that will he the number of men in file, which double, triple, quadruple, &c. and the product will be the number in rank.

292. How do you find a mean proportional between two numbers ?

PROBLEM. V. A line 36 yards long will exactly reach from the top of a fort to the opposite bank of a river, known to be 24 yards broad. The height of the wall is required ?

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Problem VI. The height of a tree growing in the centre of a circular island 44 feet in diameter, is 75 feet, and a line stretched from the top of it over to the hither edge of the water, is 256 feet. What is the breadth of the stream, provided the land on each side of the water be level ?

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ProbleM VII. Suppose a ladder 60 feet long be so planted as to reach a window 37 feet from the ground, on one side of the street, and without moving it at the foot, will reach a window 23 feet high on the other side ; I demand the breadth of the street?

Ans. 102-64 feet,

ProbleM VIII. Given the difference of two numbers, and the dif. ference of their squares, to find the numbers. ,

Rule.—Divide the difference of the squares by the difference of the numbers, and the quotient will be their sum. Then proceed by Prob. 4, p. 57,

EXAMPLES.

1. The difference of two numbers is 20, and the difference of their squares is 2000; what are the numbers ?

Ans. 60 the greater. 40 the less.

2. Said Harry to Charles, my father gave me 12 apples more than he gave brother Jack, and the difference of the squares of our separate parcels was 288; Now, tell me how many he gave us, and you shall have half of mine. Harry's share 12. Ans. }o share 6.

EXTRACTION OF THE CUBE ROOT.

A CUBE is any number multiplied by its square. To extract the cube root, is to find a number which, being multiplied into its square, shall produce the given number.

RULE.

1. Separate the given number into periods of three figures each, by putting a point over the unit figure and every third figure beyond the place of units.

2. Find the greatest cube in the left hand period, and put its root in the quotient.

3. Subtract the cube thus found, from the said period, and to

the remainder bring down the next period, and call this the dividend.

4. Multiply the square of the quotient by 300, calling it the triple square, and the quotient by 30, calling it the triple quotient, and the sum of these call the divisor.

5. Seek how often the divisor may be had in the dividend, and place the result in the quotient.

6. Multiply the triple square by the last quotient figure, and write the product under this dividend; multiply the square of the last quotient figure by the triple quotient, and place this product under the last ; under all, set the cube of the last quotient figure and call their sum the subtrahend.

7. Subtract the subtrahend from the dividend, and to the remainder bring down the next period for a new dividend, with which proceed as before, and so on till the whole be finished.”

* The process for extracting the cube root may be illustrated in the same manner as that for the square root. Take the same number 37, and multiply as before, collecting the twice 21 into one sum, as they belong to the same place, and the operation will be simplified, 373 =50653.

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293. What is a Cube 3–294. What is the method of extracting the cube root of a given number 7 295. I wish you to illustrate the process under this rule, by one of the tramples given.

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