To extract the square root of a mixed number.

RULE. 1. Reduce the fractional part of the mixed number to its lowest term, and then the mixed number to an improper fraction.

2. Extract the roots of the numerator and denominator for a new numerator and denominator.

If the mixed number given be a surd, reduce the fractional part to a decimal, annex it to the whole number, and extract the square root therefrom.

1. There is an army consisting of a certain number of men, who are placed rank and file, that is, in the form of a square, each side having 576 men, I desire to know how many the whole square contains ? Ans. 331776.

2. A certain pavement is made exactly square, each side of which contains 97 feet, I demand how many square feet are contained therein ? Ans. 9409.

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To find a mean proportional between any two given numbers.

RULE. The square root of the product of the given numbers is the mean proportional sought.

EXAMPLES.

1. What is the mean proportional between 3 and 12 ? Ans. 3 x 12 36 then ✓ 366 the mean proportional.

2. What is the mean proportional between 4276 and 842 ? Ans. 1897,4+

To find the side of a square equal in area to any given superfices. RULE. The square root of the content of any given superfices, is the square equal sought.

EXAMPLES.

3. If the content of a given circle be 160, what is the side of the square equal? Ans. 12,64911.

4. If the area of a circle is 750, what is the side of the square equal? Ans. 27,38612.

The area of a circle given to find the diameter.

RULE. AS355: 452, or as 1: 1,273239 :: so is the area: to the square of the diameter;-or, multiply the square root of the area by 1,12837, and the product will be the diameter.

EXAMPLE.

5. What length of cord will fit to tie to a cow's tail, the other end fixed in the ground, to let her have liberty of eating an acre of grass, and no more, supposing the cow and tail to be 5 yards and a half? Ans. 6,136 perches.

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The area of a circle given to find the periphery, or circumference.

RULE. As 113: 1420, or as 1 : 12,56637 :: the area: to the square of the periphery, or multiply the square root of the area. by 3,5449, and the product is the circumference..

EXAMPLES.

6.

When the area is 12, what is the circumference ?
Ans. 12,2798.

7. When the area is 160, what is the periphery?

Ans. 44,84.

Any two sides of a right angled triangle given to find the third

side.

1. The base and perpendicular given to find the hypothe

nuse.

RULE. The square root of the sum of the squares of the base and perpendicular is the length of the hypothenuse.

EXAMPLES.

8. The top of a castle from the ground is 45 yards high, and is surrounded with a ditch 60 yards broad; what length must a ladder be to reach from the outside of the ditch to the top of the castle?

Ans.

75 yards.

9. The wall of a town is 25 feet high, which is surrounded by a moat of 30 feet in breadth, I desire to know the length of a ladder that will reach from the outside of the moat to the top of the wall. Ans. 39,05 feet.

The hypothenuse and perpendicular given to find the base.

RULE. The square root of the difference of the squares of the hypothenuse and perpendicular is the length of the base.

The base and hypothenuse given to find the perpendicular.

RULE. The square root of the difference of the hypothenuse and base is the height of the perpendicular.

N. B. The two last questions may be varied for examples to the two last propositions.

Any number of men being given to form them into a square battle, or to find the number of ranks and files.

RULE. The square root of the number of men given, is the number of men either in rank or file.

10. An army consisting of 331776 men, I desire to know how many in rank and file?

Ans. 576. 11. A certain square pavement contains 48841 square stones, all of the same size, I demand how many are contained in one of the sides?

Ans. 221.

EXTRACTION OF THE CUBE ROOT.

To extract the Cube Root is to find out a number which be ing multiplied into itself, and then into that product, produceth the given number.

RULE 1. Point every third figure of the cube given, beginning at the unit's place, seek the greatest cube to the first point and subtract it therefrom, put the root in the quotient, and bring down the figures in the next point to the remainder for a resolvend.

2. Find a divisor by multiplying the square of the quotient by 3. See how often it is contained in the resolvend, rejecting the units and tens, and put the answer in the quotient.

3. To find the subtrahend. 1. Cube the last figure in the quotient. 2. Multiply all the figures in the quotient by 3. except the last, and that product by the square of the last. 3. Multiply the divisor by the last figure. Add these products togeth er, gives the subtrahend, which subtract from the resolvend; to the remainder bring down the next point and proceed as before.

ROOTS. 1. 2. 3. 4. 5. 6. 7. 8. 9. CUBES. 1. 8. 27. 64. 125. 216. 343. 512. 729.

Another new and more concise method of extracting the Cube Root.

RULE. 1. Point every third figure of the cube given beginning at the unit's place, then find the nearest cube to the first point, and subtract it therefrom, put the root in the quotient, bring down the figures in the next point to the remainder for a resolvend.

2. Square the quotient and triple the square for a divisor— as, 4 x4x3=48. Find how often it is contained in the resolvend, rejecting units and tens, and put the answer in the quotient.

3. Square the last figure in the quotient, and put it on the right hand of the divisor:

As 6×6=36 put to the divisor 484836. 4. Triple the last figure in the quotient, and multiply by the former, put it under the other, units under the tens, add them together, and multiply the sum by the last figure in the quotient, subtract that product from the resolvend, bring down the next point and proceed as before.

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