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to the number given. Such operation is called Extraction of the Square Root, to perform which observe the following

RULE. 1st. You must point your given numbers, that is, make a point or dot over the unit's placé, another over the hundred's, and upon every second figure throughout.

2nd. Then seek the greatest square number in the first point, towards the left hand, placing the square number under the first point, and the root thereof in the quotient, and subtract the said square number from the first point, and to the remainder bring down the next point, and call that the resolvend.

3d. Then double the quotient, and place it as a divisor on the left hand of the resolvend, and seek how often the divisor is contained in the resolvend, (always rejecting the unit's place,) and put the answer in the quotient, and also on the right-hand side of the divisor; then multiply by the figure last put in the quotient, and subtract the product from the resolvend, (as in common division,) and bring down the next point to the remainder, (if there be any more,) and proceed as before.

A Table of Squares, Cubes, and their Roots.

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1. Let 4489 be a number given, the square root of which is required ?

4489(67 Ans.
36

127)889 resolvend.

889 product. Demonstration. First point the given number, as before directed, and then, by the above table, seek the greatest square number in 44, (the first point to the left hand,) which you will find to be 36, and 6 the root." Put 36 under 44, and 6 in the quotient, and subtract 36 from 44, and there remains 8, to which bring down the other point, 89, placing it on the right hand, so as to make 889 for a resolvend; then double the quotient 6, and it makes 12, which place on the left hand for a divisor, and seek how often 12 is in 88, (reserving the unit's place, the answer is 7 times, which put in the quotient, and also on the right hand of the divisor, and multiply 127 by 7, as in common division, and the product is 889, which subtracted from the resolvend there remains nothing, and your work is finished. The square root of 4489 is 67, which root, if multiplied by itself, that is, 67 by 67, the product will be 4489, equal to the given square number, Work all the rest according to this demonstration,

2. Let 106929 be a square number, whose root is required?

Ans. 327.

NOTE. If remainders occar, you may annex ciphers, by two at a time, to the remainder, and so prosecate the work to any degree of exactness, or to as many places of decimals as you please. 3. Required the square root of 144?

Ans. 12. 4. Required the square root of 2176782336

Ans. 466,56 5. Required the square root of 1296 ?

36 6. Required the square root of 56644?

23,8 7. Required the square root of 119025?

345 8. Required the square root of 2268741? 1506,23+ 9. Required the square root of 36373961?

6031 10. Required the square root of 430467213 6561

If the given number be a mixed number, consisting of a whole number and a decimal together, make the number of decimal places even, as 2, 4, 6, 8, 10, &c. that so there may a point fall on the unit's place of the whole numbers. 11. Required the square root of 656714,37512?

Ans. 810,379. Note. In this question there are five places of decimals, therefore you must put a cipher on the right side of the 2, to make it even, so that the point may fall on 4, the unit's place of the whole number. 12. Required the square root of 751417,5745 ?

Ans. 866,4, 13. Required the square root of 761,801216?

Ans. 27,6007+ 14. Required the square root of 3,172181121

Ans. 1,78106+

To extract the square root of a vulgar fraction. RULE. Reduce the fraction to its lowest terms, and then extract the square root of the numerator for a new numerator, and the square root of the denominator for a new denominator. If the fraction be a surd, that is, a number whose root cannot be exactly found, reduce it to a decimal, and extract the root from it.

Ans.

15. What is the square root of ?
16. What is the square root of
17. What is the square root of ?
18. What is the square root of

ooja fotool

SURDS.

19. What is the square root of 37 ?
20. What is the square root of ?
21. What is the square root of 37?

Ans. 72414+

,93308+ ,86602+

545

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To extract the square root of a mixed number. RULE. Reduce the fractional part of the mixed number to its lowest terms, and the mixed number to an improper fraction; then extract the roots of the numerator and denominator for a new numerator and denominator. If the given mixed number be a surd, reduce the fractional part to a decimal, annex it to the whole number, and extract the square root from it.

Ans. 4}

64 3 31 911

22. What is the square root of 1726 ?
23. What is the square root of 3745?
24. What is the square root of 941?
25. What is the square root of 814?

SURDS.
26. What is the square root of 7614?
27. What is the square root of 851 ?
28. What is the square root of 7 ja

Ans. 8,7649+

9,27 + 8,7961+

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Application. 29. 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 containing 472 men; I demand how many men the whole square contains ?

Ans. 222784. 30. The floor of a large house is exactly square, each side of which contains 75 feet; I want to know how many square feet are contained therein?

Ans. 5625. 31. A certain general has an army of 5184 men; how many must he place in rank and file, to form them into a square?

✓ 5184=72 Ans. To find a mean proportional between two numbers. Rule. Multiply the two given numbers together, and extract the square root of that product. 32. What is the mean proportional between 4 and 9?

4x9=36, and 36=6 Ansı

To form any body of soldiers, so that they may be double,

triple, qc. as many in rank as in file. RULE. Divide the given number of men by 2, 3, &c. according to the nature of the question, and extract the square root of the quotient, and that will be the number of men in file, which double, triple, &c. and the product will be the number in rank.

33. Suppose 13122 men be so formed, that the number in "rank may be double the number in file; how many in each?

13122--2=6561, 6561=81 in file, 81x2=162 in rank.

EXTRACTION OF THE CUBE ROOT. To extract the Cube Root is nothing else than to find such a number, as being first multiplied into itself, and then that product, produceth the given number, to perform which there are many rules.

Rule 1. First point your given number, beginning with the unit's place, and make a point or dot over every third figure, towards the left hand. Secondly, seek the greatest cube number in the first point, towards the left hand,

putting the root thereof in the quotient, and the said cube number under the first point, and subtract it therefrom, and to the remainder bring down the next point, and call that the resolvend. Thirdly, triple the quotient, and place it under the resolvend, the unit's place of this under the ten's place of the resolvend, and call this the triple quotient. Fourthly, square the quotient, and triple the square, and place it under the triple quotient, the unit's of this under the ten's place of the triple quotient, and call this the triple square. Fifthly, add these two together, in the same order as they stand, and the sum will be the divisor. Sixthly, seek how often the divisor is contained in the resolvend, rejecting the unit's place of the resolvend, (as in the square root,) and put the answer in the quotient. Seventhly, cube the figure last put in the quotient, and put the unit's place thereof under the unit's place of the resolvend. Eighthly, multiply the square of the figure last put in the quotient, into the triple quotient, and place the product under the last, one place more to the left hand. Ninthly, multiply the triple square by the figure last put in the quotient, and place it under the last, one place more to the left hand. Tenthly, add the

three last numbers together, in the same order as they stand, and call that the subtrahend. Lastly, subtract the subtrahend from the resolvend, and if there be another point, bring it down to the remainder, and call that a new resolyend, and proceed in all respects as herein directed.

RULE 2. Take the nearest root, not too great, of the first period, for the first figure of your root, subtract its cube from said period, and to the remainder bring down the next period for a resolvend. Take three times the square of the root for a defective divisor, and seek how often it is contained in the resolvend, abating the two right-hand figures. Set down the result of this trial for the second figure of the root, and its square for the two right-hand figures of the divisor: complete the divisor by adding thereto 30 times the product of the last figure of the root into the rest. Multiply, subtract, and to the remainder bring down the next period for a new resolvend : add together the last complete divisor, the number that completed it, and twice the square of the last figure of the root, for a new defective divisor, and thus proceed throughout.

Rule 3. Point every third figure of the given cube, beginning at the unit's place, and seek the greatest cube tu the first point, subtract it therefrom, put the root in the quotient, and bring down the figures in the next point to · the remainder, for a resolvend. Find a divisor by multiplying the square of the quotient by 3, and see how often it is contained in the resolvend, rejecting the units and tens, and put the answer in the quotient. Cube the last figure in the quotient, and multiply all the figures in it by 3, except the last, and that product by the square of the last; then multiply the divisor by the last figure. Adding these products together gives the subtrahend, which subtract from the resolvend, and to the remainder bring down the next point, and proceed as before. Rule 4. Point off

every third figure of the given number, 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 next period or point to the remainder, for a resolvend; then square the quotient, and triple the square for a divisor, and find how often it is contained in the resolvend, rejecting units and tens in the resolvend, and put the answer in the quotient. Square the last figure in the quotient, and put it on the right hand of the divisor; triple the last tigure in the quotient, and mul-, tiply by the former, put it under the other, units under the tens, add them together, and multiply the sum by the last

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