:

6. A certain castle, which is 45 yards high, is surrounded by 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.

7. A line 27 yards long will exactly reach from the top of a fort to the opposite bank of a river, which is known to be 23 yards broad; what is the height of the fort? Ans. 14.142+ yards.

8. Suppose a ladder 40 feet long be so planted as to reach a window 33 feet from the ground, on one side of the street, and without moving it at the foot, will reach a window on the other side 21 feet high; what is the breadth of the street? Ans. 56.64+ feet. 9. Two ships depart from the same port; one of them sails due west 50 leagues, the other due south 84 leagues: how far are they asunder ?

Ans. 97.75+ Or, 972+ leagues.

THE CUBE ROOT.

The cube of a number is the product of that number multiplied into its square.

Extraction of the cube root is the finding of such a number, as, being multiplied into its square, will produce the number proposed.

RULE.

1. Separate the given number into periods of three figures each, beginning at the units place.

2. Find the greatest cube contained in the left hand period, and set its root on the right of the given number: subtract said cube from the left hand period, and to the remainder bring down the next period for a dividual.

3. Square the root and multiply the square by 3 for a defective divisor.

4. Reserve mentally the units and tens of the dividual, and try how often the defective divisor is contained in the rest place the result of this trial to the root, and its square to the right of said divisor, supplying the place of tens with a cypher, if the square be less

than ten.

5. Complete the divisor by adding thereto the product of the last figure of the root by the rest and by 30.

6. Multiply and subtract as in Simple Division, and bring down the next period for a new dividual; for which find a divisor as before, and so proceed till all the periods are brought down.

*

**See note under the rule for extracting the square root: it applies equally to this rule.

Note.-Defective divisors, after the first, may be more concisely found thus: To the last complete divisor add the number which completed it with twice the square of the last figure in the root, and the sum will be the next defective divisor.

PROOF.

Involve the root to the third power, adding the remainder, if any, to the result.

EXAMPLES.

1. What is the cube root of 99252.847 ?

99252.847(46.3

64

Defective divisor and square of 6=4836)35252

+720 complete divisor

5556)33336

Defective divisor & square of 3=634809)1916847 +4140 complete divisor

638949)1916847

2. What is the cube root of 16194277 ? 3. What is the cube root of 389017?

4. What is the cube root of 5735339 ?

5. What is the cube root of 34328125?

6. What is the cube root of 22069810125 ?

Ans. 253.
Ans. 73.
Ans. 179.

· Ans. 325.

Ans. 280.5

7. What is the cube root of 12.977875? Ans. 2.35 8, What is the cube root of 36155.027576 ?

Ans. 33.06+

9. What is the cube root of 15926.972504 ?

Ans. 25.16+

10. What is the cube root of .001906624 ?

Ans. .124

Note 1.-The cube root of a vulgar fraction is found by reducing it to its lowest terms, and extracting the root of the numerator for a numerator, and of the denominator for a denominator. If it be a surd, extract the root of its equivalent decimal.

3000

Ans. 3
Ans.
Ans..

2. A mixed number may be reduced to an improper fraction, or a decimal, and the root thereof extracted. 1. What is the cube root of 648 ? 2. What is the cube root of 250? 3. What is the cube root of 1520 ? 4. What is the cube root of 12 192 5. What is the cube root of 31

SURDS.

686

5130

6. What is the cube root of 71? 7. What is the cube root of 91?

APPLICATION.

15 ?

*/crca

Ans. 21.

Ans. 34.

Ans. 1.93+

Ans. 2.092+

1. The cube of a certain number is 103823; what is that number? Ans. 47. 2. The cube of a certain number is 1728; what number is it?

Ans. 12.

3. There is a cistern or vat of a cubical form which contains 1331 cubical feet; what are the length, breadth, and depth of it? Ans. each 11 feet. 4. A certain stone of a cubical form contains 474552 solid inches; what is the superficial content of one of its sides? Ans. 6084 inches.

A GENERAL RULE FOR EXTRACTING THE ROOTS OF ALL POWERS.

1. Point the given number into periods agreeably to the required root.

2. Find the first figure of the root by the table of powers, or by trial; subtract its power from the left hand period, and to the remainder bring down the first figure in the next period for a dividend.

3. Involve the root to the next inferior power to that which is given, and multiply it by the number denoting the given power, for a divisor; by which find a second figure of the root.

4. Involve the whole ascertained root to the given power, and subtract it from the first and second periods. Bring down the first figure of the next period to the remainder, for a new dividend; to which, find a new divisor, as before; and so proceed.

Note.—The roots of the 4th, 6th, 8th, 9th, and 12th powers, may be obtained more readily thus:

For the 4th root take the square root of the square

root.

For the 6th, take the square root of the cube root. For the 8th, take the square root of the 4th root. For the 9th, take the cube root of the cube root. For the 12th, take the cube root of the 4th root.

Alligation is a rule for adjusting the prices and simples of compound quantities.

CASE 1.

To find the mean price of any part of the composition, when the several quantities and their prices are given.

RULE.

As the sum of the several quantities,
Is to any part of the composition;
So is their total value,

To the value of that part.

PROOF.

The value of the whole mixture at the mean price must agree with the total value of the several quantities at their respective prices.

EXAMPLES.

1. If 6 gallons of wine at 67 cents per gallon; 7 at 80 cents, and 5 at 120 cents per gallon, be mixed together, what will 1 gallon of the mixture be worth?

2. If 19 bushels of wheat at 6 s. per bushel; 40 bushels of rye at 4 s. per bushel, and 12 bushels of barley at 3 s. per bushel, be mixed together, what will a bushel of the mixture be worth? Ans. 4 s. 44 d.

3. If a grocer mix 2 cwt. of sugar at 56 s. per cwt.; 1 cwt. at 43 s. per cwt.; and 2 cwt. at 50 s. per cwt., what will be the value of 1 cwt. of the mixture? Ans. 2 L. 11 s.

4. A farmer mingled 20 bushels of wheat at 5 s. per bushel, and 36 bushels of rye at 3 s. per bushel, with 40 bushels of barley at 2 s. per bushel: I desire to know the worth of a bushel of this mixture. Ans. 3 s.

5. If 4 ounces of silver, worth 75 cents per ounce, be melted with 8 ounces, worth 60 cents per ounce, what will 1 ounce of the mixture be worth? Ans. 65 cts.

6. A wine merchant mixes 12 gallons of wine at 4 s. 10 d. per gallon, with 24 gallons at 5 s. 6 d., and 16 gallons at 6 s. 34 d. ; what is a gallon of the mixture worth? Ans. 5 s. 7 d.