5. 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.

6. 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

7. 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.

8. 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, 973+lea.

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 cipher, 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 sun will be the next defective divisor.

FROOF.

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

Defective divisor & square of 3634809)1916847 +4140 complete divisor

638949)1916847

Ans. 253.

Ans. 73.

Ans. 179.

Ans. 325.

Ans. 280.5

Ans. 2.35.

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? 7. What is the cube root of 12.977875? 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 Not 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 If it be a surd, extract the root of its

for a denominator.

equivalent decimal.

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 64? 2. What is the cube root of 250 ?

3000

686

130

3. What is the cube root of 1529 ? 4. What is the cube root of 1219 ? 5. What is the cube root of 31 ?

SURDS.

343

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

APPLICATION.

Ans.

Ans. . Ans. Ans. 21.

Ans. 31.

Ans. 1.93+ Ans. 2.092+

1. If a block of marble be 47 inches long, 47 inches broad, and 47 inches deep, how many cubical inches does it contain ? Ans. 103823. 2. There is a cellar dug 12 feet long, 12 feet deep, and 12 feet broad; how many solid feet of earth were taken out of it? Ans. 1728.

3. How many cubes of three inches are contained in a cubical foot?

Ans. 64.

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 pow er, 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, 3th, 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.

2. What is the fourth root of 140283207936? Ans. 612. 3. What is the sixth root of 782757789696? Ans. 96 4. What is the seventh root of 194754273881? Ans. 41 5. What is the ninth root of 1352605460594688 ?

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.

P

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 I gallon of the mixture be worth?

As 18:1:: 1562 : 86.77+ Answer.

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

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

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

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 cents.

6. A wine merchant mixes 12 gallons of wine at 4s. 10d. per gallon, with 24 gallons at 5s. 6d. and 16 gallons at 6s. 31d.; what is a gallon of the mixture worth?

CASE 3.

Ans. 5s. 7d.

When the prices of several simples are given, to find how much of each, at their respective rates, must be taken to make a compound or mixture at any proposed price.