The side of a cube being given, to find the side of that cube which shall be double, triple, &c., in quantity to the given cube.

RULE.

Cube your given side, and multiply it by the given propertion between the given and required cube, and the cube root of the product will be the side sought.

30. If a cube of silver whose side is 4 inches, be worth D50, I demand the side of a cube of the like silver, whose value shall be 4 times as much?

Ans. 4x4x4=64, and 64×4=256 : 3/256=6·349+inches. 31. There is an oblong cellar, the content of which is 1953.125 cubic feet; what is the side of a cubical cellar that shall contain just as much? Ans. 12.5 feet. 32. What is the difference between a solid half foot and half of a solid foot?

Ans: 3 half feet.

Ans. 1 cord.

33. In 221184 solid inches how many cords? 34. Admitting a room to be 11 feet high, 21 feet in length, and 16 feet in width, what number of cubic feet of space in it? Ans. 3696 cubic feet. 35. The diameter of a bushel measure being 18 inches, and the height 8 inches, what is the side of a cubic box which shall contain that quantity? Ans. 12.907+inches. 36. In a cubic foot, how many cubes of 6 inches, and how many of 4, of 3, of 2, of 1, are contained therein ?

Ans. 8 of 6 inches; 27 of 4 inches; 64 of 3 inches; 216 of 2 inches; 1728 of 1 inch.

37. Suppose a cubical cellar to contain 1728 solid feet, what will one of its cubic sides measure?

38. In a square box that will contain 1000 marbles, how many will it take to reach across the bottom of the box, in a straight row ? Ans. 10.

39. What is the difference between the cube root of 27 and the square root of 9? Ans. O

40. If a globe of silver 3 inches in diameter be worth D160, what is the value of one 6 inches in diameter ?

Ans. 33 63 :: D160: D1280.

41. If the diameter of the planet Jupiter is 12 times as much as the earth, how many globes of the earth would it take to make one as large as Jupiter? Ans. 1728.

42. There are two small globes; one of them is one inch in diameter, and the other two inches; how many of the smaller globes will make one of the larger?

Ans. 8.

RULE III.

1. Find the quotient root of the left hand period, which subtract from the same, and then bring down the next period. 2. Multiply the square of the quotient figure by 300 for a divisor; then find the next figure; square this quotient figure; multiply that square by the other quotient figure, and then by 30; find the cube of this last quotient figure; add both these products to the product of the divisor and the quotient figure, the sum of which subtract from the dividend. 3. Then bring down the next period, which will complete the next dividend; square your two quotient figures, and multiply by 300 for your next divisor, and so continue till the operation is completed.

NOTE. The roots of the 4, 6, 8, 9, and 12 powers may be obtained in the following manner:—

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, &c.

REVIEW.

What is a cube? What is the cube root? What will you first do to extract the cube root of a whole number? Repeat the rule? How do you extract the cube root of a decimal fraction? When there is a decimal and whole number, how will you point them off? Why do you point decimals from the left or decimal point toward the right? In extracting a root, if there be a remainder, what may be done? What is the rule for decimals? How do you extract the cube root of a vulgar fraction? What is the rule? How do you extract the cube root of a mixed number? What is the difference between a cube and the cube root? 43. What is the cube root of 3628? Ans. 3.32+

ALLIGATION MEDIAL.

ALLIGATION is used when the quantities and prices of several things are given, to find the mean price of the mixture composed of those materials; or the method of mixing two or more simples of different qualities, so that the composition may be of a mean or middle quality. There are two kinds, Alligation Iedial and Alligation Alternate.

RULE.

Divide the entire cost of the whole mixture by the sum of the simples; the quotient will be the price of the given mixture.

1. If 19 bushels of wheat at 75c. per bushel, 40 bushels of rye at 50c. per bushel, and 12 bushels of barley at 37.5c. per bushel, be mixed together, what is a bushel of the mixture worth? 19x75 40 X 50

12 x 37

71

=

14.25D.

20.00

4.50

71)38.75(54c. 6m. Ans.

2. If 4 ounces of silver, worth 62.5c. per ounce, be melted with 8 ounces at 50c. per ounce, what is an ounce of this mixture worth? Ans. 54c.

3. A goldsmith melted together 8 ounces of gold, 22 carats fine; 1 lb. 8 oz. of 21 carats fine; and 10 oz. of 18 carats fine; what is the quality of fineness of this composition?

Ans. 20

carats fine. 4. In buying tea, I pay 90c. per lb. for 12 lbs. and D1.20 per Ib. for 16 lbs.; what is the mixture worth per lb.? Ans. D1.07.1+. 5. A wine merchant mixed 12 gallons of wine, at 75c. per gallon, with 24 gallons at 90c. per gallon, and 16 gallons at D1.10; what is 1 gallon worth?

Ans. 92c. 6m. 6. On a certain day the mercury in the thermometer was observed to average the following heights: from 6 in the morning to 9, 64°; from 9 to 12, 740; from 12 to 3, 84°; and from 3 to 6, 70°; what was the mean temperature of the day? Ans. 730.

ALLIGATION ALTERNATE

Is the method of finding what quantity of each of the ingredients whose rates are given, will compose a mixture of a given rate; so that it is the reverse of Alligation Medial, and may be proved by it.

To find the proportion in which several simples of given prices must be mixed together, that the compound may be worth a given price.

RULE I.

1. Set down the prices of the simples under each other in the order of their values, beginning with the lowest

2. Link the least price with the greatest, and the next least with the next greatest, and so on, until the price of each simple which is less than the price of the mixtures is linked with one or more that is greater, and every one that is greater with one or more that is less.

3. Write the difference between the price of the mixture and that of each of the simples opposite that price with which the particular simple is linked; then the difference standing opposite any one price, or the sum of the differences when there is more than one, will express the quantity to be taken at that price.

1. If you would mix wines worth 16c., 18c., and 22c. per quart in such a way that the mixtures be worth 20c. quart, how much must be taken of each sort?

2. How much corn at 42c., 60c., 67c., and 78c., per bushel, must be mixed together that the compound may be worth 64c. per bushel?

Ans. 14 bushels at 42c., 3 bushels at 60c., 4 bushels at 67c., and 22 bushels at 78c.

3. What quantity of oats at 50c. per bushel, and 30c. per bushel, must be mixed together, that the compound may be worth 40c. per bushel? Ans. an equal quantity of each sort. 4. How much rye at 50c. per bushel, barley at 37c. per bushel, and oats at 25c. per bushel, will make a mixture worth 31c. per bushel?

Ans. 6 bushels at 50c., 6 at 371c., aud 25 at 25c. Note. If all the given prices are greater or less than the mean or given price, they must be linked to a cipher. A variety of answers may be obtained, according to the method of linking.

RULE II.

1. Find the proportional quantities of the simples as in rule 1st. 2. Then say as the number opposite the simples whose quantity is given, is to the given quantity, so is either proportional quantity to the part of its simple to be taken.

5. What quantity of coffee at 20c. and at 16c. must be mixed with 35 lbs. at 14c. per lb. to make a mixture worth 18c. per lb.! 2: 35:: 2 : 35lbs. at 16c. 2 : 35 :: 6 : 105lbs. at 20c.

18

{

14

16

20

6. A farmer would mix 14 bushels of wheat at D1.20 per bushel, with rye at 72c., barley at 48c., and oats at 36c.; how much must be taken of each sort to make the mixture worth 64c. per bushel ?

Ans. 14 bushels of wheat, 8 bushels of rye, 4 bushels of barley, and 28 bushels of oats.

7. A person wishes to mix 10 bushels of wheat at 70c. per bushel, with rye at 48c., corn at 36c., and barley at 30c. per bushel, so that a bushel of this mixture may be worth 38c.; what quantity of each must be taken?

Ans. 2 bushels of rye, 12 bushels of corn; 40 bushels of barley.

8. How much water must be mixed with 100 gallons of wine worth 90c. per gallon, to reduce it to 75c. per gallon.

Ans. 20 gallons.

When the quantity of the compound is given as well as the price.

RULE III.

1. Find the proportional quantities as in rule 1st.

2. Then say, as the sum of the proportional quantities is to the given quantity, so is each proportional quantity to the part.

to be taken of each.

9. A grocer has currants at 4c. 6c., 9c., and 11c., per pound, and he wishes to make a mixture of 240lb., worth 8c. per pound; what quantity of each kind must be taken ?

10. How much water, at 0 per gallon, must be mixed with wine at 80c. per gallon, so as to fill a vessel of 90 gallons, which may be offered at 50c. per gallon?

Ans. 56 gallons of wine, 336 gallons of water.

11. A goldsmith has several sorts of gold, namely: of 15, 17, 20, and 22 carats fine, and would melt together, of all these sorts, so much as may make a mass of 40oz., 18 carats fine; how much of each sort is required?

Ans. 16oz. of 15 carats fine, 8oz. of 17, 4oz. of 20, 12oz. of 22. 12. How much barley at 40c. per bushel, rye at 60c., and wheat at 80c., must be mixed together that the compound may be worth 62 c. per bush.? Ans. 17 bu. barley, 17 rye, 25 wheat