that the length of the solid required to fill each of these deficiencies, must be the same as the length of the cube before any addition was made; therefore 2, the first figure of the root, must represent the length of each of these deficiencies, because it represents the length of each side of the cube before any addition was made; but this 2 expresses 20, for it stands in the place of tens; then 20 must be the

length of each of these deficiencies, and the width and thickness of each, must be equal to the last quotient figure, 4; therefore the solid feet necessary to fill the deficiencies, n, n, n, may be found by multiplying 16, the square of the last quotient figure, 4, by the length of all the deficiencies. And it has been shown that the length of each deficiency, is 20 feet, then the length of the three deficiencies must be 60 feet; therefore 16, the product of the breadth and thickness of each deficiency, multiplied by 60, the length of the whole, must give the solid contents required to fill the three deficiencies, n, n. n; thus, 16X60-960 solid feet, the solidity required to fill the three deficiencies n,n, n. Or the solidity required to fill the three deficiencies, may be obtained thus, 4x4-16, the product of the breadth and thickness of each deficiency, or the square of the last quotient figure, then 16x2= 32 which multiplied by 30, thus, 32X30-960 solid feet,

the same result as before. But perhaps it would have rendered the work more plain, if we had multiplied 16, the square of the last quotient figure, or the product of the breadth and thickness of each deficiency, by 20, the length of each deficiency, which would have given the solidity required to fill one deficiency, and then have multiplied that solidity by 3, the number of defiiencies in the corners, n, n, n; thus, 16x20-320, the solidity of one deficiency, then, 320×3=960 solid feet, the solidity in the three deficiencies, n, n, n, the same as before. And now having supplied the deficiencies n, n, n, we find on inspecting FIG. III, that there still remains a deficiency in the corner where the three last pieces meet. Now it is plain, that this last deficiency must be a small cube, because the solid required to fill this

deficiency, must be eqcal in length, breadth and thickness, to the last

24 feet.

quotient figure, 4. Then by cubing this last quotient figure, we will have the solid feet required to fill the vacancy, thus, 4X4X4-64 solid feet, the contents of the corner piece, which will be seen in its place in FIG IV.

The number of solid feet in these several additions must be added to compose the sub.. trahend, thus, 4800-+-960-64

5824, the number of solid feet in the subtrahend, that is, in all the additions, which, subtracted from the dividend, leaves no remainder, and the work is now finished.

Then FIG. IV represents the cubick pile, each side of which is 24 feet, and contains 13824 solid feet, which may be proved by cubing one of the sides of the cube, that is, the root,thus 24x24X24=13824 solid feet, the given number, that is, multiplying the length, breadth and thickness togethto find the solidity; or it may be proved by adding together the solid feet contained in the 24 feet. several parts, thus: contents of the first cube, FIG. I. 4800 first addition to the sides a, b, c, FIG. II. 960 2d addition, to fill deficiencies n, m, n, FIG. III. 64 3d addition to fill the corner e, e, e, FIG. IV.

Feet. 8000

13824-contents of the whole pile, FIG. IV, and 24 feet, the cube root or length of each side.

2. What is the cube root of 34645976 ?

34645976)326 Answer.

27

32X300 2700)7645 first dividend.

5400 Carried over.

DEM.-It has been stated that the root, at all stages of the work, represents the side of a cube, formed from what has been taken from the given number. Now 3, the first figure of the root, having a local value, represents the side of a cube, each side of which must be 300, for if we raise 300 to the third power, the product will exactly agree with the number subtracted from the given number. When we have obtained two figures, 32, in the root, they likewise having a local value, represent the side of a cube formed from what has been subtracted from the dividend or given number; and each side of the cube, at this stage of the work, must be 320. And when we have obtained three figures in the root, as 326, our root, 326, must represent the side of a cube, the solid contents of which are equal to the given number. Then it is evident, that after every subtraction our root represents the side of a new cube, so that for every figure in the root, after the first, there must be additions made similar to that which was made in obtaining the second figure of the root under example 1st; the reason of which is evident from the demonstration there given.

Ans. 9.

Ans. 23. Ans. 169.

Ans. 209.

Ans. 250.

Ans. 691.

Ans. 999.

3. What is the cube root of 729 ? 4. What is the cube root of 12167? 5. What is the cube root of 4826809? 6. What is the cube root of 9129329 ? 7. What is the cube root of 15625000? 8. What is the cube root of 329939371? 9. What is the cube root of 997002999 ? 10. What is the cube root of 729000000? NOTE. To extract the cube root of a Vulgar Fraction, first extract the cube root of the numerator for a new numerator, and then extract the cube root of the denominater for a new denominator. Or reduce the fraction to a decimal and extract the cube root of the decimal.

Ans. 900.

Ans.

Ans.

Ans.

When there are decimals in the given number, point off the whole numbers the same as if there were no decimals belonging to the given

number, then place a period over tenths in decimals, and one over every third figure beyond it, counting to the right, and if the right hand period should not be complete, annex ciphers to complete the period; then extract the root the same as in whole numbers. The periods over whole numbers, show that the root must have so many figures in whole numbers; the rest will be decimals.

14. What is the cube root of 41421,736 ? 15. What is the cube root of 85766,121? 16. What is the cube root of 117,649 ?

17. What is the cube root of 84,604519?

Ans. 34,6.
Ans. 44,1.
Ans. 4,9.

Ans. 4,39.

NOTE.-If there be a remainder after all the periods are brought down, the operation may be continued, at pleasure, by annexing periods of ciphers.

18. What is the cube root of 2?

19. What is the cube root of 1?

20. What is the cube root of 3?

Ans. 1,2599.

Ans. 1.

Ans. 1,442.

Application and use of the Cube Root. CASE I.-To find two mean proportionals between any two given numbers.

RULE.-Divide the greater extreme by the less, and the cube root of the quotient, multiplied by the less extreme, gives the less mean; multiply the said cube root by the less mean, and the product will be the greater mean proportional.

EXAMPLES.

1. What are the two mean proportionals, between 4 and 256?

Ans. 16 the less, 64 the greater. NOTE.-Sixteen aud 64 are the two mean proportionals between 4 and 256; because 4: 16:: 64: 256, and the product of the extremes, is equal to the product of the means. It will be seen that the extremes are the given numbers, and the means, the two numbers in the answer to the question.

2. What are the two mean proportionals between 8 and 4096? Ans. 64 less mean, 512 greater mean. CASE II. To find the side of a cube that shall be equal in solidity to any given solid, as a globe, cylinder, prism, cone, &c. RULE.-Extract the cube root of the solid contents of the given body, and the said root will be the side of the cube required.

EXAMPLES.

1. The statute bushel contains 2150,4252 cubick or solid inches; what must be the side of a cubick box, that shall contain the same quantity? Ans. 12,907 inches.

2. There is a certain cistern, 24 feet long, 18 feet wide, and 4 feet high; required the side of a cistern of a cubick form, that shall hold the same quantity.

Ans. 12 feet.

NOTE. The solid contents of similar Figures are in proportion to each other, as the cubes of their similar sides or diameters.

3. If a ball, 2 inches in diameter, weigh 3lb.; what will a ball of similar metal weigh, whose diameter is 4 inches?

2X2X2=8; 4X4X4 64. As 8: 3lb.:: 64: 241b. Ans. 24lb.

4. If a globe of gold 1 inch in diameter be worth $100; what is the value of another globe 4 inches in diameter ? Ans. $6400.

CASE III-The side of a cube being given, to find the side of another cube which shall be double, triple, &c. in quantity to the given cube: Or one fonrth, one fifth, &c. of the given cube.

RULE.-Cube the given side, and multiply it by the given proportion, and the cube root of the product will be the side sought. Or if required to be less; divide the cube of the given side, by the given proportion, and extract the cube root of the quotient; the root will be the side of the cube sought.

EXAMPLES.

1. There is a cubical vessel whose side is 3 feet; required the side of another cubical vessel which shall contain 8 times as much. Ans. 6 feet. 2. There is a cubical vessel whose side is 6 feet; required the side of one, that shall contain only one eighth as much. Ans. 3 feet.

Extraction of the Biquadrate Root,

Is finding a number, which being multiplied by its cube, or involved four times into itself, will be equal to the given number.

RULE. You may first extract the square root of the given numnumber, and then the square root of that root, and the last root will be the required biquadrate root.

EXAMPLES.

1. What is the biquadrate root of 33362176 ?
2. What is the biquadrate root of 5719140625?

Ans. 76.

Ans. 275.

A general Rule for extracting the Roots of all powers.

1. Prepare the given number for extracting, by pointing off from the unit's place, as the required root directs, that is, a period over every second figure, for the square root, and one over every third, for the cube root; one over every fourth, for the biquadrate, and one over every fifth for the fifth root, and so on.

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 an imperfect dividend.

3. Involve the root to the next inferiour 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 sub