figure, or tens of the root, which must be extracted from the left hand period, 13, (thousands.) The greatest cube in 13 (thousands) we find by inspection, or by the table of powers, to be 8, (thousands,) the root of which is 2, (tens ;) therefore, we place 2 (tens) in the root. As the root is one side of a cube, let us form a cube, (Fig. I.,) each side of which shall be regarded 20 feet, expressed by the root now obtained. The contents of this cube are 20 X 20 X 20=8000 solid feet, which are now disposed of, and which, consequently, are to be deducted from the whole number of feet, 13824. 8000 taken from 13824 leave 5824 feet. This deduction is most readily performed by subtracting the cubic number, 8, or the cube of 2, (the figure of the root already found,) from the period 13, (thousands,) and bringing down the next period by the side of the remainder, making 5824, as before.

2d. The cubic pile A D is now to be enlarged by the addition of 5824 solid feet, and, in order to preserve the cubic form of the pile, the addition must be made on one half of its sides, that is, on 3 sides, a, b, and Now as each side is 20 feet square, its square contents are 400 square feet, and the square contents of the 3 sides are 1200 square feet. Hence, an addition of 1 foot thick would require 1200 solid feet, and

C.

dividing 5824 solid feet by 1200 solid feet, the contents of the addition 1 foot thick, and we get the thickness of the addition. It will be seen that the quotient figure must not always be as large as it can be. There might be enough, for

Divisor, 1200) 5824 Dividend. instance, to make the three addi

tions now under consideration 5 feet thick, when there would not then be enough remaining to complete the additions.

The divisor, 1200, is contained in the dividend 4 times; consequently, 4 feet is the thickness of the addition made to each of the three sides, a, b, c, and 4 X 1209

4800, is the solid feet contained in these additions; but there are still 1024 feet left, and if we look at Fig. II., we shall perceive that this addition to the 3 sides does not complete the cube; for tnere are deficiencies in the 3 corners, n, n, n. Now the length of each of these deficiencies is the same as the length of each side, that is, 2 (tens) 20, and their width and thickness are each equal to the last quotient figure, (4;) their con tents, therefore, or the number of feet required to fill these deficien cies, will be found by multiplying the square of the last quotient figure, (42)=16, by 20; 16 X 20 =

C

20

24 feet.

Fret. 8000 4800: 960

64:

=

=

=

320 solid feet, required for one de ficiency, and multiplying 320 by 3, 320 X 3960 solid feet, required for the 3 deficiencies, n, n, n.

Looking at Fig. III., we perceive there is still a deficiency in the corner where the last blocks meet. This deficiency is a cube, each side of which is equal to the last quotient figure, 4. The cube of 4, therefore, (4 X 4X4 64,) will be the solid contents of this corner, which in Fig. IV. is seen filled.

=

Now, the sum of these several additions, viz., 4800+960 +64 5824, will make the subtrahend, which, subtracted from the dividend, leaves no remainder, and the work is done.

Fig. IV. shows the pile which 13824 solid blocks of one foot each would make, when laid together, and the root, 24, shows the length of one side of the pile. The correctness of the work may be ascertained by cubing the side now found, 243, thus, 24 X 24 X 24: 13824, the given number; or it may be proved by adding together the contents of all the several parts, thus,

addition to the sides, a, b, and c, Fig. I.
= addition to fill the deficiencies n, n, n, Fig. II.
addition to fill the corner, e, e, e, Fig. IV.

13824: contents of the whole pile, Fig. IV., 24 feet on each side.

T212. From the foregoing example and illustration we lerive the following

Questions.-T211. How is the length of one side of a cube found, when the contents are knowr? Why, Ex. 3, is the number pointed off as it Is? How many figures in the cube of any number? Illustrate by cubing some numbers. What is 2, the first figure of the root? Of what is it the root? For what is the subtraction? What is to be done with the remainder? On how many sides is it to be added, and why! What is the divisor, 1200? What is the object in dividing? The quotient expresses what? Why should it not be made as large as it can be? What additions are next made, and what are the contents of each? How are the contents found! What deficiency yet remains, and how large? Of what parts of the last figure does the subtrahend consist? Describe Fig. I.; Fig. II.; Fig. III.;- Fig. IV. How is the work proved?

RULE

FOR EXTRACTING THE CUBE BOOT.

I. Place a point over the unit figure, and over every third figure at the left of the place of units, thereby separating the given number into as many periods as there will be figures in

the root.

II. Find the greatest complete cube number in the left hand period, and place its cube root in the quotient.

III. Subtract the cube thus found froin the period taken, and bring down to the remainder the next period for a divi dend.

IV. Calling the quotient, or root figure now obtained, so many tens, multiply its square by 3, and use the product for a divisor.

V. Seek how many times the divisor is contained in the dividend, and diminishing the quotient, if necessary, so that the whole subtrahend, when found, may not be greater than the dividend, place the result in the root; then multiply the divisor by this root figure, and write the product under the dividend.

VI. Multiply the square of this root figure by the former figure or figures of the root, regarded as so many tens, and the resulting product by 3, add the product thus obtained, together with the cube of the last quotient, to the former product for a subtrahend.

VII. Subtract the subtrahend from the dividend, and to the remainder bring down the next period for a new dividend, with which proceed as before, till the work is finished.

NOTE 1. If it happens that the divisor is not contained in the dividend a cipher must be put in the root, and the next period brought down for a dividend.

NOTE 2. The same rule must be observed for continuing the operation, and pointing off for decimals, as in extracting the square root.

EXAMPLES FOR PRACTICE.

4. What is the cube root of 1860867?

Questions -¶ 212, What is the general rule? -- note 3?

NOTE 3. divided by the cube root of the denominator. (¶ 209.)

The cube root of a fraction is the cube root of the numerat

11. What is the cube root of?

12. What is the cube root of AS?

13. What is the cube root of 50?

14. What is the cube root of TS?

Ans. .

Ans. Tz.

Ans. '125+.
Ans.

PRACTICAL EXERCISES IN EXTRACTING THE CUBE

ROOT.

T213. 1. What is the side of a cubical mound, equal to one 288 feet long, 216 feet broad, and 48 feet high? Ans 144 feet.

2. There is a cubic box, one side of which is 2 feet; how many solid feet does it contain?

Ans. 8 feet.

3. How many cubic feet in one 8 times as large? and what would be the length of one side?

Ans, 64 solid feet, and one side is 4 feet.

125 times as much?

4. There is a cubical box, one side of which is 5 feet, what would be the side of one containing 27 times as much? 64 times as much? Ans. 15, 20, and 25 feet. 5. There is a cubical box, measuring 1 foot on each side; what is the side of a box 8 times as large? 27 times?

64 times? Ans. 2, 3, and 4 feet.

NOTE.It appears from the above examples that the sides of cubes are as the cube roots of their solid contents, and their solid contents as the cubes of their sides. It is also true that if a globe or ball have a certain contents, the contents of one whose diameter is double are 8 times as great, or having a treble diameter are 27 times as great, and so on; that is, the contents are proportional to the cubes of their diameters. The same proportion is true of the similar sides, or of the diameters of all solid figures of similar forms.

6. If a ball, weighing 4 pounds, be 3 inches in diameter, what wil. be the diameter of a ball of the same metal, weighing 32 pounds? 4:32:: 33: 63. Ans. 6 inches.

7. If a ball 6 inches in diameter weigh 32 pounds, what will be the weight of a ball 3 inches in diameter? Ans. 4 lbs. 8. If a globe of silver, 1 inch in diameter, be worth $6, what is the value of a globe 1 foot in diameter ? Ans. $10368.

9. There are two globes; one of them is 1 foot in diameter, and the other 40 feet in diameter; how many of the smaller globes would it take to make 1 of the larger?

Ans. 64000.

10. If the diameter of the sun is 112 times as much as the diameter of the earth, how many globes like the earth would it take to make one as large as the sun? Ans. 1404928. ́

11. If the planet Saturn is 1000 times as large as the earth, and the earth is 7900 miles in diameter, what is the diameter of Saturn? Ans. 79000 miles. 12. There are two planets of equal density; the diameter of the ess is to that of the larger as 2 to 9; what is the ratio of their solidities? Ans. 7; or, as 8 to 729.

T214. Review of Involution and Evolution.

Questions. What is involution? What are powers? How are the different powers represented? How is a number involved? What is evolution? What is a root? How do you find the square root, or the cube root of a number? What is a rational, and what a surd number? How is the square root indicated? — the cube root? Give briefly the solution of the example in the extraction of the square root; - rule. How are decimals pointed off? How is the operation continued, when rere is a remainder? Why cannot the precise root be ascertained? How is the square root of a vulgar fraction found? What is said of the relation between the sides and contents of squares? the diameters and

Questions. T213.

-

What proportion exists between the sides of cubes, and their solid contents? Illustrate. What betweer the diameters of globes and their contents? If you increase the diameter of a ball 5 times how much are its contents increased?