in length is each side of the field? and how many rods apart are the opposite corners? Answers, 120 rods, and 1697+ rods. 23. There is a square field containing 10 acres; what distance is the centre from each corner? Ans. 28 28+ rods.

Extraction of the Cube Root.

T211. 1. How many feet in length is each side of a cubic block, containing 125 solid feet?

SOLUTION. As the solid contents of a cubical body are found, when one side is known, by involving the side to the third power, or cube, ( 206,) so when the solid contents are known, we find the length of one side by extracting the cube root, a number, which, taken as a factor 3 times, will produce the given number, (207.) The cube root of 125 we find by inspection, or by the table, ¶ 206, to be 5. Ans. 5 feet.

216 solid feet?

2. What is the side of a cubic block, containing 64 solid feet? 27 solid feet? 512 solid feet? Answers, 4 ft., 3 ft., 6 ft., and 8 ft

3. Supposing a man has 13824 feet of timber, in separate blocks of 1 cubic foot each; he wishes to pile them up in a cubic pile; what will be the length of each side of such a pile?

20 D

SOLUTION.It is evident that, as in the former examples, we must find the length of one side of a cubical pile which 13824 such blocks will make by extracting the cube root of 13824. But this number is so large, that we cannot so easily find the root as in the former examples; - we will endeavor, however, to do it by a sort of trial; and,

1st. We will try to ascertain the number of figures, of which the root 20 will consist. This we may do by pointing the number off into periods of three figures each. For the cube of any figure will contain 3 times as many, or 1 or 2 less than 3 times as many figures as the number itself. The cube of contains 1 figure; the cube of 5 con ains 2 figures; the cube of 9 contains 3 figures; the cube of 10 contains 4 figures, and so on.

Pointing off, we see that the root will consist of two figures, a ten and a unit. Let us, then, scek for the first

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 forin 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 120C 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 there 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 contents, therefore, or the number of feet required to fill these deficiencies, will be found by multiplying the square of the last quotient figure, (42,) 16, by 20; 16 X 20=

FIG. III.

20

+ 4

20

20

320 solid feet, required for one deficiency, and multiplying 3 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 2000+701440 or additions, viz., 4800 +960 + 64

4800

=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.

960 addition to fill the deficiencies n, n, n, Fig. II.

64

=

=

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 derive the following

Questions.-T211. How is the length of one side of a cube found, when the contents are known? 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? Wny 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. Ill.; - Fig. IV. How is the work proved?

RULE

FOR EXTRACTING THE CUBE ROOT.

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 from the period taken, and bring down to the remainder the next period for a dividend.

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.

1202 × 3 = 43200) 132867 second Dividend.

?

Ans. 72.

Ans. 276.

Ans.. 4'39.

Ans. '07. Ans. 1'25+. Ans..

NOTE 3. The cube root of a fraction is the cube root of the numerator divided by the cube root of the denominator. ( 209.)

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 inany 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.