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the length of each side, which is expressed by the former quotient figure, 2, (tens.) 3 times 2 (tens) are 6 (tens) = 60; or, what is the same in effect, and more convenient in practice, we may multiply the quotient figure, 2, (tens,) by 30, thus, 2 X 30= 60, as before; then, 60 X 16=960, con- tents of the three deficiencies n, n, n.
13824 ach side
given number; or it may be
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 4 X 464,) will be the solid contents of this corner, which in Fig. IV. is seen filled.
Now, the sum of these sev eral 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 proved by adding together
addition to the sides a, b, and c, Fig. 1.
From the foregoing example and illustration we derive the following
FOR EXTRACTING THE CUBE ROOT.
I. Separate the given number into periods of three figures each, by putting a point over the unit figure, and every third figure beyond the place of units.
II. Find the greatest cube in the left hand period, and pu its root in the quotient.
III. Subtract the cube thus found from the said period, and to the remainder bring down the next period, and call this the dividend.
IV. Multiply the square of the quotient by 300, calling it the divisor.
V. Seek how many times the divisor may be had in the dividend, and place the result in the root; then multiply the divisor by this quotient figure, and write the product under the dividend.
VI. Multiply the square of this quotient figure by the former figure or figures of the root, and this product by 30, and place the product under the last; under ail write the cube of this quotient figure, and call their amount the 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; and so on, till the whole 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 the square
Note 3. The pupil will perceive that the number which we call the divisor, when multiplied by the last quotient figure, does not produce so large a number as the real subtrahend; hence, the figure in the root must frequently be smaller than the quotient figure.
EXAMPLES FOR PRACTICE.
6. What is the cube root of 1860867?
1o X 300 = 300) 860 first Dividend.
22 X 1 X 30 = 120
728 first Subtrahend.
27 132867 second Subtrahend 000000
32 X 12 X 30 =
7. What is the cube root of 373248? 8. What is the cube root of 21024576? 9. What is the cube root of 84'604519? 10. What is the cube root of '000343 ? 11. What is the cube root of 2 ? 12. What is the cube root of 2?
Note. See 105, ex. 10, and ¶ 108, ex. 14.
13. What is the cube root of 128? 14. What is the cube root of ? 15. What is the cube root of 500? 16. What is the cube root of Th?
Ans. 1'25 +
SUPPLEMENT TO THE CUBE ROOT.
1. What is a cube ? 2. What is understood by the cube root? 3. What is it to extract the cube root? 4. Why is the square of the quotient multiplied by 300 for a divisor? 5. Why, in finding the subtrahend, do we multiply the square of the last quctient figure by 30 times the former figure of the root? 6. Why do we cube the quotient figure? 7. How do we prove the operation?
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. 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? 125 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? times? 64 times?
Ans. 2, 3, and 4 feet.
¶ 111. Hence we see, that the sides of cubes are as the cube roots of their solid contents, and, consequently, their contents are as the cubes of their sides. 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 will 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 ?
9. There are two globes; one of them is 1 foot in diame ter, and the other 40 feet in diameter; how many of the smaller globes would it take to make 1 of the larger?
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 less is to that of the larger as 2 to 9; what is the ra tio of their solidities? Ans. ; or, as 8 to 729.
Note. The roots of most powers may be found by the square and cube root only: thus, the biquadrate, or 4th root, is the square root of the square root; the 6th root is the cube root of the square root; the 8th root is the square root of the 4th root; the 9th root is the cube root of the cube root, &c. Those roots, viz. the 5th, 7th, 11th, &c., which are not resolvable by the square and cube roots, seldom oc our, and, when they do, the work is most easily performed by logarithms; for, if the logarithm of any number be divided by the index of the root, the quotient will be the logarithm of the root itself.
112. Any rank or series of numbers, more than two, increasing or decreasing by a constant difference, is called an Arithmetical Series, or Progression.
When the numbers are formed by a continual addition of the common difference, they form an ascending series; but when they are formed by a continual subtraction of the common difference, they form a descending series.
Thus, ( 3, 5, 7, 9, 11, 13, 15, &c. is an ascending series. 15, 13, 11, 9, 7, 5, 3, &c. is a descending series.
The numbers which form the series are called the terms of the series. The first and last terms are the extremes, and the other terms are called the means.
There are five things in arithmetical progression, any three of which being given, the other two may be found :
1st. The first term.
2d. The last term.
3d. The number of terms.
4th. The common difference. 5th. The sum of all the terms.
1. A man bought 100 yards of cloth, giving 4 cents for the first yard, 7 cents for the second, 10 cents for the third, and so on, with a common difference of 3 cents; what was the cost of the last yard?
As the common difference, 3, is added to every yard except the last, it is plain the last yard must be 99 × 3, 297 cents, more than the first yard. Ans. 301 cents.