300. Let us now see how the cube of any number, as 16, is formed. Sixteen is composed of 1 ten and 6 units, and may be written 10+6. To find the cube of 16, or of 10+6, we must multiply the number by itself twice

To do this we place the number thus

16=10+ 6 10+ 6 60+ 36

1. By examining the parts of this number it is seen that the first part 1000 is the cube of the tens; that is,

10 x 10 x 10=1000.

2. The second part 1800 is three times the square of the tens multiplied by the units; that is,

3 × (10)3×6=3× 100 × 6=1800.

3. The third part 1080 is three times the square of the units multiplied by the tens; that is,

3 x 62 x 10=3 × 36 × 10=1080.

4. The fourth part is the cube of the units; that is, 636 × 6 × 6=216.

1. What is the cube root of the number 4096 ?

ANALYSIS.-Since the number

contains more than three figures, we know that the root will contain at least units and tens.

Separating the three righthand figures from the 4, we know that the cube of the tens

OPERATION.

4 096(16

1

12x3=3)30 (9-8-7-6 163-4 096

will be found in the 4; and 1 is the greatest cube in 4.

299. What is the cube root of a number? How many perfect cubes are there between 1 and 1000?

300. Of how many parts is the cube of a number composed? What are they?

Hence, we place the root 1 on the right, and this is the tens of the required root. We then cube 1 and subtract the result from 4, and to the remainder we bring down the first figure 0 of the next period.

We have seen that the second part of the cube of 16, viz. 1800, is three times the square of the tens multiplied by the units: and hence, it can have no significant figure of a less denomination than hundreds. It must, therefore, make up a part of the 30 hundreds above. But this 30 hundreds also contains all the hundreds which come from the 3d and 4th parts of the cube of 16. If it were not so, the 30 hundreds, divided by three times the square of the tens, would give the unit figure exactly

Forming a divisor of three times the square of the tens, we find the quotient to be ten; but this we know to be too large. Placing 9 in the root and cubing 19, we find the result to be 6859. Then trying 8 we find the cube of 18 still too large; but when we take 6 we find the exact number. Hence the cube root of 4096 is 16.

301. Hence, to find the cube root of a number,

RULE.-I. Separate the given number into periods of three figures each, by placing a dot over the place of units, a second over the place of thousands, and so on over each third figure to the left; the left hand period will often contain less than three places of figures.

II. Note the greatest perfect cube in the first period, and set its root on the right, after the manner of a quotient in division. Subtract the cube of this number from the first period, and to the remainder bring down the first figure of the next period for a dividend.

III. Take three times the square of the root just found for a trial divisor, and see how often it is contained in the dividend, and place the quotient for a second figure of the root. Then cube the figures of the root thus found, and if their cube be greater than the first two periods of the given number, diminish the last figure, but if it be less, subtract it from the first two periods, and to the remainder bring down the first figure of the next period for a new dividend.

IV. Take three times the square of the whole root for a second trial divisor, and find a third figure of the root. Cube the whole root thus found and subtract the result from the first three periods of the given number when it is less than that number, but if it is greater, diminish the figure of the root; proceed in a similar way for all the periods.

302. To extract the cube root of a decimal fraction.

Annex ciphers to the decimal, if necessary, so that it shall consist of 3, 6, 9, &c., places. Then put the first point over the place of thousandths, the second over the place of millionths, and so on over every third place to the right; after which extract the root as in whole numbers.

NOTES.-1. There will be as many decimal places in the root as there are periods in the given number.

2. The same rule applies when the given number is composed of a whole number and a decimal.

3. If in extracting the root of a number there is a remainder after all the periods have been brought down, periods of ciphers may be annexed by considering them as decimals.

301. What is the rule for extracting the cube root? 302. How do you extract the cube root of a decimal fraction? How many decimal places will there be in the root? Will the same rule apply when there is a whole number and a decimal? If in extracting the root of any number you find a decimal, how do you proceed?

303. To extract the cube root of a common fraction.

I. Reduce compound fractions to simple ones, mixed numbers to improper fractions, and then reduce the fraction to its lowest terms.

II. Extract the cube root of the numerator and denominator separately, if they have exact roots; but if either of them has not an exact root, reduce the fraction to a decimal and extract the root as in the last case.

1. What must be the length, depth, and breadth of a box, when these dimensions are all equal and the box contains 4913 cubic feet?

2. The solidity of a cubical block is 21952 cubic yards: what is the length of each side? What is the area of the surface?

3. A cellar is 25 feet long 20 feet wide, and 81 feet deep: what will be the dimensions of another cellar of equal capacity in the form of a cube?

4. What will be the length of one side of a cubical that shall contain 2500 bushels of grain?

granary

5. How many small cubes of 2 inches on a side can be sawed out of a cube 2 feet on a side, if nothing is lost in sawing?

6. What will be the side of a cube that shall be equal to the contents of a stick of timber containing 1728 cubic feet?

7. A stick of timber is 54 feet long and 2 feet square: what would be its dimensions if it had the form of a cube?

NOTES.-1. Bodies are said to be similar when their like parts are proportional.

2. It is found that the contents of similar bodies are to each other as the cubes of their like dimensions.

303. How do you extract the cube root of a vulgar fraction?

3, All bodies named in the examples are supposed to be simi. lar.

8. If a sphere of 4 feet in diameter contains 33.5104 cubic feet, what will be the contents of a sphere 8 feet in diameter ? 43 : 83 : : 33.5104 : Ans.

9. If the contents of a sphere 14 inches in diameter is 1436.7584 cubic inches, what will be the diameter of a sphere which contains 11494.0672 cubic inches?

10. If a ball weighing 32 pounds is 6 inches in diameter, what will be the diameter of a ball weighing 2048 pounds?

11. If a haystack, 24 feet in height, contains 8 tons of hay, what will be the height of a similar stack that shall contain but 1 ton?

ARITHMETICAL PROGRESSION.

304. An Arithmetical Progression is a series of numbers in which each is derived from the preceding one by the addition or subtraction of the same number.

The number added or subtracted is called the common difference.

305. If the common difference is added, the series is called an increasing series.

Thus, if we begin with 2, and add the common difference, 3, we have

2, 5, 8, 11, 14, 17, 20, 23, &c.,

which is an increasing series.

If we begin with 23, and subtract the common difference, 3, we have

23, 20, 17, 14, 11, 8, 5, &c.,

which is a decreasing series.

304. What is an arithmetical progression? What is the number added or subtracted called?

305. When the common difference is added, what is the series called? What is it called when the common difference is subtracted? What are the several numebrs called? What are the first and last called? What are the intermediate ones called?