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 third and fourth 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 8 still too large; but when we take 6, we find the exact number. Hence, the cube root of 4096 is 16. 372. 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.

373. To extract the cube root of a decimal fraction. Rule.-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 rout, 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.

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

Rule.-I. Reduce compound fractions to simple ones, mixed numbers to improper fractions, and then rèduce the fraction to its lowest terms:

II. Extract the cube root of the numerator and denomi nator 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 83 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 granary that shall contain 2500 bushels of grain?

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.

3. All bodies named in the examples below, are supposed to be similar.

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.

375. 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.

376. 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.

The several numbers are called, the terms of the progression or series: the first and last are called, the extremes, and the intermediate terms are called, means.

377. In every arithmetical progression, there are five parts: 1st, The first term;

2d,

The last term;

3d, The common difference;

4th, The number of terms;

5th, The sum of all the terms.

If any three of these parts are known or given, the remaining ones can be determined.

CASE I.

378. Knowing the first term, the common difference, and the number of terms, to find the last term.

1. The first term is 3, the common difference 2, and the number of terms 19: what is the last term?

ANALYSIS. By considering the manner in which the increasing progression is formed, we see that the 2d term is obtained by adding the common difference to the 1st term; the 3d, by adding the common difference to the 2d; the 4th, by adding the common difference to the 3d, and so on; the number of additions being 1 less than the number of terms found.

OPERATION.

18, No. less 1. 2, com. dif.

36

3, 1st term.

39, last term.

But instead of making the additions, we may multiply the common difference by the number of additions, that is, by 1 less than the number of terms, and add the first term to the product: Hence,

Rule.-Multiply the common difference by 1 less than the number of terms; if the progression is increasing, add the product to the first term, and the sum will be the last term; if it is decreasing, subtract the product from the first term, and the difference will be the last term.