4. Extract the square root of 10 to 8 places of decimals.

II. TO EXTRACT THE CUBE ROOT OF A GIVEN NUMBER.

Rule-1. Divide the number into periods of three figures, beginning with the units' place, by placing a dot over every third figure.

3.

2. Find the largest number, whose cube does not exceed the first period. Write this as the first figure in the cube root, and subtract its cube from the first period.

4. To the remainder annex the next period; call the number so formed the 1st dividend.

5. Treble the first figure of the root, and write the result some little distance to the left of the dividend; multiply it by the first figure, and write the product nearer the dividend, leaving room for two figures between them. Call this last number the 1st trial-divisor.

6. Divide the dividend, with the last two figures cut off, by the trialdivisor; and write the quotient as the 2nd figure in the root, and also to the right of three times the 1st figure. Call the number so formed, the 1st multiplicand.

7. Multiply the 1st multiplicand by the 2nd figure in the root, and

write the product under the 1st trial-divisor, keeping two figures to the right; and add the two together; call the sum the 1st divisor.

8. Multiply the 1st divisor by the 2nd figure of the root, and subtract the product from the 1st dividend.

9. To the remainder annex the next period, forming the 2nd dividend. 10. Treble the figures of the root, or add twice the last figure to last multiplicand, and write the product as in 5; add the 1st divisor to the line above it, taking in the square of the last figure of the root; thus forming the 2nd trial-divisor.

11. Proceed in the same manner, as directed in 6, 7, &c. and repeat the process till all the periods have been used, when, if there is no remainder, the cube root will have been formed; but if there be one, there is no cube root.

12. Whenever any quotient obtained as directed, gives a subtrahend greater than the dividend, a less number must be taken.

13. This rule applies to decimals as well as to whole numbers; only in decimals, the number of decimal places must be a multiple of 3, and the root will contain one third as many decimal places as the number.

14. The cube root of a fraction may be obtained by extracting the cube root of numerator and denominator; or by reducing the fraction to a decimal, and then extracting the cube root.

15. In finding the cube root of a number to a certain number of decimal places, when one more than half the number of figures required in the root are found, the remaining figures may be found by dividing by the trial-divisor the last dividend with the last two figures cut off.

3. Extract the cube root of 12 to 8 places of figures.

Def. 1. Numbers are said to form an Arithmetic Progression or series, or to be in Arithmetic Proportion, when the differences between any two consecutive terms of the series are the same. This difference is called the common difference.

Def. 2. The first and last terms of a series are called the extremes, the other terms the means.

1. TO FIND ANY REQUIRED TERM OF AN ARITHMETIC SERIES, OF WHICH THE FIRST TERM AND THE COMMON DIFFERENCE ARE KNOWN.

Rule. To the first term add the product of the common difference multiplied by a number less by one than the number of the term.

EXAMPLE.

Find the 9th term of the series, of which the 1st term is 13, and common difference is 2.

9th term 13 + 2 × 8 = 13 +16

= 29.

11. TO FIND THE SUM OF ANY NUMBER OF TERMS OF A GIVEN ARITHMETIC SERIES.

Rule. Find the last term, and multiply half the sum of the first and last terms by the number of terms.

EXAMPLE.

Find the sum of 20 terms of the series 1, 3, 5, &c.

III. TO FIND ANY NUMBER OF ARITHMETIC MEANS BETWEEN TWO

NUMBERS.

Rule. Divide the difference between the last and first term by a number greater by one than the required number of means. The quotient is the common difference, which being known, the terms of the series are readily found.

Def. 1. Numbers are said to form a Geometric Progression or Series, or to be in Geometric Proportion, when the ratios of any one term to the preceding are the same. This ratio is called the Common Ratio.

Def. 2. The limit of the sum of any number of terms of an infinite Geometric Series, is the number to which the sum continually approaches, as the number of terms is increased, but which the sum of any number of terms whatever can never equal or exceed. The common ratio here is always a proper fraction.

I. TO FIND ANY REQUIRED TERM OF A GIVEN GEOMETRIC PROGRESSION.

Rule. Raise the common ratio to a power one less than the number of the required term, and multiply by the first term.

EXAMPLE.

Find the 7th term of 2, 4, 8, &c.

Here 1st term 2: common ratio = 2,

... 7th term = 2 × 26 = 27

= 16 X 8 128.

II. TO FIND THE SUM OF ANY NUMBER OF TERMS OF A GEOMETRIC

SERIES.

Rule. Divide the difference between the first term, and the last term multiplied by the common ratio, by the difference between 1 and the common ratio.

EXAMPLES.

1. Find the sum of 8 terms of the series 1, 3, 9, &c.

2. Find the sum of 9 terms of the series 1, 1, 1, &c.

III. TO FIND THE LIMIT OF THE SUM OF AN INFINITE GEOMETRIC

SERIES.

Rule. Divide the first term by the difference between 1 and the common ratio.

EXAMPLE.

Find the limit of the sum of the series 1, 1, 1, &c.

IV. TO FIND ANY NUMBER OF GEOMETRIC MEANS BETWEEN TWO

NUMBERS.

Rule. Divide the last by the first term; extract a root of the quotient greater by one than the number of means; the root is the common ratio; which being found, the means are readily obtained.