NOTE 1.-When there is a remainder after all the periods have been brought down, annex a period of three nothings, to form a new dividend; and then proceed with the further extraction of the root the figure of the root thus obtained is a decimal. The process may be carried to any degree of minuteness by annexing more nothings.

There are always as many figures in the root as there are periods in the given number; and those figures are decimals in the root, which are extracted from the decimals in the given number.

NOTE 2.-If at any time, on bringing down a new period to form a dividend, the partial divisor is found to be greater than the dividend, a nothing must be placed in the root, and two nothings annexed to the partial divisor: the next period is then brought down, and annexed to the dividend, and the extraction of a new figure of the root proceeded with as before.

NOTE 3.-The cube root of a fraction is found by extracting the roots of the numerator and denominator.

THE REASON of the Rule is shewn in 'Algebra,' p. 105.

10. A box, whose length, breadth, and depth are equal, contains 216 cubic feet; what are its dimensions? Ans. 6 feet each way.

11. A person has a box 5 feet long, 4 feet broad, and 64 feet deep, and wishes another box to contain the same number of cubic feet, whose length, breadth, and depth shall be equal; what are the required dimensions? . Ans. 5 feet each.

GLOBES are in proportion to each other as the cubes of their diameters: hence to find the diameter of a globe that shall contain 8 times more than another whose diameter is 2 feet, multiply the cube of 2 feet, the given diameter, by 8, and extract the cube root of the product.

CUBES are in proportion to each other as the cubes of a side of each. 12. If a globe, whose diameter is 4 inches, weigh 5 lb., what is the diameter of another globe which weighs 40 lb.? Ans. 8 inches. 13. If the side of a box, whose length, breadth, and depth are equal, is 4 feet long, what is the length of a side of another cubical box, that contains 27 times as many cubic feet? . Ans. 12 feet.

14. The cube root of the solid content of a body is the side of a cube of equal content or volume; find the side of a cube that will contain a bushel = 2218·192 cubic inches, Ans. 13.041665 inches. 15. A French kilolitre is 61023-79179 cubic inches; find the side of a cube that will contain a kilolitre,

Ans. 39-37008 inches

1 mètre. 16. The diameter of a sphere is the cube root of the quotient of its solidity divided by 5236: find the diameter of a sphere whose solidity is 4188800 cubic inches, Ans. 200 inches. 17. Find the diameter of a sphere whose solid content is 5254725974-3748 cubic miles, Ans. 2157 miles.

18. Kepler's third law is, that the squares of the periodic times of any two planets are to each other in the same proportion as the cubes of their mean distances from the sun. The earth's periodic time is 365-256384 days, and that of Mars 686-076119 days; if the earth's distance from the sun be 1, what is that of Mars? Ans. 1.522357. 19. From the facts stated in the last question, find the mean distance of Jupiter from the sun, his periodic time being 4332-5693 days nearly, Ans. 5 20114.

SERIES.

A SERIES is a succession of numbers, that are derived from one another according to a certain law.

The first and last terms of any series are called the extremes, and the others the means.

I. EQUIDIFFERENT SERIES, OR ARITHMETICAL PROGRESSION.

A SERIES, in which the difference between any two consecutive terms is the same, is called an equidifferent series. If the terms increase, it is called an increasing series; when they decrease, a decreasing series.

THE CONSTANT DIFFERENCE between any two successive terms is called the common difference. Thus, 1, 5, 9, 13, 17, &c., is an increasing equidifferent series, the common difference of which is 4; and 30, 27, 24, 21, &c., is a decreasing equidifferent series, the common difference being 3.

THE LAST TERM OF AN EQUIDIFFERENT SERIES is equal to the first term, increased, when the series is increasing, by the product of the common difference multiplied into a number which is one less than the number of terms; but diminished by it, when the series is decreasing.

224

Take any equidifferent series, as 3, 7, 11, 15, 19, 23, whose It is obvious that this series may be common difference is 4. arranged thus: 3, 3+1 x 4, 3 + 2 × 4, 3 + 3 × 4, 3 + 4 × 4, 3+5 x 4, where it is plain that any term as the 6th is equal to the first 3, increased by the common difference, 4, multiplied by 5, a number 1 less than 6, the number of terms. Similarly when the series is decreasing.

Exercises.

1. The first term of an increasing equidifferent series is 5, the Ans. 1094. common difference 11, and the number of terms 100. What is the last term? 2. The first term of a decreasing equidifferent series is 59, the common difference 2, and the number of terms 24. Required the last term,

Ans. 13.

THE SUM OF THE TERMS OF AN EQUIDIFFERENT SERIES is equal to the sum of the first and last terms multiplied by half the number of terms.

Take any equidifferent series, as 3, 7, 11, 15, 19, 23, consisting of any number of terms, as 6; then if s denote the sum of the (3 + 23) × 6 series, s =

2

For, s 3+ 7+11+15+19+23, then arranging the series backwards s=23+19+15+11+ 7+ 3 adding vertically, we have 2s26+26+26+26+26+26=26 × 6.

26 × 6 (3+23) × 6

or

2

2

Exercises.

3. The first term of an equidifferent series is 17, the last 85, What is the sum of the series? and the number of terms 17.

Ans. 867. 4. The first term of an equidifferent series is 100, the last 21, and the number of terms 85. What is the sum of the series?

Ans. 51425.

5. How many times does a common clock strike in a day?

Ans. 156.

6. A debt can be discharged in 52 weeks by paying 1s. the first week, 3s. the second, 5s. the third, and so on. Required the amount of the debt, and the last payment,

Ans. Amount, £133, 18s., and last payment £5, 3s. 7. From two towns, A and B set out to meet each other; A went 3 miles the first day, 5 the second, 7 the third, and so on; B went 4 miles the first day, 6 the second, 8 the third, and so on; they met in 8 days. What was the distance between the towns? Ans. 168 miles.

II. EQUIRATIONAL SERIES, OR GEOMETRICAL PROGRESSION.

A SERIES, in which the ratio of any two consecutive terms is the same, is called an equirational series. It is called an increasing or decreasing series, according as the terms increase or decrease.

THE RATIO of any term to the preceding one is called the common ratio.

Thus, 7, 14, 28, 56, is an increasing equirational series, whose common ratio is 2; and 1, 1, 4,,, is a decreasing one, whose common ratio is .

THE LAST TERM OF AN EQUIRATIONAL SERIES is equal to the first term multiplied by that power of the common ratio whose index is one less than the number of terms.

Take any equirational series, as 3, 6, 12, 24, 48, 96, whose common ratio is 2; it is obvious that this series may be written thus-3, 3 × 2, 3 × 22, 3 × 23, 3 × 21, 3 × 25; where it is plain that any term, as the 6th, is equal to the first 3, multiplied by 2, the common ratio, raised to that power whose index is 5-that is, one less than 6, the number of terms.

Exercises.

8. The first term of an equirational series is 4, the common ratio 3, and the number of terms 13. What is the last term?

Ans. 2125764.

9. The first term of an equirational series is 100, the common ratio, and the number of terms 17. What is the last term? Ans. 14

THE SUM OF THE TERMS OF AN EQUIRATIONAL SERIES is found by multiplying the first term by the difference between 1 and that power of the common ratio whose index is equal to the number of terms, and then dividing the product by the difference between 1 and the common ratio.

Take any equirational series, as 2, 12, 72, 432, 2592, whose (6-1) x 2

common ratio is 6; let s represent the sum, then s = 6-1 For, as 1 time s = 2 + 2 × 6 + 2 × 62 + 2 × 63 + 2 x 6'; multiply by com. ratio, times s=2×6+2×62+2× 63 + 2 × 6* + 2 × 65;

Exercises.

What is the sum of the

10. The first term of an equirational series is 5, the common ratio 3, and the number of terms 14. series?

Ans. 11957420.

11. The first term of an equirational series is 8, the common ratio 4, and the number of terms 12. Required the sum of the series, Ans. 44739240.

12. The first term of an equirational series is 1, the common ratio, and the number of terms 10. What is the sum of the

series?

Ans. 148828

1953125'

when the number of terms is indefinitely great, the fraction in the numerator of the sum becomes indefinitely small, and its value may be regarded as 0; hence, when the series is decreasing, and the number of terms infinite (as it is called), the sum is found by dividing the first term by the difference between 1 and the common ratio.

LOGARITHMS.

THE RULES by which the following exercises are to be worked, with an explanation of Logarithms, will be found at pages v. to xv. of the Introduction to 'Mathematical Tables' of this Course.

5. Find the number whose log. is 2.7169325,.

" 2.2656950.

"

1.8669651.

" 2.6756198.

9. Multiply 73-627, 01852, and 9.31654 together, 10. Find the continued product of '68715, 00354,

839 612, and 5183,

11. Divide 68.3167 by 983.7,

" ⚫0694487.

" 07760669.

" 1.115835.