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the numerator for a numerator, and of the denominator for a denominator. If it be a surd, extract the root of its equivalent decimal.

2. A mixed number may be reduced to an improper fraction, or a decimal, and the root thereof extracted. 1. What is the cube root of 648?

Ans. 2. What is the cube root of 250?

Ans. 3. What is the cube root of 1520 ?

Ans. ģ 4. What is the cube root of 1217? 5. What is the cube root of 313?

Ans. 37.

3000

686

5130

Ans. 21

SURDS.

6. What is the cube root of 7?

Ans. 1.93+ 7. What is the cube root of 92?

Ans. 2.092+ APPLICATION. 1. If a block of marble be 47 inches long, 47 inches broad, and 47 inches deep, how many cubical inches does it contain ?

Ans. 103323. 2. There is a cellar dug 12 feet long, 12 feet deep, and 12 feet broad; how many solid feet of earth were taken out of it?

Ans. 1728. 3. How many cubes of 3 inches are contained in a cubical foot?

Ans. 64. 4. A certain stone of a cubical form contains 474552 solid inches; what is the superficial content of one of its sides?

Ans. 6084 inches.

A GENERAL RULE FOR EXTRACTING

THE ROOTS OF ALL POWERS

1. Point the given number into periods agreeably to the required root.

2. Find the first figure of the root by the table of powers, or by trial ; subtract its power from the left hand period, and to the remainder bring down the first figure in the next period for a dividend.

3. Involve the root to the next inferior power to that which is given, and multiply it by the number denoting the given power, for a divisor; by which find a second Sgure of the root.

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4. Involve the whole ascertained roợt to the given powe er, and subtract it from the first and second periods. Bring down the first figure of the next period to the remainder, for a new dividend; to which, find a new divisor, as before; and so proceed.

Note.—The roots of the 4th, 6th, 8th, 9th and 12th powers, may be obtained more readily thus :

For the 4th root take the square root of the square root.
For the 6th, take the square root of the cube l'oot.
For the 8th, take the square root of the 4th root.
For the 9th, take the cube root of the cube root.
For the 12th take the cube root of the 4th root.

EXAMPLES.
1. What is the 5th root of 916132832 ?

916132832(62 Ans.
7776

6 X 6 X 6 X 6 X 6

6 X 6 X 6 X 6 X 5 6480 13853

7776
6480 divisor.

62 x 62 x 62 x 62 x 62=916132832

916132832
916132832

2. What is the fourth root of 140%83207936? Ans. 612 3. What is the sixth root of 782757739696 ? Ans. 96. 4. What is the seventh root of 194754273881 ? Ans. 41. 5. What is the ninth root of 1352605460594688 ?

Ans. 48.

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Any rank or series of numbers, increasing or decreasing by a common difference, is said to be in arithmetical progression; as 2, 4, 6, 8, 10, and 6, 5, 4, 3, 2, 1. The numbers which form the series are called the

The first and last terms are called the extremes. Note.In

any

series of numbers in arithmetical progression, the sum of the two extremes is equal to the sum of any two terms equally distant from them; as in the latter of the above series 6 +1 4+ 3, and 5 + 2.

terms.

When the number of terms is odd, the double of the middle term is equal to the sum of the two exremes, por any two terms equally distant from the middle term; as in the former of the foregoing series 6 x 2 = 2 + 10, and = 4 + 8.

CASE 1. The first term, common difference, and number of terms given to find the last term, and sum of all the terms.

RULE. 1. Multiply the number of terms, less 1, by the common difference, and to the product add the first term, the sum is the last term.

2. Multiply the sum of the two extremes by the number of terms, and half the product will be the sum of all the terms.

EXAMPLES.

1. The first term of a certain series 'm articulas progression is 2, the common difference is 2, and the number of terms 15; what is the last term, and the sum of all the term's? 15 number of terms. 302

two extremes. 1

23

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240 Sum of all the terms. 2. Bought 15 yards of linen, at 2 cents for the first yard, 4 for the second, 6 for the third, &c. increasing 2 cents every yard; what was the cost of the last yard, and what was the cost of the whole ?

Ans. The last yard cost 30 cts. the whole cost $2.40. 3. Sold 20 yards of silk, at 3d. for the first yard, 6d. for the second, 9d. for the third, &c. increasing 3d. every yard; what sum did it amount to? Ans. L. 12s. 6d. 4. 16 persons gave charity to a poor man; the first gave 5d. the second 9d. and so on in arithmetical progression; how much did the last person give ; and what sum did the man receive ?

Ans. The last gave 5s. 5d. sum received 2L. 6s. 8d. 5. If 100 stones be laid two yards distant from each other in a right line, and a basket placed two yards from the first stone; what distance must a person travel, to gather them singly into the basket ?

Ans. 11miles Sfur. 180yds. 6. A merchant sold 1000 yards of linen, at 2 pins for the first yard, 4 for the second, and 6 for the third, &c. increasing two pins every yard; how much did the linen produce, when the pins were afterwards sold at 12 for a farthing?

Ans. 86L. 175. 10d.

CASE 2. When the two extremes and number of terms are given, to find the common difference.

RULE. Divide the difference of the extremes by the number of terms, less one ; the quotient will be the common difference.

EXAMPLES. 1. 20 and 60 are the two extremes of a certain series in arithmetical progression, and 21 is the number of terms; what is the common difference ?

Ans. 2. 60 20

}Extremes.

21-1= 20)40 Difference of extremes.

2 Common difference., 2. l'here are 21 men whose ages are equally distant from each other in arithmetical progression; the youngest is 20 years old, and the eldest 60; what is the common differenoe of their ages, and the age of each man?

Ans. Common difference 2 years.

60 years is the age of the first man. 60=2= 58

age of the second. 58 2 = 56

age of the third, & Co

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3. A debt is to be paid at 16 different payments in arithmetical progression ; the first payment to be 14 L. and the last 100 L. what is the cominon difference, each payment, and the whole debt ?

Common difference 5 L. 145, 8d. Ans. First payment 14L. Second, 19L.

14s. 8d. Third, 25 L. Is. 4d. &c. 4. A person is to travel from Philadelphia to a certain place in 16 days, and to go but 4 miles the first day, increasing every day by an equal excess, so that the last days journey may be 79 miles; what is the common difference ; and what the whole distance ?

5 Common difference 5 miles. Ans.

.

GEOMETRICAL PROGRESSION.

Any series of numbers, the terms of which increase by a common multiplier, or decrease by a common divisor, are said to be in geometrical progression ; as 3, 6, 12, 24, 48; and 48, 24, 12, 6, 3.

The number by which the series is increased or decreased is called the ratio. The last term and sum of the series are found by this

RULE. 1. Raise the ratio to the power whose index is one less than the number of terms given, which, being multiplied by the first term, will give the last term, or greater extreme.

2. Multiply the last term by the ratio, from the product şubtract the first term, and divide the remainder by ratio less one for the sum of the series.

EXAMPLES 1. A thrasher wrought 20 days, and received for the first day's labour 4 grains of wheat; for the second, 12 ; for the third, 36, &c. How much did his wages amount. to, allowing 7680 grains to make a pint, and the whole to be disposed of at one dollar per bushel ?

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