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will be 3 times as great as the first; and if the remainder be divided by 3, the quotient will be the sum of the terms of the first progression. But 256 is the product of the last term of the given progression multiplied by the ratio, 1 is the first term, and the divisor, 3, is 1 less than the ratio: Hence,

Rule.-Multiply the last term by the ratio; take the difference between the product and the first term, and divide the remainder by the difference between 1 and the ratio.

NOTE. When the progression is increasing, the first term is subtracted from the product of the last term by the ratio, and the divisor is found by subtracting 1 from the ratio. When the progression is decreasing, the product of the last term by the ratio is subtracted from the first term, and the ratio is subtracted from 1.

Examples.

1. The first term of a progression is 2, the ratio 3, and the last term 4375; what is the sum of the terms?

2. The first term of a progression is 128, the ratio 1⁄2, and and the last term 2: what is the sum of the terms?

3. The first term is 3, the ratio 2, and the last term 192: what is the sum of the series?

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4. A gentleman gave his daughter in marriage on New Year's day, and gave her husband 1s. toward her portion, and was to double it on the first day of every month during the year: what was her portion?

5. A man bought 10 bushels of wheat, on the condition that he should pay 1 cent for the 1st bushel, 3 for the 2d, 9 for the 3d, and so on to the last: what did he pay for the last bushel, and for the 10 bushels?

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6. A man has 6 children: to the youngest he gives $150 to the 2d, $300; to the 3d, $600, and so on, to each twice as much as to the one before: how much did the eldest receive, and what was the amount received by them all?

MENSURATION.

386. A TRIANGLE is a portion of a plane, bounded by three straight lines. It has three sides and, three angles.

BC is called, the base; and AD, perpendicular to BC, the altitude.

387. To find the area of a triangle.

Rule.-Multiply the base by half the altitude, and the product will be the area. (Bk. IV. Prop. VI.)* .

Examples.

A

B

D

1. The base BC, of a triangle, is 40 yards, and the perpendicular AD, 20 yards: what is the area?

2. In a triangular field, the base is 40 chains, and the perpendicular 15 chains: how much does it contain? (Art. 194.)

3. There is a triangular field, of which the base is 35 rods, and the perpendicular 26 rods: what are its contents?

388. A SQUARE is a figure having four equal sides, and all its angles right angles.

389. A RECTANGLE is a four-sided figure, like a square, in which the sides are perpendicular to each other, but the adjacent sides are not equal.

390. A PARALLELOGRAM is a four-sided figure which has its opposite sides equal and parallel, but its angles not right angles.

The line DE, perpendicular to the base, is called, the altitude.

*Davies' Legendre.

D

E

391. To find the area of a square, rectangle, or parallelogram.

Rule.-Multiply the base by the perpendicular height, and the product will be the area. (Bk. IV., Prop. V.)

Examples.

1. What is the area of a square field, of which the sides are each 33.08 chains?

2. What is the area of a square piece of land, of which the sides are 27 chains?

3. What is the area. of a square piece of land, of which the sides are 25 rods each?

4. What are the contents of a rectangular field, the length of which is 40 rods, and the breadth 20 rods?

5. What are the contents of a field 40 rods square?

6. What are the contents of a rectangular field, 15 chains long, and 5 chains broad?

7. How many acres in a field 27 chains long and 69 rods broad?

8. The base of a parallelogram is 271 yards, and the perpendicular height 360 feet: what is the area?

392. A TRAPEZOID is a four-sided figure, ABCD, having two of its sides, AB, DC, parallel. The perpendicular, CE, is called, the altitude.

D

E B

393. To find the area of a trapezoid. Rule.-Multiply half the sum of the two parallel lines by the altitude, and the product will be the area. (Bk. IV., Prop. VII.)

Examples.

1. Required the area of the trapezoid ABCD, having given AB = 321.51 ft., DC = 214.24 ft., and CE 171.16 ft. 2. What is the area of a trapezoid, the parallel sides of which are 12.41 and 8.22 chains, and the perpendicular distance between them 5.15 chains?

3. Required the area of a trapezoid whose parallel sides. are 25 feet 6 inches, and 18 feet 9 inches, and the perpendicular distance between them 10 feet and 5 inches.

4. Required the area of a trapezoid whose parallel sides are 20.5 and 12.25, and the perpendicular distance between them 10.75 yards.

5. What is the area of a trapezoid whose parallel sides are 7.50 chains and 12.25 chains, and the perpendicular height 15.40 chains?

6. What are the contents, when the parallel sides are 20 and 32 chains, and the perpendicular distance between them. 26 chains?

394. A CIRCLE is a portion of a plane, bounded by a curved line, called the circumference. Every point of the circumference is equally distant from a certain point within, called the centre: thus, C is the centre, and any line, as ACB, passing through the centre, is called, a diameter.

C

B

If the diameter of a circle is 1, the circumference will be 3.1416. Hence, if we know the diameter, we may find thẻ circumference by multiplying by 3.1416; or, if we know the circumference, we may find the diameter by dividing by 3.1416.

Examples.

1. The diameter of a circle is 4: what is the circumference? 2. The diameter of a circle is 93: what is the circumference? 3. The diameter of a circle is 20: what is the circumference? 4. What is the diameter of a circle whose circumf. is 78.54? 5. What is the diameter of a circle whose circumference is 11652.1944?

6. What is the diameter of a circle whose circumf. is 6850?

395. To find the area or contents of a circle.

Rule.-Multiply the square of the radius by the decimal, 3.1416. (Bk. V., Prop. XIV., Cor. 2.)

Examples.

1. What is the area of a circle whose diameter is 6? 2. What is the area of a circle whose diameter is 10? 3. What is the area of a circle whose diameter is 7? 4. How many square yards in a circle whose diam. is 3 ft.?

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Rule.-Multiply the square of the diameter by 3.1416. (Bk. VIII., Prop. X., Cor.)

Examples.

1. What is the surface of a sphere whose diameter is 12? 2. What is the surface of a sphere whose diameter is 7? 3. Required the number of square inches in the surface of a sphere whose diameter is 2 feet, or 24 inches.

4. How many square miles on the earth's surface, supposing it a sphere, whose diameter is 7912 miles ?

398. To find the contents of a sphere.

Rule.-Multiply the surface by the radius, and divide the product by 3: the quotient will be the contents. (Bk. VIII., Prop. XIV.)

Examples.

1. What are the contents of a sphere whose diameter is 12? 2. What are the contents of a sphere whose diameter is 4? 3. What are the contents of a sphere whose diam. is 14 in. ? 4. What are the contents of a sphere whose diam. is 6 ft.?

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