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GEOMETRICAL PROGRESSION.

418. Geometrical progression is a series of numbers which increase by a common multiplier, or decrease by a common divisor; as 2, 4, 8, 16, 32, &c.; or 32, 16, 8, 4, 2.

OBS. 1. If the series increases, it is called ascending; if it decreases, descending. The numbers which form the series, are called the terms of the progression. The common multiplier, or divisor, is called the ratio.

2. In an ascending series, each succeeding term is found by multiplying the preceding by the ratio. Thus, if the first term is 2, and the ratio 3, the series is 2, 6, 18, 54, &c.

In a descending series, each succeeding term is found by dividing the preceding by the ratio. If the first term is 54 and the ratio 3, the series is, 54, 15, 6, 2.

3. If the first term and ratio are the same, the progression is simply a series of powers; as 2; 2×2; 2×2×2; 2×2×2×2, &c.

4. In Geometrical as well as in Arithmetical progression, there are five parts to be considered, viz: the first term, the last term, the number of terms, the ratio, and the sum of all the terms. These parts have such a relation to each other, that if any three of them are given, the other two may be easily found.

419. To find the last term, when the first term, the ratio, and the number of terms are given.

Multiply the first term into that power of the ratio whose index is 1 less than the number of terms, and the product will be the last term required.

OBS. The several amounts in compound interest, form a geometrical series of which the principal is the 1st term; the amount of $1 for 1 year the ratio, and the number of years+1 the number of terms. Hence the amount of any sum at compound interest, may be found in the saine way as the last term of a geometrical series.

1. If the first term of a geometrical progression is 4, and the ratio 3, what is the 5th term? Ans. 324.

2. If the first term is 48, and the ratio, what is the 5th term.

3. The first term of a series is 3, the ratio 4: what is the 7th term?

QUEST.-418. What is geometrical progression? Obs. When the series in creases, what is it called? When it decreases, what? What are the terms of the progression? In an ascending series, how is each succeeding term found? How in a descending series? If the first term and the ratio are the same, what is the series? 419. When the first term, the ratio, and the number of terms are given, how do you find the last term? Obs. How find the amount of any sum at compound interest by geometrical progression ?

4. The first term of a series is 2, the ratio 2: what is the 23d term?

5. If a scholar receives 1 credit mark for the 1st example he solves, 2 for the 2d, 4 for the third, and so on, the number being doubled for each example, how many marks will he receive for the 12th?

6. What is the amount of $225, at 6 per cent. compound interest for 4 years?

7. What is the amount of $310.50, at 7 per cent. compound interest for 5 years?

420. To find the sum of the series, when the ratio and the extremes are given.

Multiply the greatest term by the ratio, from the product subtract the least term, and divide the remainder by the ratio less 1. OBS. 1. When the first term, the ratio, and the number of terms are given, to find the sum of the series we must first find the last term, then proceed as above. 2. The sum of an infinite series whose terms decrease by a common divisor, may be found by multiplying the greatest term into the ratio, and dividing the product by the ratio less 1. The least term being infinitely small, is of no comparative value, and is therefore neglected.

8. If the extremes are 4 and 972, and the ratio 3, what is the sum of the series? Ans. 1456.

9. The first term is 3, the ratio 2, and the number of terms 9: what is the sum of the series?

10. The extremes of a series are 24 and 48144, and the ratio 11: what is the sun of the series?

11. What is the sum of the infinite series,, },
1, &c.
12. What is the sum of the infinite series,,, &c.
13. What is the sum of the series .1; .01; .001, &c.

14. A man bought a garden 3 rods wide and 4 rods long and agreed to pay 1 cent for the 1st sq. rod, 4 cents for the 2d, 16 cents for the 3d, and so on, quadrupling each sq. rod: how much did his garden cost him?

15. A lady bought a dress containing 12 yards, agreeing to pay 1s. for the 1st yard, 2s. for the 2d, 4s. for the 3d, and so on: how much did her dress cost?

QUEST.-420. When the ratio and the extremes are given, how find the sum of the series? Obs. How find the sum of an ite series, whose terins decrease by a common divisor?

MENSURATION.

421. MENSURATION is the art of measuring magnitudes. · OBS. The term magnitude, denotes that which has one or more of the three dimensions, length, breadth, and thickness.

422. A line is length without breadth.

A surface is that which has length and breadth, without height or thickness.

423. In measuring surfaces, it is customary to assume a square as the measuring unit, whose side is a linear unit of the same name; as a square foot, a square rod, &c. (Thomson's Legendre's Geometry, IV. 4. Sch.)

Note. For the demonstration of the following principles, see references.

424. A square is a figure which has four equal sides, and all its angles right angles. (Art. 153. Obs.)

A parallelogram is a quadrilateral figure whose opposite sides are equal and parallel. It may be right-angled, or oblique angled. (Figs. 1, 3.)

Fig. 1.

A rectangle is a right-angled parallelogram. (Fig. 1.) 425. To find the area of a rectangle, and a square. Multiply the length by the breadth. (Leg. IV. 5.)

Note. When the area and one side of a rectangle are given, the other side is found by dividing the area by the given side. (Art. 291. Note.)

1. How many acres are there in a field 120 rods long, and 90 rods wide?

Ans. 67 acres.

2. How many acres in a field 800 rods long, and 128 rods wide? 3. Find the area of a square field whose sides are 65 rods in length. 4. A man fenced off a rectangular field containing 3750 sq. rods, the length of which was 75 rods: what was its breadth ?

5. One side of a rectangular field is 1 mile in length, and it contains 160 acres: what is the length of the other side?

426. A rhombus is a quadrilateral figure whose sides are equal and its opposite sides parallel, but its angles not right angles. (Fig. 2.)

A rhomboid is an oblique angled paral lelogram. (Fig. 3.)

OBS. The term altitude, denotes perpendicular height; Bs A, B, Fig. 3.

Fig. 2.

B

Fig. 3.

427. To find the area of a rhombus, and rhomboid. Multiply the length by the altitude. (Leg. IV. 5.)

6. The length of a rhombus is 17 ft., and its perpendicular height 16 ft.: what is its area? Ans. 272 sq. ft.

7. What is the area of a rhomboid whose altitude is 25 rods, and its length 28.6 rods?

428. A trapezium is a quadrilateral figure, having only two of its sides parallel. (Fig. 4.)

OBS. A diagonal is a straight line joining two opposite angles; A

as A B, Fig. 4.

Fig. 4.

429. To find the area of a trapezium. (Leg. IV. 7.) Multiply half the sum of the parallel sides by the altitude.

B

8. The parallel sides of a trapezium are 15 feet and 21 feet, and its altitude 12 feet: what is its area? Ans. 216 feet. 9. Find the area of a trapezium whose parallel sides are 25 rods and 37 rods, and its altitude 18 rods.

430. To find the area of a triangle. (Art. 354. Leg. IV. 6.) Multiply the base by half the altitude.

OBS. 1. The base of a triangle is found by dividing the area by half the altiude. (Art. 355.)

2. The altitude of a triangle is found by dividing the area by half the base.

10. What is the area of a triangle whose base is 45 feet, and its altitude 20 eet? Ans. 450 q. ft. 11. What is the area of a triangle whose base is 156 feet, and its altitude 63 feet?

431. To find the area of a triangle, when the three sides are given.

From half the sum of the three sides subtract each side respecively; then multiply together half the sum and the three remainders, and extract the square root of the product.

12. What is the area of a triangle whose sides are respectively 10 feet, 12 feet, and 16 ft. Ans. 59.92+.

13. What is the area of a triangle whose sides are each 12 yds.?

432. A circle is a plane figure bounded by a curve line, every part of which is equally distant from a certain point within, called the centre. (Fig. p. 147.)

The circumference is the curve line by which it is bounded. The diameter is a straight line which passes through the sentre, and is terminated on both sides by the circumference.

The radius or semi-diameter is a straight line drawn from the centre to the circumference.

OBS. From the definition of a circle, it follows that all the radii are equal; also, that all the diameters are equal.

433. To find the circumference of a circle, when the diameter is given. (Leg. V. 11. Sch.)

Multiply the given diameter by 3.14159.

14. What is the circumference of a circle whose diameter is 15 ft. ?

Ans. 47.12385 ft.

15. What is the circumference of a circle whose diameter is 100 rods ?

434. To find the diameter of a circle, when the circumference is given.

Divide the given circumference by 3.14159.

OBS. The diameter of a circle may also be found by dividing the area by .7854, and extracting the square root of the quotient.

16. What is the diameter of a circle whose circumf. is 94.2477 ft.? Ans. 30 ft. 17. What is the diameter of a circle whose circumference is 628.318 yards? 435. To find the area of a circle. (Leg. V. 11.)

Multiply half the circumference by half the diameter; or, multiply the circumference by a fourth of the diameter.

OBS. The area of a circle may also be found by mu'tiplying the square of its diameter by the decimal .7854.

18. What is the area of a circle whose diameter is 100 ft.? Ans. 7854 sq. ft. 19. What is the area of a circle whose diameter is 120 rods?

20 What is the area of a circle whose circumference is 160 yards?
21. Required the diameter of a circle containing 50.2656 square rods!
22. Required the diameter of a circle containing 201.0624 square feet?

436. A solid is a magnitude which has length, breadth, and thickness.

437. In measuring solids, it is customary to assume a cube as the measuring unit, whose sides are squares of the same The sides of a cubic inch, are square inches; of a cubic foot, are square feet, &c. (Art. 154. Obs.)

name.

438. To find the solidity of bodies whose sides are perpes dicular to each other. (Art. 164. Leg. VII. 11. Sch.)

Multiply the length, breadth, and thickness together.

OBS. When the contents of a solid and two of its sides are given, the oth side is found by dividing the contents by the product of the two sides. (Art. 294.)

23. How many cubic feet are there in a stick of timber 60 feet long, 3 feet wide, and 2 feet thick? Ans. 400 cubic feet

T.P.

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