229. Properties of the Right-angled Triangle, with Problems.

SQUARE OF THE HYPOTHENUSE.

(a.) The side opposite the right angle of a right-angled triangle is called the hypothenuse.

In figure 2 the side A C is the hypothenuse.

A

B

(b.) The square of the hypothenuse equals the sum of the squares of the other two sides.

(c.) The annexed figure will illustrate the meaning of this. Its truth can be rigidly demonstrated by geometry.

A B C represents a right-angled triangle right-angled at B. Let A B be 4 ft., and B C 3 ft. Then will A C be 5 ft., and A B2 + B C2 A C2, or 16 +9 = 25.

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1. What is the hypothenuse of a right-angled triangle of which one side is 5 ft. and the other 11 ft.?

Suggestion. The hypothenuse equals 36 +121/157 ft.

2. What is the third side of a right-angled triangle of which the hypothenuse is 12 ft. and the given side 9 ft.?

Suggestion. 1448163 Ans.

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3. What is the distance from one corner of a floor to the opposite corner, if the floor is 24 ft. long and 18 ft. wide?

4. What is the diagonal of a square 30 ft. on a side?

5. A boy, flying his kite, found that he had let out 845 yards of string, and that the distance from where he stood to a point directly under the kite was 676 yards. How high was the kite?

6. A certain window is 20 ft. from the ground. How long must a ladder be which, having its foot 15 ft. from the bottom of the building, will just reach the window?

7. How long is a side of the greatest square which can be inscribed in a circle 3 feet in diameter ?

NOTE. The diagonals of a square bisect each other at right angles.

8. A tower, 60 feet high, stands on a mound 30 feet above a horizontal plane. On this plane, and directly south of the tower, is a spring, and directly east from the spring, and at a distance of 160 feet from it, on the same plane, stands a large oak tree. Now, allowing that the distance in a direct line from the top of the tower to the spring is 150 feet, what is the distance from the top of the tower to the foot of the oak?

9. Two men started from the same place, and travelled, one north at the rate of 4 miles per hour, and the other east at the rate of 3 miles per hour. After travelling 7 hours, they turned, and travelled directly towards each other at the same rate as before, till they met. How many miles did each

travel?

10. There is a rectangular field 100 rods long and 80 rods wide, the sides of which run north and south. A man started from the south-west corner, and travelled due north along the western boundary of the field for 60 rods, when he travelled across the field in a straight line to the north-east corner. How much farther did he travel than he would if he had gone in a straight line all the way?

230. Solids.

(a.) A SPHERE is a solid bounded by a curved surface, every part of which is equally distant from a point within, called the centre.

(b.) A line drawn from the centre to the surface is called a RADIUS, and a line drawn from any point in the surface through the centre to the opposite point is called a DIAMETER.

(c.) A PRISM is a solid having its several faces parallelograms, and its bases two equal and parallel polygons.

(d.) A CUBE (see 41, b.) is a kind of prism.

(e.) A CYLINDER is such a solid as would be formed by revolving a rectangle about one of its sides. It has also been defined to be " & round body with circular ends."

(f) A PYRAMID is a solid body bounded laterally by triangles, of which the vertices meet at a common point, and the bases terminate in the sides of a polygon, which forms the base of the pyramid.

(9.) A CONE is a solid which has a circular base, and tapers regularly to a point called the vertex.

(h.) A FRUSTUм of a cone or pyramid is a part cut off by a plane parallel to the plane of its base.

(i.) Similar solids have the same shape, i. e., the angles of one of them equal the corresponding angles of the other, and the sides about the equal angles are proportional.

All spheres are similar. Two cones, or two cylinders, are similar when their altitudes are to each other as the radii or diameters of their

(j.) Figure 1 represents a sphere; figure 2 a prism; figure 3 a cylinder; figure 4 a pyramid; figure 5 a cone; figure 6 a frustum of a

cone.

(k.) The SURFACE of a sphere equals the square of its diameter multiplied by 3.1416.*

(1.) The surfaces of spheres are to each other as the squares of their radii or diameters.

(m.) The SOLIDITY, or SOLID CONTENTS, of a sphere equals the product of the surface multiplied by of the radius, or by of the diameter, or it equals of the cube of the diameter multiplied by 3.1416.*

(n.) The solidities of spheres are to each other as the cubes of their radii or diameters.

See foot note, page 343.

(o.) The solidities of similar solids are to each other as the cubes of their like dimensions.

(p.) The solidity of a prism equals the area of its base multiplied by its altitude.

(9.) The solidity of a cylinder is equal to the area of its base multiplied by its altitude.

(r.) The convex surface of a cylinder is equal to the circumference of its base multiplied by its altitude.

(s.) The solidity of a cone or of a pyramid equals the area of its base multiplied by of its altitude.

(t.) The solidity of a frustum of a cone, or of a pyramid, equals § of the product of its altitude multiplied by the sum of its upper base, plus its lower base, plus the mean proportional between the two bases. NOTE. - The mean proportional of two numbers is the square root of their product. Thus, the mean proportional of 4 and 9 = √4 × 9

= 6.

231. Problems.

1. What is the solidity of a sphere the diameter of which is 3 feet?

2. What is the surface of a sphere the radius of which is 1 foot?

3. What is the diameter of a sphere of which the solidity is 10 feet?

4. What is the circumference of a sphere the solidity of which is 12 feet?

5. What is the diameter of a sphere of which the surface is 6 feet?

6. What is the solidity of a prism of which the altitude is 9 feet, and the base contains 10 square feet?

7. What is the solidity of a cylinder of which the altitude is 6 feet and the radius of the base 2 feet?

8. What is the convex surface of a cylinder of which the diameter of the base is 5 feet and the altitude 4 feet?

9. What is the solidity of a cone of which the altitude is 9 feet and the circumference of the base 10 feet?

10. What must be the diameter of a sphere which contains 8 times as many cubic feet as one 3 feet in diameter ?

SECTION XVIII.

PROGRESSIONS.

232. Arithmetical Progression.

(a.) A SERIES OF NUMBERS IN ARITHMETICAL PROGRESSION, or an ARITHMETICAL SERIES, is a series of numbers each of which differs from the preceding by the same number. (b.) Such a series would be obtained by continually adding the same number to, or subtracting it from, any given number. Thus we should have

By adding 2's to 1,

.....

By adding 7's to 3,..

By subtracting 4's from 29,

By subtracting 3's from 56,

...

1, 3, 5, 7, 9, 11, &c.

3, 10, 17, 24, 31, 38, 45, &c. 25, 21, 17, 13, 9, &c.

53, 50, 47, 44, 41, &c.

(c.) If the series is formed by addition, it is called an INCREASING SERIES; if by subtraction, it is called a DE

CREASING SERIES.

(d.) The numbers composing a series are called the TERMS of the series.

(e.) The difference between the consecutive terms of any series is called the COMMON DIFFERENCE, and is always the number by the addition or subtraction of which the series is formed.

(f) Since the terms of a series are formed by continual additions or subtractions of the same number, it follows that the second term of any series equals the first, plus or minus the common difference; that the third equals the first, plus or minus twice the common difference; that the fourth term equals the first, plus or minus three times the common difference; &c.

(g.) Hence, any term of an arithmetical series is equal to the first term, plus or minus the common difference taken one less times than there are terms in the series ending with the required term.

(h.) Moreover, if the first term of an increasing arithmetical series be subtracted from the last, or if the last term of a decreasing series be subtracted from the first, the remainder will be the product of the common difference multiplied by one less than the number of terms.