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SUGGESTIONS TO TEACHERS. Teachers who desire to make this subject attractive and instructive will do well to apply its principles in some practical manner. An empty barrel, and a gauger's rod, will furnish a better exercise for a class of scholars than whole pages of abstract examples.

A visit to a public park in the city, or to the fields in the country, with a surveyor's chain, staff, and compass, requiring the pupils to make notes and measurements, for school work, will fix principles in the mind that would otherwise be soon forgotten. Measurements in the schoolyard or play-ground will be of equal value. A neighboring pile of stone, or the excavation of a cellar, will furnish materials for better examples than any book can prescribe.

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DEFINITIONS. A Straight Line is a line that does not change its direction at any point; as the line A B.

A Curved Line is one that is continually changing its direction; as the line A CB.

A Perpendicular Line is one that meets another, making the angles next to each other equal; thus, if the angles A B C and A B D are equal, the line A B is said to be perpendicular to the 'ine CD.

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An Angle is a figure formed by

B two straight lines drawn from the same point; as the angle B A C.

Note.-In speaking of an angle we name the letters or figures that stand at the ends of the lines which form it, placing in the middle the letter or figure which stands at the junction of the lines.

A Right Angle is either of the equal angles formed by two lines perpendicular to each other; as the angles A B C and A B D.

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An Acute Angle is one that is less than a right angle; as the angle B A C.

An Obtuse Angle is one that is A greater than a right angle; as the angle A B C.

A Surface is the limit or boundary of a solid, and has only two dimensions, length and breadth; as the surface A B C D.

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A Plane Surface, or a Plane, is a surface in which if any two points are taken, the straight line which joins these points will lie wholly in the surface.

A Curved Surface is a surface no part of which is plane.

The Area of a figure is the number of units of any required denomination included within its boundary line or lines.

A Solid is that which has length, breadth, and thickness; thus, the figure A B C D represents a solid.

The Volume of a solid is the number of units of any required denomination contained within the surfaces which bound it.

SURFACES. A Plane Figure is a plane surface which derives other names from the nature of its boundary line or lines.

A Polygon is a plane figure bounded by straight lines.

A polygon of three sides is called a Triangle; of four sides, a Quadrilateral; of five sides, a Pentagon; of six sides, a Hexagon, etc.

TRIANGLES. A Triangle is a plane figure bounded by three straight lines.

A Scalene Triangle has no two of its sides equal.
An Isosceles Triangle has two of its sides equal.
An Equilateral Triangle has its three sides equal.

A Right Angled or Right Triangle is one that has a right angle.

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The Base of a triangle, or of any figure, is the side on which it is supposed to stand; and the Altitude is a line drawn from the angle opposite the base and perpendicular to it. Thus, B C is the base of either of the triangles immediately preceding, and A D is the altitude of the equilateral triangle A B C.

In a right triangle, the side opposite to the right angle is called the Hypotenuse, and the side which forms a right angle with the base is called the Perpendicular.

In the preceding right triangle, A B is the hypotenuse, and A C, the perpendicular.

To find the area of a triangle.

RULE. From half the sum of the three sides, subtract each side separately; multiply the half sum and the three remainders together, and the square root of the product will be the area required.

Note.—When the base and altitude are given, the area is equal to the base multiplied by half the altitude.

1. What is the area of a triangular piece of ground whose sides are 3, 4, and 5 rods?

3 +4 +5 = 6 6—3=3 6–4=2 V6x3x2x1= V36=6 sq. rd., area. 6-5=1 2. What is the area of a triangle, the sides being 100, 150, and 200 rods?

3. The base of a triangle is 75 yards, and the altitude 20 feet; what is the area?

4. A triangular lot of ground is to be sold at the rate of $3 per square foot; what is the value of the lot, the sides being 150, 200, and 250 feet?

5. How many acres in a field whose sides are 350, 400, and 600 yards?

6. An irregular field whose sides are 20, 30, 40, and 50 rods respectively, measures 30 rods from opposite corners, dividing the first two sides from the third and fourth; what is the area of the field ?

19. What is the area of a field, in the form of an isosceles triangle, the equal sides measuring 30 rods each, and the remaining side 20 rods?

8. The base of a right triangle is 27 chains, and the perpendicular is 36 chains; what is the area?

9. A field 160 feet in length, and 120 feet in width, is divided into two equal triangles by a line 200 feet in length, joining opposite corners; what is the area of the field?

Note.— It is established by Geometry that “The square of the hypotenuse of a right triangle is equal to the sum of the squares of the other two sides.” Hence the following:

To find the hypotenuse of a right triangle..

RULE. Extract the square root of the sum of the squares of the base and perpendicular.

To find the base or perpendicular.

RULE. Extract the square root of the difference between the square of the hypotenuse and the square of the given side.

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