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various surfaces, and show their length and breadth ; as those of a card, pocket-handkerchief, blackboard. All surfaces are bounded by lines or edges, and the children should next touch or point to the edges or boundary lines of various surfaces.
Solid bodies have three dimensions, all crossing each other. The largest is called length; the next, breadth ; the smallest, thickness. A box, or other object of some size, may be used as an illustration, and its different dimensions measured ; and it may be easily explained that it occupies some space, and that many such objects would fill the room. Other illustrations should be given, and the children encouraged to point out solid objects, and guess at their different dimensious. Let the children repeat these definitions together.
A line has length only.
Familiar illustrations should be given of these properties ; as, any number of lines put together would not make the thickness of the smallest thread; the whole surface of the floor is no part of the substance of the floor, but only the outside or boundary, and has no weight, or thickness.
Lines. Lines define the shape and boundary of things, and by lines all things are measured. A line is the distance from one point to another. These points are called its ends. Lines are divided into right lines, or the shortest distance between two points, as when a string is stretched tightly; and curved lines. Curved lines are of many varieties, as circular and elliptical curves. Illustrations must be given on the blackboard, and the children required to find examples for themselves, in various objects, of straight, curved, waved, spiral, and other lines. The direction of lines should next be taught, as horizontal, perpendicular, oblique, parallel, converging, and diverging lines.
Angles. When lines meet or cross each other, they form angles or corners. Give examples as the corner of the room, of a book, a board, a table. Draw on the blackboard the three varieties of angles, right, acute, and obtuse; require the children to point them out frequently,
and to find other angles or corners answering to them. Make the children form the different angles for themselves with the gonigraph, or draw them on the blackboard, or on slates held in the lap; show how many angles can be formed with two lines; with three, four,
five. These figures should be drawn on a large scale, and the children required to count and point to the different angles.
Plane Figures. Lines are said to be parallel when they are at the same distance from each other in every part; if ever so long, they will never meet. Two lines in any other position, on the same plane, converge, and will meet or cross each other; but in no case will they form a polygon or enclose a space. This must be easily illustrated with two rulers, or two school forms, which can not be made to enclose a space between them. A farmer could not enclose a field with two straight hedges : two straight walls would not make a house or room; but three straight lines will enclose a space, and form a triangle. Draw an accurate equilateral triangle on the blackboard, measure each side with a string or compasses, and prove it to be equal-sided. Allow some of the children to form the same with the gonigraph, or to attempt to draw it, or to form it with three laths or rulers of equal length. Explain to them that only one kind of triangle can be formed with the same sides. A triangle may have only two of its sides equal, and is then called isosceles. Prove to the children the equality of two sides in each of these figures, and lead them to point out their differences, and to distinguish the different kinds of angles. A triangle may have all its sides unequal, and is then called scalene. A similar proof should be gone through of the inequality of the sides, and the children required to point out the acute, right, or obtuse angles, and the longest and shortest sides of each figure.
In describing an equilateral triangle to little children, it may bu said consist of three equal straight lines, one leaning to the right, ono to the left, and one horizontal; it may also be dived into three equal acute angles; one opening downwards, one to the right, and one to the left. All the other triangles should be analyzed in the same simple manner, and representations of various objects in which they occur should be sketched, and the intelligence of the children exercised in distinguishing them.
A square has four equal sides, and four right angles : if its two opposite sides are horizontal, the other two will be vertical. The opposite sides of a square are parallel : the distance from the corner
A to the corner c is equal to the distance from the corner B to the corner . A square may be described as four right angles. If a square is first formed with a gonigraph, and the opposite angles pressed toward each other, a rhomb is produced ;
the sides are still equal, but the angles are no square. Jonger right angles, two opposite ones being acute, and the other two obtuse. Many representative figures may now be formed for the amusement and observation of the children, composed of the triangle, square and rhomb.
A rectangle lias four ri ht angles, and its op: osite sides are equal but its adjacent sides may be unequal. It may thus be resolved into four right angles with unequal legs. As this is a form of frequent occurrence, sufficient illustrations may be found in surrounding objects, as windows, doors, slates, books, &c. The oblique parallelogram or rhomboid has its opposite sides and angles equal ; but its adjacent angles and sides unequal. It may be separated into two acute and two obtuse angles with unequal legs.
The other four-sided figures are those with three equal sides, with two, and those in which all the sides are unequal : they are called trapeziums. A pentagon has five equal sides and five equal obtuse
angles, and may be said to consist of five obtuse angles. The other regular polygons are, the hexagon, six sides; heptagon, seven sides ; octagon, eight sides ; nonagon, nine sides ; and decagon, ten sides. All these should be carefully constructed before the children, by first drawing a circle, and then dividing the circumference into the proper number of parts, and uniting the points so obtained by lines. These figures can also be formed with the greatest facility with the gonigraph, and should be thoroughly learned and analyzed in every way before we proceed further.
A circle is a plain figure bounded by a single curved line, called its circumference, every part of which is at the same distance from the center. The diameter of a circle is a straight line passing through the center, and bounded by the circumference. The radias is a straight line drawn from the center to the circumference. The parts of the circle baving been repeatedly drawn and explained, it should
be divided into semicircle, quadrant, segment, and octant. The nature of the ellipse is best illustrated by constructing it before the children, and varying the proportionate axes.
A spiral line may be illustrated by a slip of card rolled up and allowed to uncoil by its elasticity; by a piece of watch-spring ; by the tendrils of plants; and its occurrence may be pointed out in uvivalve shells. The line may be drawn for illustration, by tying a piece of chalk to a string, and winding the string about a fixed spindle as a center, and tracing the line as you unwind it. Waved lines are shown by the moving surface of water, or by a cord shaken, and by drawing.
Solids.--Definitions. A tetrahedron is a figure bounded by four equilateral triangles. It has six edges, four solid angles, and twelve plane angles.
A square pyramid is bounded by four triangular sides and a square base. It has eight edges, five solid angles, and sixteen plane angles.
A triangular prism is bounded by two equal and parallel triangles and three rectangles. It has nine edges, six solid angles and eighteen planė angles.
A cube is bounded by six square sides, and has twelve edges, eight solid angles, and twenty-four plane angles.
A cylinder is bounded by two equal plane circles, parallel to each other, and united by one curved surface.
A cone is a figure having a circle for its base, its side being a curved surface ending in a point, called its apex.