perties; 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 or boundary of Horizontal. 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 Perpendicular. 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 black board, 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.

Oblique.

Parallel.

Curved.

Waving.

ANGLES.

Spiral.

Diverging.

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 black board the three varieties of angles, right, acute,

and obtuse; require the children to point them out fre

Right angle.

Acute angle.

Obtuse angle.

quently, 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 black board, 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 may be easily illustrated with two rulers, or two school forms, which cannot 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 black board, 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 Equilateral laths or rulers of equal length. Explain Triangle. to them that only one kind of triangle can

Isosceles triangle.

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 Scalene triangle figure.

In describing an equilateral triangle to little children, it may be said to consist of three equal straight lines, one leaning to the right, one to the left, and one horizontal; it may also be divided into three equal acute angles; one opening downwards, one to the right, and one to the left. All the other triangles should be analysed 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 D. A square may be described as four right angles. If a square is first formed with a gonigraph, and the opposite angles pressed towards each other, a rhomb is produced; the sides are

Square

с

Rhomb.

still equal, but the angles are no longer 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.

1

A rectangle has four right angles, and its opposite sides 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.

Rectangle.

Rhomboid.

Trapezium.

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 analysed in every way before we proceed further.

Pentagon.

Hexagon.

Heptagon.

Octagon.

A circle is a plane figure, bounded by a single curved line, called its circumference, every part of which is at the same distance from the centre. The diameter of a circle is a straight line passing through the centre, and bounded by the circumference. The radius is a straight

Circle.

1

line drawn from the centre to the circumference. The parts of the circle having 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 2 varying the proportionate axes.

A spiral line may be illustrated 1 Quadrant. by a slip of card rolled up and allow2 Octant.. ed to uncoil by its elasticity; by a 3 Segment. piece of watchspring; by the tendrils of plants; and its occurrence may be pointed out in univalve shells. The line may be drawn for illustration, by tying a piece of chalk to a string, and winding the