brad-awl inserted at the other, to form a centre, by shifting the place of which, circles of different diameters may be accurately delineated. With two centres and a loop of twine ellipses can be drawn ; and the sight of these simple contrivances is instructive to the children. Another means of illustrating geometrical forms is by the gonigraph, an instrument consisting of ten short rulers or joints of iron hinged together; the facility with which various lines and forms can be represented by this contrivance, renders it very popular in Infant Schools; and it has the additional advantage, that it can be used by the children themselves. We now proceed to give such hints as are necessary for the order and succession of the lessons on this subject, which the teacher must further expand and illustrate. Length. The first step is to give a clear conception of extension in one direction. Draw a fine straight line, and explain that it has length only: measure it with a string or compasses, and then give various illustrations of length or distance from one point to another; stretch a string or tape, divided into feet, along the room, and show that the room is so many feet long; remove the string, and explain that the length of the room is still there, and would be the same whether the room was wide or narrow. Make the different children tell where they live, and point out that some have far to come to school, and others a less distance that in each case we speak only of the length of the way, not of its width. Extend these illustrations: as the length of a stick, of a a road, a street, a table, the play the like. Also draw proportiona them. Distance from one place said to be so far, or so long; and thick. A road would be just as 1 good road or a bad one; whether w it, or went by a railroad, we shou distance, although in different peri Length and breadth.-A surf breadth, but no thickness; it is the anything, as the surface of the floo walls, the play-ground, and so fort sure of the floor is called length; t or width. One child may be, mad room, and another across it. It n that if either the length or breadt made less, the surface of the floo and if the length of the playground surface would be greater. The ch to point out various surfaces, and sh breadth; as those of a card, pocket board. All surfaces are bounded b the children should next touch or boundary lines of various surfaces. Solid bodies have three dimension other. The largest is called length the smallest, thickness. A box, or size, may be used as an illustration, an sions measured; and it may be eas occupies some space, and that many fill the room. Other illustrations s the children encouraged to point ou guess at their different dimensions. repeat these definitions together. A line has length only. A surface has length and breadth. th and dary of ing, the est mea breadth ong the d out, n were zaller; ed, its made and black es or ach dth; ome at it uld nd zd en 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. Spiral. Diverging. 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 black board the three varieties of angles, right, acute, quently, and to find other angles to them. Make the children for for themselves with the gonigraph, black board, or on slates held in many angles can be formed with four, five. These figures should scale, and the children required to the different angles. PLANE FIGURES Lines are said to be parallel w same distance from each other in ev long, they will never meet. Two position, on the same plane, conver cross each other; but in no case w gon or enclose a space. This may with two rulers, or two school form made to enclose a space between th not enclose a field with two st straight walls would not make a l three straight lines will enclose a triangle. Draw an accurate equ the black board, meas string or compasses, a equal-sided. Allow so to form the same with attempt to draw it, or t Equilateral laths or rulers of equa gle. m out fre nswering at angles n on the now how th three, a large point to at the ver so other eet of poly rated ot be ould but 7 a on ha be 20 to 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 Isosceles triangle, 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 в 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, C |