faces, or by plane and curved faces, or by only one curved surface.

M.-Now examine the boundaries of the faces of the first group you have mentioned. What do you observe?

P. They are all straight lines.

M.-And the boundaries of the other group?
P.-Curved lines and straight lines, or only curved

lines.

The substance of the lesson is then written on the school-slate by the master, and the pupils are required to commit it to memory. Thus:

1. All objects discernible by the senses are extended in three dimensions: namely, in length, in breadth, and in depth or thickness.

2. Objects considered with reference to these three dimensions are called solids.

3. The surface of a solid is its length and breadth considered without reference to its depth.

4. Every solid is bounded either by one surface only, or by several faces.

5. Solids are either bounded by plane faces, or by both plane and curved faces, or by only curved faces.

6. The boundaries of faces are either straight lines, or both straight and curved lines, or only curved lines.

When the above is committed to memory, it is effaced from the slate, and the pupils are required to write it from memory and in the same order.

LESSON II.

At the beginning of this and every following lesson, the pupils ought to be required to recapitulate the preceding lesson, first viva voce, and then by writing it out on their slates.

M.-We will now first examine those solids which are bounded by plane faces only. See in what respect their faces differ.

P.-In size, in shape, and in the number of straight lines which bound them.

M.-Speaking of the boundaries of faces, it is usual to call them sides, instead of lines. I have brought here a considerable number of solids which are bounded by plane faces. Arrange them according to the number of sides by which some of their faces are bounded, beginning with the least. What is the least number of sides by which some of the faces are bounded?

P. By three straight lines-by three sides. M.-And by what word will you express the space which three straight lines inclose?

P.-A three-sided face-a triangle.

M.-Imitate a triangle on your slates. How many lines are necessary to inclose a space? Try one, two, three. If a space is inclosed by two lines, what sort of lines must these be?

P.-Either a straight and a curved line, or two curved lines.

M.-Which face have you placed next in succession to the triangle?

P.-One which is bounded by four sides.

M.-Imitate it on your slates. Which of the faces come next?

P.-The five-sided face; then the six, seven, and eight-sided face.

M.-Imitate all these faces on your slates. Examine the three-sided figure on your slates: in how many points do its three sides meet?

P. In three points.

M.-(Draws a triangle upon the school-slate).

a

с

I will put the letters a, b, c at the three points, in order that we may be able to distinguish one side from the others. By what word will you express the position of the line a b, to the line b c ?

P. The line a b is inclined to the line b c.

M.-And how many inclinations have the three sides to each other?

1

P.-Three inclinations.

M.-The inclination which one line has to another

line is called an angle. How many angles are in a

three-sided figure?

P.-Three angles.

M. See how many angles there are in each of the figures on your slates.

P.-A four-sided figure has four angles; a five-sided, five; a six-sided, six; a seven-sided, seven; and an eight-sided has eight angles.

M.-Can you imagine a figure having nine, ten, eleven, etc. sides? Describe them on your slates, and observe how many angles each figure has.

P.-Every figure has as many angles as it has

sides.

M. You have mentioned another word instead of three-sided figure.

P.-Yes, a triangle.

M. From what circumstance do you think it is called thus ?

P. From its having three angles.

M.-The names of these several faces are derived sometimes from the number of their angles, and sometimes from the number of their sides. Thus, a three-sided face is sometimes called a trilateral figure (from the Latin tres, three, and latus, a side), or a triangle; a four-sided face is called a quadrilateral figure (from the Latin quatuor, four, and latus, a side); a five-sided face, a pentagon (from the Greek TéνTS, five, and ywvía, angle); a six-sided face, a hexagon, (from the Greek , six, and yovía, angle); a sevensided face, a heptagon (from the Greek Tra, seven,

and ywvía, angle); an eight-sided face, an octagon (from the Greek ỏкT, eight, and ywvía, angle). And if this mode of expression be extended to faces which are bounded by many sides, they are called polygons, (from the Greek woλus many, and ywvía, angle).

As before, the pupils are called upon to reproduce the lesson on their slates; the substance of which is then arranged into sentences, and written by the master on the large school-slate, the pupils committing them to memory.

1.-Solids bounded by plane faces differ in shape and in the number of their faces.

2. Their faces differ in the number of their sides.

3.—A face bounded by three sides is called trilateral; by four sides, quadrilateral; by five sides, a pentagon; by six sides, a hexagon; by seven sides, a heptagon; by eight sides, an octagon; by nine or more sides, a polygon.

4.-An angle is the inclination of two lines to one another which meet in a point.

5.—A trilateral face has three angles, it is therefore called a triangle; a quadrilateral has four angles; a pentagon has five, a hexagon six, a heptagon seven, an octagon eight; a polygon has as many angles as it has sides.