faces of the cylinder is everywhere equally distant from a point within the figure.

M.-Right. And, what would you call this point? P. The centre of the figure.

M.-Yes; and the curved line is called the circumference of the figure, (from the Latin circum, around, and ferens, carrying,)-the figure itself being called a circle.-How would you define a circle?

P.—A circle is a plane figure, bounded by one curved line everywhere equi-distant from a point within the figure. This point is called, the centre of the circle; and the curved line, the circumference.

M.-A straight line passing through the centre of a circle, and terminated, each way, by the circumference, is called a diameter; and a straight line drawn from the centre to the circumference, is called a radius.-When circles are equal, what may be said of their diameters, and what of their radii? If circles are unequal, what may be said of their diameters, and what of their radii? If the diameters of several circles be equal, what may be said of these circles? If the radii or diameters of different circles be unequal, what may be said of these circles?

(The pupils are, here, required to illustrate these several cases by drawing circles corresponding with each.)

M.-What happens, if part of a sphere be cut-off? P.-Each of the parts is a solid, bounded by one curved and one plane surface: the plane surface is a circle.

M.-If several parts be cut-off from a sphere, are the circles obtained by these sections always equal ?— When are they equal?

P.-No. If the sphere be cut-through the centre, or if it be cut-through at equal distances from the centre, they are, then, equal.

M.-Imagine a line drawn so as to connect the centres of the circular bases of this cylinder. Examine the angles which this line makes with the faces. P.-The angles are right angles.

M.-And, in this cylinder-what are the angles? P. They are not right angles.

M.-In the former case, the cylinder is called a right cylinder; in the latter case, an oblique cylinder, (from the Latin obliquus, indirect).

SUBSTANCE OF THIS LESSON.

1. The cylinder is a solid bounded by two opposite plane faces and one curved face.

2. Each of the plane faces is called a circle.

3. A circle is a plane figure, bounded by one curved line equi-distant, everywhere, from a certain point within the figure.

4. This point is called the centre of the circle. 5. The curved line is called the circumference. 6. A straight line passing through the centre, and terminating, each way, in the circumference, is called a diameter.

7. A straight line drawn from the centre to the circumference is called a radius.

8. When the straight line joining the centres of the circular bases of a cylinder is at right angles to either of them, the cylinder is called a right cylinder; when the straight line which joins the centres of the circular bases of a cylinder is not at right angles to either of them, the cylinder is called an oblique cylinder.

LESSON XII.

THE CONE.

M.-Examine these solids: they are called cones. What do you discover?

P.-They are bounded by one plane surface and one curved surface. The plane surface is a circle; the curved surface terminates in a point. This point, in one of the cones, is directly over the centre of the centre: the straight line joining the centre and that point is at right angles to the base; in the other cone it is not at right angles to the base.

M.-The point, you have mentioned, is called the summit or vertex of the cone. By what name would you distinguish the one cone from the other?

P. The one is a right, the other an oblique, cone. M.-Compare the cone with the other solids. To which of these has it the greatest resemblance?

P. It is most like the pyramid.

M.-What sort of pyramid would be almost a cone? P.-That pyramid whose base is a polygon of the greatest number of sides: the greater the number of

the sides of the base, the more nearly does the pyra

mid approach the cone.

M.-Which of the other solids, besides the pyramid, resembles the cylinder?

P. The prism.

M.-What sort of prism is most like a cylinder? P.-That prism whose bases are bounded by the greatest number of sides.

M.-And, which of the solids resembles the sphere ?

P.-The trapezohedron, or that sort of solid which is bounded by a great number of plane faces.

M.-By what name would you describe a solid bounded by a great number of plane faces?

P.-Polyhedron-(from the Greek woλus, many, and dpa, a seat).

M.-What plane figure, bounded by straight lines, would most nearly resemble the circle?

P.-One bounded by the greatest number of sides.

SURFACES.

CHAPTER I.*

STRAIGHT LINES: ANGLES.

SECTION I.-ONE AND TWO STRAIGHT LINES.

M.—Mention the different parts of a solid.

P.-Solid angles, edges, faces, sides or lines, plane angles.

M.-In this and the following lessons, we shall endeavour to discover the properties of some of these parts. For this purpose we shall begin with the lines; and, first, with one straight line. Draw one straight line, and state all you discover respecting it.

P.—A straight line has length-it extends in a certain direction. This is not really a straight line, but a solid; it merely represents a straight line-it has two ends or extremities.

M.—What are the extremities of a straight line? P.-Points.

M.—Tell me, now, what can be done with a straight

line.

P.-It can be lengthened, so as to become very

* This division, into chapters and sections, is adopted, now, for the sake of preserving the continuity of the subject. In the course of tuition, however, each section may be found to comprise several lessons.