M.-Calculate the number of faces, edges, and the number of solid angles of a triangular pyramid.

P.—A triangular pyramid has four triangular faces, six edges, and four solid angles. The tetrahedron is a triangular pyramid.

M.-Calculate the number of faces, edges, and solid angles of a quadrangular pyramid.

P.-A quadrangular pyramid has five faces, four triangular faces, and one which is quadrilateral, which is called the base of the pyramid: it has eight edges, and nine solid angles. The octahedron is formed by two quadrangular pyramids, being joined base to

base.

M.-Look again at the bipyramidal dodecahedron. What do you now observe?

P.-It is formed by two hexangular pyramids joined base to base.

M.-Thence its name.

Give a description of a

pyramid whose base is a polygon of twelve sides.

P.-It has thirteen faces, twenty-four edges, and

thirteen solid angles.

M.-How many plane angles are there in all the faces, taken together?

P.-Forty-eight plane angles.

M.-What is the number of sides in all the faces ? P-Forty-eight sides.

LESSON IX.

THE PRISM.

M. (placing several prisms before his pupils)—Examine these solids. What do you observe?

P.-Two of the faces are polygons, the others are rectangles or parallelograms.

M.-Such solids are called prisms. Which of the faces would you call the base of the prism?

P.-Either of the polygons may be its base.

M.—Examine the solid angles. How are they formed, and what must be their number in each prism? P.-Each is formed by three plane angles; their number must be double the number of the sides of the polygon.

M.—In what manner may you distinguish one prism from another?

P.-By its bases. A prism whose bases are triangles we would call triangular; one which has quadrilateral bases we would call a quadrangular prism, and so on.

M.-Enumerate the faces, edges, and solid angles of a quadrangular prism.

P.-It has six faces, twelve edges, and eight solid angles.

M.-Are you not already acquainted with a quadrangular prism? What is its name?

P.-Hexahedron, or cube.

M. (showing the parallelopiped) - Compare this prism with the hexahedron.

C

P.-The faces of the hexahedron are all equal; they are all squares. In this solid, only two faces are squares; the others are rectangles.

M.-This prism is called a parallelopiped. Enumerate the faces, edges, and solid angles of a prism whose bases are polygons of twelve sides.

P.-This prism must have fourteen faces, thirty-six edges, and forty-eight solid angles.

LESSON X.

SOLIDS BOUNDED BY CURVED FACES.

THE SPHERE.

M.—Examine this solid. What can you say of it? P.-It is bounded only by one curved surface.

M.-Imagine a point within this solid, exactly in the middle of it. What may be said of the surface relatively to this point?

P. It is everywhere equally distant from this point.

M.-This solid is called a sphere; the point which is equally distant from every point of the surface, is called the centre of the sphere. Imagine a straight line drawn from any point of the surface of a sphere, through its centre, to the opposite surface: such a line is called a diameter (from the Greek dia, through, and μéτpov, measure).- How many diameters can be drawn in a sphere?

P.-As many as you please.

M.-What may be said, on comparing the diameters of the same sphere?

P.-Diameters of the same sphere are equal. M.-When are diameters of different spheres equal? P.-When the spheres themselves are equal. M.-And, if two spheres are equal, what may be said of their diameters?

P. Their diameters must also be equal.

M.-Imagine a straight line drawn from any point in the surface of a sphere to its centre: such a straight line is called a radius (from the Latin radius, a ray). How many radii can be drawn in a sphere ? -Compare a radius with a diameter of the same sphere.

P.-An indefinite number.-A diameter is just twice as long as a radius;—a radius is only half a diameter.

M.-If two or more spheres are equal, what may be said of their radii? And if the radii of different spheres be unequal, what may be said of the spheres ?

P.-If two or more spheres are equal, their radii must be equal; and if the radii of different spheres be unequal, the spheres must be unequal.

SUBSTANCE OF THIS LESSON.

1. A sphere is a solid bounded by one curved surface, which is everywhere equi-distant from a point, within the solid, called the centre.

2. A straight line drawn from any point in the surface of a sphere, through the centre, to the opposite surface, is called a diameter.

3. A straight line drawn from any point of the surface of a sphere to the centre, is called a radius.

4. A diameter of a sphere is double the length of a radius of the same sphere.

5. Diameters and radii of the same sphere, or of equal spheres, are equal.

6. If spheres are equal, their diameters and radii are equal.

7. If spheres are unequal, their diameters and radii are unequal.

8. If the diameters or radii of different spheres are unequal, the spheres themselves are unequal.

LESSON XI.

THE CYLINDER.

M.-Examine this solid: it is called a cylinder. What can you say of it?

P. It is bounded by two opposite plane faces, and by one curved face.

M.-Represent the plane faces on your slates. What do you observe?

P.-Each is bounded by one curved line.
M.-Draw, on your slates, a figure,

which is also bounded by only one
curved line, and yet dissimilar to these
figures. Compare, now, these several
figures.

P. The curved line which bounds one of the plane