LESSON III.

M. What other parts do you discover on these solids?

P.-Corners and edges.

M.-How are the corners formed?

P.-By several angles of different planes meeting in one point; or by several edges meeting in one point. M.-How many edges or angles of different faces are at least required to form a corner or solid angle? Try, one-two-three.

P.-Three at least.

M.—Instead of "corners," say solid angles; how are the edges formed?

P.-By the meeting of two faces.

DESCRIPTION OF THE FIVE REGULAR SOLIDS.*

M.-Which of these five solids is bounded by the least number of faces? By how many faces is it bounded? This solid is therefore called Tetrahedron (from the Greek Terpa, four, and dpa, seats). M.-What are the four faces?

P.-Four triangles.

M.-How many sides have four triangles?

P.-Twelve.

M.-How many of these sides are there to each of the edges?

P.-Two sides.

* Tetrahedron, Hexahedron Octahedron, Dodecahedron, Icosa

hedron.

M.-How many edges therefore must this solid have?

P.- Six edges; because there are six twos in twelve.

M.-Now take the solid, examine it, and see whether it is so. How many angles are there about each corner or solid angle?

It is important that the pupils be convinced by actual examination of the solid, that the calculation which they have made is strictly true.

P.-Three angles.

M.-These angles are called plane angles; can you tell why?

P.-Because they are the angles of the plane faces. M.-How many plane angles are there in the four triangular faces ?

P.-Twelve plane angles.

M.-How many solid angles must the tetrahedron have?

P.-Four solid angles; because about every solid angle there are three plane angles, and there are four threes in twelve.

M.-See whether it is so.

SUBSTANCE OF THE LESSON.

1. Two faces meeting laterally form an edge. 2. Three or more edges meeting in one point form a solid angle.

3. The tetrahedron is a solid bounded by four triangular faces: it has six edges, and four solid angles.

LESSON IV.

M.-Compare the sides of the faces of the tetrahedron. What do you observe ?

P.—They are of the same length; they are equal. M.-How will you call a triangle which has three equal sides?

P.-An equal-sided triangle.

M.-Call it an equi-lateral triangle (from the Latin æquus, equal, and latus, side). Describe an equi-lateral triangle on your slates; put letters at the angles. -Are all triangles necessarily equi-lateral ?

P.-No; for two sides of a triangle may be equal to each other, and the third unequal; or the three sides may be unequal.

The master desires the pupils to draw such triangles upon their slates; after which, he may describe an equilateral, an isosceles, and a scalene triangle upon the school-slate, and, pointing to them, continue.

M.-A triangle having only two of its sides equal to each other is called an isosceles (from the Greek ioos, equal, and σkéλos, a leg); and the unequal side is called its base. And a triangle having none of its sides equal to each other is called a scalene (from the Greek okaw, to limp, and σkaλnvos, unequal) triangle. Compare the angles of the faces of the tetrahedron.

P. They are all equal to each other.

M.-How will you call a triangle of which the angles are equal to each other?

P.-Equiangular triangle.

M.-What are the faces of this solid (showing the octahedron)? By how many such faces is it boundHow many plane angles are there about each

ed?

solid angle?

P. It is bounded by eight plane equilateral and equiangular triangles. There are four plane angles about each of its solid angles.

M.—The name of this solid is octahedron (from the Greek OxT, eight, and dpa, a seat). Can you find out how many edges the octahedron has, without actually looking at the solid?

P.-It must have twelve edges; because, since it is bounded by eight triangles, there are twenty-four sides to them, two of which belong to each edge; consequently the solid must have twelve edges.

M.-And how many solid angles has the octahedron ?

P.—It must have six solid angles; because in its eight faces there are twenty-four plane angles, four of which are about each solid angle, and there are six fours in twenty-four; consequently the octahedron must have six solid angles.

M. See whether it is so.-What, then, is sufficient to observe in a solid, in order to ascertain the other parts ?

P. It is sufficient to know the number and kind of faces by which the solid is bounded, and also the number of plane angles which are about each of its solid angles.

SUBSTANCE OF THE LESSON.

1. A triangle having equal sides is called an equilateral triangle.

2. A triangle having equal angles is called an equi-angular triangle.

3.-A triangle having two of its sides equal is called an isosceles triangle: the unequal side is called its base.

4. A triangle having unequal sides is called a scalene triangle.

5.-The faces of the tetrahedron are equilateral and equiangular triangles.

6. The octahedron is a solid bounded by eight equilateral and equiangular triangles: there are four plane angles about each of its solid angles.

7.—The number and kind of faces, and also the number of plane angles, being known, the number of its edges and solid angles can be ascertained therefrom.

LESSON V.

M.-Is there another among these solids which is bounded by triangles? What is their number? It is therefore called icosahedron (from the Greek ikoσi, twenty, and dpa, a seat). See how many plane angles there are about each of its solid angles, and then calculate the number of its edges and solid angles.