In the horizontal they are as follows, viz.:

Note. The preceding tables of differences are given as an example of the manner in which a subject should be treated. The scholar should be accustomed to examine a subject in all its relations.

42. Into how many triangles can a rectilineal figure be divided by diagonals not intersecting one another?

The vertex of each angle is already connected by the bounding lines with the vertices of 2 other angles; there remain then just as many vertices with which each vertex is not connected, as the figure has sides less 3. Consequently there may be as many diagonals not intersecting one another, as the figure has sides less 3.

1. The triangle. This can have no diagonal; for 3-3=0.

2. The quadrilateral. Here we have 4-3-1 diagonal, which will divide the figure into 2 triangles. 3. The pentagon. Here we have 5-3-2 diagonals, dividing the figure into 3 triangles.

In each case we have one more triangle than we have diagonals, and hence we may infer the general rule. If as many diagonals be drawn in a rectilineal figure as can be without one intersecting another, there will be formed as many triangles as the figure has sides less 2.

43. We will now consider the number and kind of figures into which a rectilineal figure may be divided by straight lines drawn at pleasure.

1. The triangle, (fig. 43.) A line drawn from the vertex of an angle to the opposite side divides the triangle into 2 triangles. A line drawn from one side to another side divides the triangle into 1 triangle and 1 quadrilateral.

2. The quadrilateral, (fig. 44.) A line drawn from the vertex of an angle may meet a side, or the vertex of another angle. In the former case 1 triangle and 1 quadrilateral, in the latter case 2 triangles, are formed.

A line drawn from one side of the quadrilateral may meet either the adjacent or the opposite side. In the former case 1 triangle and 1 pentagon, in the latter case, 2 quadrilaterals, are formed. Fig, 45.

Note. In the preceding exercises one or two examples only have been given, to show the manner of proceeding. Each teacher can increase the number at his pleasure.

V. SOLIDS.

44. We will now find by calculation the number of lines and angles in a prism.

Rule. Multiply the number of sides in the base by 3.

That is, a prism has twice as many solid angles as

its base has sides.

45. We will now find the number of lines and angles in a pyramid.

I. The Lines. Edges or Sides.

Triangular pyramid has

(3x3)+3

=

2

Triangular pyramid has (3 × 3) + 3=12

46. We will now find by calculation the number of

lines and angles in the regular solids.

I. Lines. 1. Exterior.

Tetraedron has

Edges or Sides.

4x3

6

2