er dimension, called the breadth or width; and the distance between the front face and the back face is the third dimension, called the length of the cube.

Thus a cube is called a magnitude of three dimensions.

The three terms most commonly applied to the dimensions of a cube are length, breadth, and thickness.

1. Place a cube with one face flat on a table, and with another face toward you, and say which dimension you consider to be the thickness, which the breadth, and which the length.

2. Show to what objects the word height is more appropriate, and to what objects the word depth, and to what the word thickness.

As a surface has no thickness, it has two dimensions only, length and breadth. Thus a surface is called a magnitude of two dimensions.

3. Show how many faces a cube has.'

1 The surfaces of a cube are considered to be plane sur faces.

When a surface is such, that a line placed anywhere upon it will rest wholly on that surface, such surface is said to be a plane surface.'

As a line has neither breadth nor thickness, it has one dimension only, that of length.

Thus a line is called a magnitude of one dimension.

4. Count how many lines are formed on a cube by the intersection of its six plane surfaces.

If that which has neither breadth, nor thickness, but length only, can be said to have any form, then a line is such, that if it were turned upon its extremities, each part of it would keep its own place in space.

We cannot with a pencil make a line on paper-we represent a line.

The boundaries or ends of a line are called points, and the intersection of two lines gives a point.

As a point has neither length, breadth, nor

1 When the word line is used in these definitions and ques tions a straight line is always meant.

thickness, it is said to have no dimension. It

has position only.

A point is therefore not a magnitude.

5. Name the number of points that are made by the intersection of the twelve lines of a cube. We cannot with a pencil make a point on paper-we represent a point.

When any two straight lines meet together from any other two directions than those which are perfectly opposite, they are said to make an angle.

And the point where they meet is called the angular point.

Thus two lines that meet each other on a cube make an angle.

6. Represent on paper a rectilineal angle.

7. Can two lines meet together without being in the same plane?

8. Point out two lines on a cube that exist on the same surface, and yet do not make an angle.

9. Name the number of plane angles on

the six surfaces of a cube, and the number of angular points, and say why the angular points are fewer than the plane angles.

The meeting of two plane surfaces in a line -for example, the meeting of the wall of a room with the floor, or the meeting of two of the surfaces of a cube-is called a dihedral angle.'

10. Say how many dihedral angles a cube has.

The corner made by the meeting of three or more plane surfaces is called a solid angle. 11. Say how many solid angles there are in a cube.

When a surface is such that a line, when resting upon it in any direction, will be touched by it toward the middle of the line only, and not at both ends, such surface is called a convex surface.

12. Give an example of a convex surface.

When a surface is such that a line while rest

ing upon it, in any direction, will be touched by

1 Dihedral means two-surfaced.

it at the ends, and not toward the middle of the line, such surface is called a concave surface.

13. Give an example of a concave surface.

A simple curve is such, that on being turned on its extremities, every point along it will change its place in space; so that, in a simple curve, no three points are in a straight line.

14. Give an example of a simple curve.

Lines or curves grouped together by way of illustration, or for ornament, without regard to magnitude or surface, take the name of diagrams.

15. Give an example of a diagram.

When a surface' is spoken of with regard to its form and size, it takes the name of figure.

If the boundaries of a surface are straight lines, the figure is called a rectilinear figure, and each boundary is called a side.

Thus we have rectilinear figures of four sides, of five sides, of six sides, etc.

16. Make a few rectilinear figures.

1 In the definitions and questions of this work, when the word surface is used, a plane surface is meant.