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ing from corner to corner, through the middle point of the cube; 6 line axes, or imaginary straight lines passing through the cube as before and joining the middle. points of two lines.

Each of these surfaces is a plane surface, or simply a plane; that is, if we take two points in any part of it and connect them by a straight line, such line will touch the surface through its whole length. The surfaces of the cube, and indeed of every solid bounded by planes, are called the faces of the solid; the lines where the faces meet are called edges or sides. The angles made by the meeting of two faces are called plane angles; the corners, or angles made by the meeting of 3 faces, are called solid angles.

2. Let us seek if any of the parts which have been mentioned are equal one to another. Each corner is of the same size and shape as each of the others. All 6 faces are equal one to another. No side is longer or shorter than another side; all are equally long. All the 24 line angles are equal. Each face has 4 equal angles.

A figure having 4 angles is called a tetragon, from two Greek words, meaning four and angle; or more commonly a quadrilateral, which means four-sided, because it has also 4 sides. Every quadrilateral in which all the sides and angles are respectively equal, one to another, is called a square; thus, the faces of the cube are squares. Every angle of the square is a right angle; that is, such an angle as is made by one straight line meeting another so as to make the adjacent angles, (that is, the angles made on each side of itself,) equal; such lines are perpendicular one to the other. Thus, the adjacent sides of the cube are perpendicular to one another, because they make right angles with each other. The

faces also make right angles with one another, and are therefore perpendicular.

The opposite faces of the cube are parallel to each other; that is, they are in all parts at the same distance from each other, and would never meet however far they were extended in every direction. If the cube lies upon one of its faces upon a table, the surface of which is parallel to water at rest, the position of such face and of its bounding lines is horizontal; the position of the opposite face and its bounding lines is also horizontal; the faces have like positions, they are parallel. The other faces of the cube have an upright or vertical position; that is, such a position as a leaden weight, hanging freely, gives to a string.


1. What relative positions have the faces of the cube? Which are opposite to one another?

2. To how many edges is each edge of the cube perpendicular?

3. To how many faces of the cube is each edge perpendicular?

4. To how many faces is each face perpendicular? 5. How many faces of the cube are parallel one to another?

6. If the cube lies on a horizontal surface, how many of the edges will have a horizontal, and how many a vertical position?

3. You may now draw upon your slate all that has been remarked in the cube.

1. Straight lines. Draw straight lines from above to

below, from below to above, from the left to the right, from the right to the left, from left below to right above, from right below to left above, from left above to right below, from right above to left below. Draw, in each of the foregoing 8 directions, 6 straight lines parallel to one another, and of equal length.

Draw 2 straight lines which shall touch at one point.

Draw many pairs of lines which shall come together, if produced, that is, made longer.

Draw many lines, which, if produced, will form one line.

2. Angles. Draw two straight lines which shall be perpendicular to one another, and thus make a right angle.

Make right angles in all the positions in which you have drawn the first straight lines.

Point out the lines on the walls or furniture of this room which appear to form right angles.

3. Faces. Draw lines forming a square.

Draw many squares in different positions.

Draw 6 squares of which each succeeding one shall be larger than the one preceding it.

Draw many pairs of squares, in each of which one square shall be equal to the other.


4. We will now examine this upright triangular prism. It is bounded by 5 faces; 2 of these are triangles, that is, figures having 3 angles; and 3 are quadrilaterals. The 2 triangles are placed opposite one to the other, are parallel, and of equal size, and their sides

are of equal length; they are called the bases of the prism. From the form of the bases the prism takes its distinctive name; thus the cube is a quadrangular prism. In the triangular prism the 3 quadrilaterals touch one another; each joins the other two; together they form the convex surface of the prism. Upon the triangles it is said to stand; upon the quadrilaterals to lie.

The triangular prism has 6 corners, or solid angles. If the prism stands, 3 corners are placed above and 3 below; if it lies, 4 corners will be below, 2 above. At each corner 3 line angles meet.

Each triangle is bounded by 3 edges or sides; each quadrilateral by 4. There are 9 sides; each side belongs to two figures; 6 of them, each to one triangle and one quadrilateral; the other 3, each to two quadrilaterals. At each corner 3 sides meet. In this prism the 3 sides belonging solely to the quadrilaterals are perpendicular to those which belong also to the triangles; the sides of the triangles are not perpendicular one to another. The opposite sides of the quadrilaterals are parallel each to each. The 6 sides common to the triangles and quadrilaterals are equal one to another; the 3 sides which belong to the quadrilaterals alone are likewise equal one to another.

5. We will now consider the angles.

1. The plane angles. At each side there is a plane angle. The triangular prism has 9 plane angles; 6 at the sides common to the triangles and quadrilaterals, and 3 at those which belong to the quadrilaterals alone. In this prism the former are right angles, the latter are not right angles.

2. The line angles. Each triangle has 3 line angles, each quadrilateral has 4. Thus in the 2 triangles there

are 6; in the 3 quadrilaterals there are 12; consequently in the whole prism there are 6+1218 line angles. Three of these angles meet at each corner. The 12 angles of the quadrilaterals are right angles. The 6 angles of the triangles are not right angles. The corresponding angles of the two triangles in all prisms must have a like position; and the lines which form the corresponding angles are respectively parallel.

In the triangles each side is opposite to an angle; in the quadrilaterals each side is opposite to another side, and each angle to another angle.

In the triangular prism we can suppose 1 surface axis; 3 edge-face axes; 6 corner-edge axes.

We will now examine triangular prisms which are not upright, those whose bases have unequal sides, and those which are truncated so that their bases are no longer parallel to one another.

6. Angles which are not right angles have the common name of oblique angles. An oblique angle may be greater or less than a right angle; if it be greater than 1, but less than 2 right angles, it is called an obtuse angle; if it is greater than 0 and less that 1 right angle, it is called an acute angle. We compare these angles with a right angle, because the magnitude of a right angle is constant, and always remains equal to itself. It follows from the definition of a right angle, that all right angles are equal. But all obtuse angles are not equal; neither are all acute angles equal.

The magnitude of an angle does not depend upon the length of the lines which form it, and which are called the sides or sometimes the legs of the angle; but upon their inclination one to the other. We may make the sides longer or shorter, still the magnitude of the angle re

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