55. Def. The join of opposite vertices in a ppd. is a diagonal. These are four in number, viz. AA', BB', CC", and DD' (Fig. of 53).

Since AD is to BC, is to D'A' and equal to it, AD'A'D is a parallelogram, and its diagonals bisect one another. Hence AA' and DD' bisect one another; and similarly, AA' and BB' bisect one another, etc.

Therefore, all the diagonals of a ppd. pass through a common point, and are bisected at that point. The common point of the diagonals is the centre.

56. Theorem. Every line-segment passing through the centre of a parallelepiped, and having its end-points upon the figure, is bisected at the centre.

Proof. PQ (Fig. 53) is a line-segment passing through the centre, O, and having its end-points P, Q in the face AC and A'C' respectively.

Join AP and A'Q. Then AP and A'Q are complanar, since PQ passes through 0; and the plane of AP and A'Q cuts the parallel faces AC and A'C' in parallel lines (Art. 21. Cor. 1).

.. AP is to A'Q.

Cor. The centre of a ppd. is the centre of every cen

tral section.

57. As a parallelepiped has three direction edges, three sections may be made normal to each of these edges respectively. These sections will be forms of the parallelogram.

Def. 1. If none of the sections are rectangles, the ppd. is triclinic, and none of its angles, whether face or dihedral, are right angles.

2. If one section is a rectangle, the ppd. is diclinic, and four dihedral angles, whose edges are parallel, are right angles.

3. If two sections are rectangles, the ppd. is monoclinic, and two sets of four dihedral angles are right angles.

4. If the three sections are rectangles, all the faces. are rectangles, and all the dihedral angles are right angles, and all the corners are right corners (Art. 40. Def.). The figure is then a cuboid.1

Cor. In the cuboid all the diagonals are equal, and the direction lines are mutually perpendicular to one another.

Def. 2. A cuboid with its edges equal is a cube. The faces of the cube are squares.

The analogues of the ppd., the cuboid, and the cube, are in plane geometry the parallelogram, the rectangle, and the square.

1 This term was proposed by Mr. Hayward. Before the appearance of Mr. Hayward's work I used the term orthopiped for a rectangular parallelepiped. But cuboid is evidently a better and a more convenient

term.

THE PYRAMID.

58. Def. 1. When a corner of any number of faces is cut by a plane which cuts all the faces, the closed figure so formed is called a pyramid.

The cutting plane is the base, and the planes which form the corner are faces of the pyramid. The edges which bound the base are basal edges, and those which belong to the corner are lateral edges. The vertex of the corner is the vertex or apex of the pyramid.

Def. 2. Pyramids are classified into triangular, square, etc., according to the character of the base. A triangular pyramid is a tetrahedron.

59. Def. If a pyramid be cut by a plane parallel to its base, the portion lying between the base and this cutting plane is called a frustum of a pyramid.

The frustum has thus two bases, a lower and an upper, or a major base and a minor base.

From Art. 28. Cor. 2, it follows that the two bases of the frustum of a pyramid are similar polygons.

THE PRISM.

60. When the vertex of a pyramid goes to infinity in a direction normal to the base, the lateral edges become parallel lines, and the resulting figure is not a closed figure. But under like circumstances the frustum becomes a closed figure with two congruent bases, and is called a prism.

If one edge of a prism is normal to a base, all the edges are normal, and the lateral faces are rectangles. This is called a right prism.

And if one of the lateral edges is inclined to the base, they are all inclined at the same angle.

oblique prism.

This is an

Prisms are usually named from the character of the right section. Thus a right rectangular prism is a cuboid, and a parallelepiped may be a right prism or an oblique prism, depending upon its kind (Art. 57).

THE REGULAR POLYHEDRA.

61. Def. A regular polyhedron is one in which all the faces are regular polygons of the same number of sides, and all the corners are formed by the same number of faces.

This implies that all the edges are equal, that all the face-angles are equal, and that all the dihedral angles are equal.

On account of the perfect symmetry of the figure, it must have a definite centre equally distant from each face and equally distant from each vertex. The normal at the centre of each face passes through the centre of the figure, and the line from a vertex to the centre is an isoclinal to the edges of that vertex and to the faces of that vertex.

One of the regular polyhedra is familiarly known as the cube.

62. Theorem. There cannot be more than five regular polyhedra.

Proof. The least number of faces which can form a corner is three, and these must not be complanar. Therefore the three face-angles must together be less than a

circumangle, or a face-angle must be less than fourthirds of a right angle (Art. 42).

The only regular polygons having their internal angles less than of a right angle are (P. Art. 133. Cor.) the equilateral triangle, the square, and the regular pentagon; and these alone can form the face of a regular polyhedron.

Equilateral Triangle.

A corner may be formed of 3, 4, or 5 equilateral triangles, and may therefore be three-, four-, or five-faced. 1. The three-faced corner gives the regular tetrahedron, with 4 faces, 4 corners, and 6 edges.

2. The four-faced corner gives the regular octahedron, with 8 faces, 6 corners, and 12 edges.

3. The five-faced corner gives the regular icosahedron, with 20 faces, 12 corners, and 30 edges.

Square.

Only one corner, a three-faced, can be formed by squares. 4. This gives the cube, with 6 faces, 8 corners, and 12 edges.

Regular Pentagon.

Only one corner, a three-faced one, can be formed.

5. This gives the regular dodecahedron, with 12 faces, 20 corners, and 30 edges.

These are the five regular polyhedra.

63. Euler's theorem, Art. 48, gives

F+CE+2.

Now the numbers denoted by F and C are evidently interchangeable, while E remains the same. That is,