8. Parallel planes are such as do not meet when produced.

9. A straight line is parallel to a plane if it does not meet the plane when produced.

10. The angle between two straight lines which do not meet is the angle contained by two intersecting straight lines respectively parallel to the two non-intersecting lines.

Thus if AB and CD are two straight lines which do not meet, and ab, bc are two intersecting lines parallel respectively to AB and CD; then the angle between AB and CD is measured by the angle abc.

A

B

11. A solid angle is that which is made by three or more plane angles which have a common vertex, but are not in the same plane.

12. A solid figure is any portion of space bounded by one or more surfaces, plane or curved.

These surfaces are called the faces of the solid, and the intersections of adjacent faces are called edges.

POLYHEDRA.

13. A polyhedron is a solid figure bounded by plane faces.

NOTE. A plane rectilineal figure must at least have three sides; or four, if two of the sides are parallel. A polyhedron must at least have four faces; or, if two faces are parallel, it must at least have five faces.

14. A prism is a solid figure bounded by plane faces, of which two that are opposite are similar and equal polygons in parallel planes, and the other faces are paral· lelograms.

The polygons are called the ends of the prism. A prism is said to be right if the edges formed by each pair of adjacent parallelograms are perpendicular to the two ends; if otherwise the prism is oblique.

15. A parallelepiped is a solid figure bounded by three pairs of parallel plane faces.

A parallelepiped may be rectangular as in fig. 1, or oblique as in fig. 2. The name cuboid is sometimes given to a rectangular parallelepiped whose length, breadth, and thickness are not all equal.

16. A pyramid is a solid figure bounded by plane faces, of which one is a polygon, and the rest are triangles having as bases the sides of the polygon, and as a common vertex some point not in the plane of the polygon.

The polygon is called the base of the pyramid.

A pyramid having for its base a regular polygon is said to be right when the vertex lies in the straight line drawn perpendicular to the base from its central point (the centre of its inscribed or circumscribed circle).

17. A tetrahedron is a pyramid on a triangular base: it is thus contained by four triangular faces.

18.

Polyhedra are classified according to the number of their faces:

thus a hexahedron has six faces;

an octahedron has eight faces;

a dodecahedron has twelve faces.

19. Similar polyhedra are such as have all their solid angles equal, each to each, and are bounded by the same number of similar faces.

20. A polyhedron is regular when its faces are similar and equal regular polygons.

21. It will be proved (see page 451) that there can only be five regular polyhedra.

They are defined as follows:

(i) A regular tetrahedron is

a solid figure bounded by four plane faces, which are equal and equilateral triangles.

22a. A sphere is a solid figure described by the revolution of a semicircle about its diameter, which remains fixed.

The axis of the sphere is the fixed straight line about which the semicircle revolves.

The centre of the sphere is the same as the centre of the semicircle.

A diameter of a sphere is any straight line which passes through the centre, and is terminated both ways by the surface of the sphere.

The following definition of a sphere, analogous to that given for a circle (1. Def. 15), may also be noted:

226. A sphere is a solid figure contained by one surface, which is such that all straight lines drawn from a certain point within it to the surface are equal: this point is called the centre of the sphere.

A radius of a sphere is a straight line drawn from the centre to the surface.

It will be seen that the surface of a sphere is the locus of a point which moves in space so that its distance from a certain fixed point (the centre) is constant.