THEOREM XI.

711. A regular solid is symmetrical with respect to all its faces, edges, and vertices.

Proof. Let ABC be a face of any regular polyhedron;

A, B, and C will then be ver

H

are all identically equal, whatever polyhedral angles take the positions A, B, C, their faces and edges will coincide with the positions of the faces and edges already marked in the figure.

Because the edges are all of equal length, the vertices at the ends of D, E, F, G will fall into the same positions where the former vertices were.

Continuing the reasoning, the whole polyhedron will be found to occupy the same space as before, every face, edge, and vertex falling where some other face, edge, or vertex was at first.

Because this is true in whatever way the positions of the faces may be interchanged, the polyhedron is symmetrical. Q.E.D.

712. Corollary. Conversely, if a polyhedron be such that, when any one face is brought into coincidence with the position of any other, every other face shall coincide with the former position of some face, the polyhedron is regular.

THEOREM XII.

713. If a plane be passed through each vertex of a regular solid, at right angles to the radius, these planes will be the faces of another regular solid.

Proof. 1. Let A, B, and C be any vertices of a regular solid, and its centre. Imagine planes passed through A, B, and C, perpendicular to OA, OB, and OC respectively, and cut off along their A lines of intersection, so as to form the faces of another solid. We call this the new solid, and the original one the inner solid.

B

2. Because the inner solid is regular, if we bring any other of its faces into the position ABC, the whole solid wlll occupy the same position as before (§ 711).

3. Because each face of the new solid is at right angles to the end of a radius to some vertex of the inner solid, and these radii all coincide with the former positions, the plane of each face of the new solid will, when the change of position is made, take the position of the plane of some other face.

4. Therefore the edges in which these planes intersect will take the positions of other edges.

5. Therefore the vertices where these edges meet will take the positions of other vertices.

6. Therefore the new solid occupies the same space as before the change, and is consequently symmetrical with respect to all its faces, edges, and vertices.

Therefore it is a regular solid (§ 712).

Q.E.D.

714. Def. A pair of polyhedrons such that each face of the one corresponds to a vertex of the other are called sympolar polyhedrons.

THEOREM XIII.

715. Every regular solid has as many faces as its sympolar has vertices, and as many edges as its sympolar has edges.

Proof. 1. Continuing the reasoning of the last theorem, it is evident that the centres of the faces of the new solid coincide with the vertices of its sympolar.

2. Because each edge of a regular solid is equally distant from the centres of two adjoining faces, each edge of the new solid will be equally distant from two adjoining vertices of the sympolar.

3. Since every two such vertices are connected by an edge of the sympolar, the new solid will have as many edges as the sympolar, each edge of the one being

at right angles and above the edge of

the sympolar. Q.E.D.

Let A, B, and C be three vertices of the sympolar, and therefore the centres of three faces of the new solid.

4. By what has just been shown, Pɑ, Pb, and Pc will be three edges of the new solid, meeting in a vertex at P. Therefore

A

B

5. The new solid will have a vertex over the centre of each side of the sympolar, and so will have as many vertices as the sympolar has faces. Q.E.D.

THEOREM XIV.

716. The sympolar of a polyhedron whose faces have S sides will have S-hedral angles.

Proof. The vertex of one polyhedron being at P over the centre of the face of its sympolar, the edges meeting at this vertex are each perpendicular to an edge of the face of the sympolar.

Therefore if the face has S sides, the polyhedral angle above it will have S edges, and therefore S faces.

Q.E.D.

717. Corollary. Conversely, the sympolar of a polyhedron whose vertices are S-hedral will have S-sided faces.

718. Corollary. What pairs of regular solids are sympolar to each other can be readily determined from the preceding theorems.

Therefore its sym

The tetrahedron has four vertices. polar has four faces, and is therefore another tetrahedron.

The cube has 8 trihedral vertices and 6 four-sided faces. Therefore its polar has 8 triangular faces and 6 four-hedral vertices. This is the octahedron.

Conversely, the sympolar of the octahedron is the cube. They each have 8 edges.

The dodecahedron has 12 pentagonal faces and 20 trihedral vertices. Therefore its sympolar has 12 five-hedral vertices and 20 triangular faces. This is the icosahedron.

Each of these polyhedrons has 30 edges.

These results are shown in the following table, where the headings at the top of each column refer to the solid on the left, and those at the bottom to its sympolar on the right.

Note that each column applies to two solids. For instance, the lefthand column shows the number of sides to each face of the solid named at the left, and the number of edges at each vertex of the solid named at the right.

BOOK X.

OF CURVED SURFACES.

CHAPTER I.

THE SPHERE.

Definitions.

19. Def. A curved surface is a surface no part of which is plane.

720. Def. A spherical surface is a surface which is everywhere equally distant from a point within it called the centre.

721. Def. A sphere is a solid bounded by a spherical surface.

NOTE 1. A spherical surface may also be described as the locus of the point at a given distance from a fixed point called the centre.

NOTE 2. In the higher geometry a spherical surface is called a sphere. We shall use this appellation when no confusion will thus arise.

722. Def. The radius of a sphere is the distance of each point of the surface from the centre.

723. Def. A diameter of a sphere is a straight line passing through its centre, and terminated at both ends by the surface.

Corollary. Every diameter is twice the radius; therefore all diameters of the same sphere are equal.

24. Def. A tangent plane to a sphere is a plane which has one point, and one only, in common with the sphere.