Since, then, the angle A will always be equal to the angle D, we infer that the inclination of the two planes APC, APB will always be equal to the inclination of the two planes DQF, DQE. In the first case, the inclination of the plane is the angle A or D; in the second case, it is the supplement of those angles. Scholium. If two trihedral angles have the three plane angles of the one equal to the three plane angles of the other, each to each, and, at the same time, the corresponding angles arranged in the same manner in the two trihedral angles, then these two trihedral angles will be equal; and if placed one upon the other, they will coincide. In fact, we have already seen that the quadrilateral PAYC will coincide with the quadrilateral QDZF. Thus the point Y falls upon the point Z, and, in consequence of the equality of the triangles AYB, DZE, the straight line YB, perpendicular to the plane APC, is equal to the straight line ZE, perpendicular to the plane DQE; moreover, these perpendiculars lie in the same direction; hence the point B will fall upon the point E, the straight line PB on the straight line QE (their extremities already coinciding), and the two trihedral angles will entirely coincide with each other.

This coincidence, however, can not take place except we suppose the equal plane angles to be arranged in the same manner in the two trihedral angles; for if the equal plane angles be arranged in an inverse order, or, which comes to the same thing, if the perpendiculars YB, ZE, instead of being situated both on the same side of the planes APC, DQF, were situated on opposite sides of these planes, then it would be impossible to make the two trihedral angles coincide with each other. It would not, however, be less true, according to the above theorem, that the planes, in which the equal angles lie, would be equally inclined to each other; so that the two trihedral angles would be equal in all their constituent parts, without admitting of superposition. This species of equality is termed symmetry.

Thus the two trihedral angles in question, which have the three plane angles of the one equal to the three plane angles of the other, each to each, but arranged in an inverse order, are termed angles equal by symmetry, or, simply, symmetrical angles.

The same observation applies to polyhedral angles formed by more than three plane angles. Thus, a polyhedral angle formed by the plane angles A, B, C, D, E, and another polyhedral angle formed by the same angles in an inverse order, A, E, D, C, B, may be such that the planes in which the equal angles are situated are equally inclined to each other. These two polyhedral angles, which would in this case be equal, although not admitting of superposition, would be termed polyhedral angles equal by symmetry, or symmetrical polyhedral angles.

In plane figures there is no species of equality to which this designation can belong, for all those cases to which the term might seem to apply are cases of absolute equality, or equality of coincidence. The reason of this is, that in a plane figure one may take the upper part for the under, and vice versâ. This, however, does not hold in solids, in which the third dimension may be taken in two different directions.

This term symmetrical is of very extensive application. Two magnitudes are said to be symmetrical with respect to a plane when the corresponding points are on opposite sides of the plane in the same perpendicular to it, and at equal distances from it.

Thus the two halves of the human body are symmetrical with respect to what anatomists call the median plane. (See Appendix V.)

A plane figure may be said to be symmetrical with. respect to a median line when points on one side of the median line are at equal perpendicular distances from it with opposite points on the other side (see Appendix II., Def. 2).

EXERCISES.

1. To make a trihedral angle with three given plane angles.

2. Prove that in a trihedral angle the sum of the diedral angles is greater than two and less than six right angles.

3. That two trihedral angles are equal when they have two plane angles and the included diedral angle equal [disposed in the same order].

4. Also, when they have one plane angle and two adjacent diedral angles.

5. Also, when they have three diedral angles equal.

6. Show that if from a point within a trihedral angle perpendiculars be drawn to each of the planes which compose it, a new trihedral will be formed whose plane angles will be supplements of the diedral angles of the first, and vice versa.

7. Prove that the three planes bisecting the diedral angles of a trihedral angle meet in the same line.

SOLID GEOMETRY.

DEFINITIONS.

1. A POLYHEDRON is a solid bounded by planes. The intersection of any two of the planes is called a side or edge of the polyhedron. Each bounding plane will be a polygon, and is called a face of the polyhedron.

2. Similar polyhedrons are such as have all their solid angles equal, each to each, and are contained by the same number of similar planes.*

3. A Pyramid is a solid figure contained by triangular planes meeting in one point, called the Vertex, and terminating in the sides of a polygon, called the Base of the pyramid.

A Regular Pyramid is one the base of which is a regular polygon, and the vertex in a perpendicular to the base at its center.

4. A Prism is a solid figure contained by plane figures, of which two that are opposite are equal, similar, and parallel to each other, called bases; and the others are parallelograms. The latter are together called the Lateral Surface of the prism.

A Right Prism is one in which the parallelograms are perpendicular to the bases.

Pyramids and prisms are called Triangular, Quadrangular, Pentagonal, &c., according as their base is a triangle, quadrilateral, pentagon, &c.

5. A Sphere is a solid figure described by the revolution of a semicircle about its diameter, which remains unmoved. The moving semicircle is called the generatrix.

6. The Axis of a sphere is the fixed right line about which the semicircle revolves.

*For a more comprehensive definition of similar solids, see Appendix V.