POLYEDRAL ANGLES.

65. Definition. When three or more planes meet in a common point, they form a polyedral angle.

Thus, the figure S-ABCD, formed by the planes ASB, BSC, CSD, DSA, meeting in the common point S, is a polyedral angle.

The point S is the vertex of the angle; the intersections of the planes, SA, SB, etc., are its edges; the portions of the planes bounded by

A

B

S

the edges are its faces; the angles ASB, BSC, etc., formed by the edges, are its face angles.

A triedral angle is a polyedral angle having but three faces, which is the least number of faces that can form a polyedral angle.

66. Definition. Two polyedral angles are equal when they can be applied to each other so as to coincide in all their parts.

Since two equal polyedral angles coincide however far their edges and faces are produced, the magnitude of a polyedral angle does not depend upon the extent of its faces; but in order to represent the angle clearly in a diagram we usually pass a plane, as ABCD, cutting all its faces in straight lines AB, BC, etc.; and by the face ASB is not meant the triangle ASB, but the indefinite surface included between the lines SA and SB indefinitely produced.

67. Definition. A polyedral angle S-ABCD is convex, when any section, ABCD, made by a plane cutting all its faces, is a convex polygon (I. 95).

C'

68. Symmetrical polyedral angles. If we produce the edges AS, BS, etc., through the vertex S, we obtain another polyedral angle S-A'B'C'D', which is symmetrical with the first, the vertex S being the centre of symmetry.

S, and then AA', BB', etc., being divided proportionally by three parallel planes (37), if any one of them is bisected at S, the others are also bisected at that point. The points A', B', etc., are, then, symmetrical with A, B, etc., the definition of symmetry in a plane (I. 138), being extended to symmetry in space.

The two symmetrical polyedral angles are equal in all their parts, for their face angles, ASB and A'SB', BSC and B'SC', are equal, each to each, being vertical plane angles; and the diedral angles at the edges SA and SA', SB and SB', etc., are equal, being vertical diedral angles formed by the same planes. But the equal parts are arranged in inverse order in the two figures, as will appear more plainly, if we turn the polyedral angle S-A'B'C'D' about, until the polygon A'B'C'D' is brought into the same plane with ABCD, the vertex S remaining fixed; the polygon A'B'C'D' is then in the position abcd, and it is apparent that while in the polyedral angle S-ABCD the parts ASB, BSC, etc., succeed each other in the order from right to left, their corresponding equal parts aSb, bSc, etc., in the polyedral angle S-abcd succeed each other in the order from left to right. The two figures, therefore, cannot be made to coincide by superposition, and are not regarded as equal in the strict sense of the definition (I. 75), but are said to be equal by symmetry.

PROPOSITION XXV.-THEOREM.

69. The sum of any two face angles of a triedral angle is greater than the third.

The theorem requires proof only when the third angle considered is greater than each of the others.

Let S-ABC be a triedral angle in which the face angle ASC is greater than either ASB or BSC; then, ASB + BSC> ASC

For, in the face ASC draw SD making the angle ASD equal to ASB, and through any point D of SD draw any straight line ADC cutting SA

and SC; take SB SD, and join AB, BC.

=

S

The triangles ASD and ASB are equal, by the construction (I. 76). whence AD AB. Now, in the triangle ABC, we have

AB + BC > AC,

and subtracting the equals AB and AD,

BC > DC;

therefore, in the triangles BSC and DSC, we have the angle BSC> DSC (I. 85), and adding the equal angles ASB and ASD, we have ASB+BSC> ASC.

PROPOSITION XXVI.-THEOREM.

70. The sum of the face angles of any convex polyedral angle is less than four right angles.

Let the polyedral angle S be cut by a plane, making the section ABCDE, by hypothesis, a convex polygon. From any point O within this polygon draw OA, OB, OC, OD, OE.

B

S

D

The sum of the angles of the triangles ASB, BSC, etc., which have the common vertex S, is equal to the sum of the angles of the same number of triangles AOB, BOC, etc., which have the common vertex 0. But in the triedral angles formed at A, B, C, etc., by the faces of the polyedral angle and the plane of the polygon, we have (69). SAE SAB> EAB,

SBA + SBC > ABC, etc.;

hence, taking the sum of all these inequalities, it follows that the sum of the angles at the bases of the triangles whose vertex is S is greater than the sum of the angles at the bases of the triangles whose vertex is 0; therefore, the sum of the angles at S is less than the sum of the angles at O, that is, less than four right angles.

PROPOSITION XXVII.-THEOREM.

71. Two triedral angles are either equal or symmetrical, when the three face angles of one are respectively equal to the three face angles of the other.

In the triedra. angles S and s, let ASB

asb, ASC =asc, and

BSC bsc; then, the diedral angle SA is equal to tle diedral

On the edges of these angles take the six equal distances SA, SB, So, sa, sb, sc, and draw AB, BC, AC, ab, bc, ac. The isosceles triangles SAB and sab are equal, having an equal angle included by equal sides, hence AB ab; and for the same reason, BC AC ac; therefore, the triangles ABC and abc are equal.

=

=

bc,

At any point D in SA, draw DE in the face ASB and DF in the face ASC, perpendicular to SA; these lines meet AB and AC, respectively, for, the triangles ASB and ASC being isosceles, the angles SAB and SAC are acute; let E and F be the points of meeting, and join EF. Now on sa take sd SD, and repeat the same construction in the triedral angle s.

=

=

=

=

The triangles ADE and ade are equal, since AD ad, and the angles at A and D are equal to the angles at a and d; hence, AE ae and DE =de. In the same manner, we have AF af and DF = df. Therefore, the triangles AEF and aef are equal (I. 76), and we have EF ef. Finally, the triangles EDF and edf, being mutually equilateral, are equal; therefore, the angle EDF, which measures the diedral angle SA, is equal to the angle edf, which measures the diedral angle sa, and the diedral angles SA and sa are equal (41). In the same manner, it may be proved that the diedral angles SB and SC are equal to the diedral angles sb and sc, respectively.

So far the demonstration applies to either of the two figures denoted in the diagram by s-abe, which are symmetrical with each other. If the first of these figures is given, it follows that S and s are equal, since they can evidently be applied to each other so as to coincide in all their parts (66); if the second is given, it follows that Sands are symmetrical (68).

BOOK VII.

POLYEDRONS.

1. DEFINITION. A polyedron is a geometrical solid bounded by planes.

The bounding planes, by their mutual intersections, limit each other, and determine the faces (which are polygons), the edges, and the vertices, of the polyedron. A diagonal of a polyedron is a straight line joining any two of its vertices not in the same face.

The least number of planes that can form a polyedral angle is three; but the space within the angle is indefinite in extent, and it requires a fourth plane to enclose a finite portion of space, or to form a solid; hence, the least number of planes that can form a polyedron is four.

2. Definition. A polyedron of four faces is called a tetraedron; one of six faces, a hexaedron; one of eight faces, an octaedron; one of twelve faces, a dodecaedron; one of twenty faces, an icosaedron.

3. Definition. A polyedron is convex when the section, formed by any plane intersecting it, is a convex polygon.

All the polyedrons treated of in this work will be understood to be convex.

4. Definition. The volume of any polyedron is the numerical measure of its magnitude, referred to some other polyedron as the unit. The polyedron adopted as the unit is called the unit of volume.

To measure the volume of a polyedron is, then, to find its ratio to the unit of volume.

5. Definition. Equivalent solids are those which have equal volumes.