SCHOLIUM. If the angles AMC and BMC are obtuse, the perpendiculars CA and CB will not meet MA and MB. In that case, prolong CM through the vertex, M; and through any point, C, of this prolongation, pass a plane perpendicular to CM. It will cut the other two edges in two points, A and B. Apply the same construction to the angle M'—A'B'C' ; and the demonstration remains the same. If AMC and BMC are right angles, the face angles, AMB and A'M'B', measure the dihedrals, AMCB and A'M'C'B', and are equal by hypothesis.

XXXV.*

Theorem. Conversely, if two trihedrals have three dihedrals of the one equal to three dihedrals of the other, each to each, their face angles will be equa leach to each, and the trihedrals will be congruent, or symmetrical.

HYPOTH. Dihedral AMCBA'M'C'B', &c.

=

TO BE PROVED. Face angle AMB A'M'B', &c. (See figures in XXXIV.)

PROOF. The face angles of the polar trihedrals are equal each to each, being supplements of the equal dihedrals. (30); hence the dihedrals of the polar trihedrals are also equal (34); and the face angles of the given trihedrals are equal, being supplements of these dihedrals (30).

M

XXXVI.

Theorem. If two face angles of a trihedral are equal, the opposite dihedrals are equal.

HYPOTH. AMC BMC.

TO BE PROVED. Dihedral AMBC = BMAC.

PROOF. From C draw CE plane AMB,
MB; join AC and BC; then / EBC

and EAC measure the dihedrals AMBC and BMAC (16).

Also hence

hence

▲ AMC

BMC (I., 21):

BEC (I., 29, Cor.):

CA = CB, and ▲ AEC

EBC=EAC; that is, the dihedral AMBC=BMAC.

XXXVII.

Theorem. Conversely, if two dihedrals of a trihedral are equal, the opposite face angles are equal.

HYPOTH. The dihedral AMBC = BMAC; or, EBC = EAC.

TO BE PROVED. Face angle AMC BMC.

PROOF. Like the last.

COR. 1. If the three face angles of a trihedral are equal, the three dihedrals are also equal (36).

COR. 2. Conversely, if the three dihedrals of a trihedral are equal, the three face angles are also equal.

BOOK VI.

POLYHEDRONS.

I.

DEF. 1. A geometrical solid is a portion of space enclosed by surfaces. When the bounding surfaces are all planes, the solid is called a polyhedron. The planes are the faces, and their intersections the edges, of the polyhedron. A diagonal of a polyhedron is a line joining any two vertices not in the same plane.

COR. The edges of a polyhedron are straight lines (V., 1, Cor. 2), and the faces are polygons.

DEF. 2. Polyhedrons are congruent, that is, may be so applied to each other as to correspond in all their parts, when their edges and angles (plane and solid) are equal each to each, and arranged in the same order.

DEF. 3. Two polyhedrons are symmetrical when the edges and angles are equal each to each, but arranged in the opposite order. If two equal faces, ABCD and A'B'C'D', be compared with each other, so that A falls on A,

A/

B

E'

E

B on B', &c., it is evident that the solids will extend in opposite directions, and hence cannot be so compared with each other as to coincide.

138

DEF. 4. Two polyhedrons are similar when the angles of the one are equal to the angles of the other, each to each, and the homologous edges proportional; that is, when the solid angles are congruent each to each, and the homologous faces are similar polygons.

THE PRISM.

II.

DEF. 1. If, from the vertices of a polygon to a parallel plane, parallel lines be drawn, and a plane be passed through each side of the polygon, and the lines drawn from its extremities, the solid enclosed is called a prism.

DEF. 2. The two parallel polygons are called the bases, the other faces taken together the lateral or convex surface, the parallel lines the lateral edges, and the perpendicular distance between the bases the altitude, of the prism.

COR. 1. The lateral faces are parallelograms (V., 22), and the bases are congruent polygons. (Give proof.) COR. 2. The lateral edges are all equal to each other.

COR.. 3. If two parallel planes cut a prism, the sections are congruent polygons; for they form the bases of a new prism.

DEF. 3. A prism is triangular, quadrangular, pentagonal, &c., according as it has three, four, five, &c., lateral faces.

DEF. 4. A right prism is one whose lateral edges are perpendicular to the bases.

DEF. 5. An oblique prism is one whose lateral edges are oblique to the bases.

DEF. 6. A regular prism is a right prism whose bases are regular polygons.

III.

DEF. 1. A parallelopiped is a prism whose bases are parallelograms.

COR. 1. Any two opposite faces of a parallelopiped are parallel and congruent. They are parallel by V., 21, Cor.,

and congruent, since they are mutually equiangular (V., 5) and mutually equilateral, any two homologous sides being the opposite sides of the same parallelogram.

COR. 2. A parallelopiped has six faces, any one of which may be taken as the base.

(II.,

DEF. 2. A rectangular parallelopiped is a right parallelopiped whose bases are rectangles. Def. 4.)

COR. 3. All the faces of a rectangular parallelopiped are rectangles.

DEF. 3. A cube is a rectangular parallelopiped, all of whose faces are squares.

IV.

Theorem. The four diagonals of a parallelopiped bisect

each other.

D

H

A

G

F

B

HYPOTH. ABCD-E is a parallelopiped. TO BE PROVED. The diagonals AG, BH, CE, and DF, bisect each other.

PROOF. Through the equal and parallel edges, AE and CG, pass a plane, cutting the parallel faces ABCD and EFGH in AC and EG. Then ACGE is a parallelogram, and the diagonals AG and CE bisect each other (I., 41). In like manner it may be shown that AG and BH, also AG and FD, bisect each other. Hence the four diagonals bisect each other. Construct figure, and show that AG bisects BH.