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If two solid angles are formed by three plane angles respectively equal to each other, the planes which contain the equal angles will be equally inclined to each other.

Let the angle ASC= DTF, the angle ASB= DTE, and the angle BSC ETF; then will the inclination of the planes ASC, ASB, be equal to that of the planes DTF, DTE.

A

B

D

E

Having taken SB at pleasure, draw BO perpendicular to the plane ASC; from the point O, at which that perpendicular meets the plane, draw OA, OC perpendicular to SA, SC; join AB, BC; next take TE=SB; draw EP perpendicular to the plane DTF; from the point P draw PD, PF, perpendicular to TD, TF; lastly, join DE, EF.

The triangle SAB is right-angled at A, and the triangle TDE at D; (6.6;) and since the angle ASB=DTE, we have SBA TED. Likewise SB=TE; therefore the triangle SAB is equal to the triangle TDE; hence SA=TD, and AB=DE. In like manner it may be shown, that SC=TF, and BC=EF. That granted, the quadrilateral SAOC is equal to the quadrilateral TDPF: for, place the angle ASC upon its equal DTF; because SA=TD, and SC TF, the point A will fall on D, and the point C on F; and at the same time, AO, which is perpendicular to SA, will fall on PD, which is perpendicular to TD, and in like manner OC on PF; wherefore the point O will fall on the

point P, and AO will be equal to DP. But the triangles AOB, DPE, are right-angled at O and P; the hypothenuse AB DE, and the side AO-DP; hence those triangles are equal; (17.1;) hence the angle OAB=PDE. The angle OAB is the inclination of the two planes ASB, ASC; the angle PDE is that of the two planes DTE, DTF; consequently those two inclinations are equal to each other. Hence, If two solid angles are formed, &c.

Scholium. If two solid angles are composed of three plane angles respectively equal to each other, and if at the same time the equal or homologous angles are disposed in the same manner in the two solid angles, these angles will be equal, and they will coincide when placed upon one another. We have already seen that the quadrilateral SAOC may be placed upon its equal TDPF; thus placing SA upon TD, SC falls upon TF, and the point O upon the point P. But, because the triangles AOB, DPE are equal, OB perpendicular to the plane ASC is equal to PE perpendicular to the plane TDF; besides, those perpendiculars lie in the same direction; therefore the point B will fall upon the point E, the line SB upon TE, and the two solid angles will wholly coincide.

This coincidence, however, takes place only when we suppose that the equal plane angles are arranged in the same manner in the two solid angles; for if they were arranged in an inverse order, it would be impossible to make these solid angles coincide with one another. The proposition would not, however, on this account, be less true. This sort of equality, which is not absolute, or such as admits of superposition, deserves to be distinguished by a particular name: we shall call it equality by symmetry.

Thus those two solid angles, which are formed by three plane angles respectively equal to each other, but disposed in

an inverse order, will be called angles equal by symmetry, or simply, symmetrical angles.

Solid angles formed by more than three plane angles, and are equal to each other without being capable of superposition, will be called solid angles equal by symmetry, or symmetrical solid angles.

Among plane figures, equality by symmetry does not properly exist, all figures which might take this name being absolutely equal, or equal by superposition; the reason of which is, that a plane figure cannot be inverted, and the upper part taken indiscriminately for the under. This is not the case with solids; in which the third dimension may be taken in two different directions.

BOOK VII.

POLYEDRONS.

DEFINITIONS.

1. THE name solid polyedron, or simply polyedron, is given to every solid terminated by planes or plane faces; which planes, it is evident, will themselves be terminated by straight lines.

The solid which has four faces is called a tetraedron; that which has six, a hexaedron; that which has eight, an octaedron; that which has twelve, a dodecaedron; that which has twenty, an icosaedron; and so on.

The tetraedron is the simplest of all polyedrons; because at least three planes are required to form a solid angle, and these three planes leave a void, which cannot be closed without one other plane.

2. The common intersection of two adjacent faces of a polyedron is called the side, or edge of the polyedron.

3. A regular polyedron is one whose faces are all equal regular polygons, and whose solid angles are all equal to each other.

4. The prism is a solid bounded by several plane parallelograms, which are terminated at both ends by two plane polygons equal, similar and parallel.

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To construct this solid, let ABCDE be any polygon; then if in a plane parallel to ABCDE, the lines FG, GH, HI, &c., be drawn equal and parallel to the sides AB, BC, CD, &c., thus forming the polygon FGHIK equal to ABCDE; if in the next place, the vertices of the angles in the one plane be joined with the homologous vertices in the other, by straight lines AF, BG, CH, &c., the faces ABGF, BCHG, &c., will be parallelograms, and ABCDEFGHIK, the solid so formed, will be a prism.

5. The equal and parallel polygons ABCDE, FGHIK are called the bases of the prism; the plane parallelograms taken together constitute the lateral or convex surface of the prism; the equal straight lines AF, BG, CH, &c., are called the sides of the prism.

6. The altitude of a prism is the distance between its two bases, or the perpendicular drawn from a point in the upper base to the plane of the lower base.

7. A prism is right when the sides AF, BG, CH, &c., are perpendicular to the planes of the bases; and then each of them is equal to the altitude of the prism. In all other cases the prism is oblique, and the altitude is less than the side.

8. A prism is triangular, quadrangular, pentagonal, hexagonal, &c., when the base is a triangle, a quadrilateral, a pentagon, a hexagon, &c.

9. A prism whose base is a parallelogram and which has all its faces parallelograms, is called a parallelopipedon. The parallelopipedon is rectangular when all its faces are rectangles.

10. Among rectangular parallelo

pipedons, we distinguish the cube, or regular hexaedron, bounded by six equal squares.

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