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429. The sum of the plane angles which form any polyedral angle is less than four right angles.
Let the polyedral angles whose
vertex is S be formed by any number of plane angles, A S B, BS C, CSD, &c.; the sum of all these plane angles is less than four right angles.
Let the planes forming the poly- A edral angle be cut by any plane, ABCDEF. From any point, O,
in this plane, draw the straight lines A O, BO, CO, DO, EO, FO. The sum of the angles of the triangles ASB, BSC, &c. formed about the vertex S, is equal to the sum of the angles of an equal number of triangles AOB, BOC, &c. formed about the point O. But at the point B the sum of the angles A BO, O B C, equal to A B C, is less than the sum of the angles ABS, SBC (Prop. XIX.); in the same manner, at the point C we have the sum of BCO, OCD less than the sum of BCS, SCD; and so with all the angles at the points D, E, &c. Hence, the sum of all the angles at the bases of the triangles whose vertex is 0, is less than the sum of all the angles at the bases of the triangles whose vertex is S; therefore, to make up the deficiency, the sum of the angles formed about the point O is greater than the sum of the angles formed about the point S. But the sum of the angles. about the point O is equal to four right angles (Prop. IV. Cor. 2, Bk. I.); therefore the sum of the angles about S must be less than four right angles.
430. Scholium. This demonstration supposes that the polyedral angle is convex; that is, that no one of the faces would, on being produced, cut the polyedral angle; if it were otherwise, the sum of the plane angles would no longer be limited, and might be of any magnitude.
4.1. If to triedral angles are formed by plane angles which cre equal each to each, the planes of the equal angles will be equally inclined to each other.
nation of the planes ASC, ASB be equal to that of the planes DTF, DTE.
For, take SB at pleasure; draw BO perpendicular to the plane ASC; from the point O, at which the perpendicular meets the plane, draw OA, OC, perpendicular to SA, SC; and join A B, B C. Next, take TE equal SB; draw EP perpendicular to the plane DTE; from the point P draw PD, PF, perpendicular respectively to TD, TF; and join DE, EF.
The triangle SA B is right-angled at A, and the triangle TDE at D; and since the angle ASB is equal to DTE, we have S B A equal to TED. Also, SB is equal to TE; therefore the triangle SAB is equal to TDE; hence SA is equal to T D, and A B is equal to D E.
In like manner it may be shown that SC is equal to TF, and BC is equal to EF. We can now show that the quadrilateral ASCO is equal to the quadrilateral DTFP; for, place the angle ASC upon its equal DTF; since SA is equal to TD, and SC is equal to 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 DP, which is perpendicular to TD, and, in like manner, CO on FP; wherefore the point O will fall on the point P, and AO will be equal to D P.
But the triangles AOB, DPE are right-angled at O and P; the hypotenuse A B is equal to D E, and the side AO is equal to DP; hence the two triangles are equal (Prop. XIX. Bk. I.); and, consequently, the angle OAB is equal to the angle PDE. The angle OA B is the inclination of the two planes ASB, ASC; and the angle PDE is that of the two planes DTE, DTF; hence, those two inclinations are equal to each other.
432. Scholium 1. It must, however, be observed, that the angle A of the right-angled triangle O A B is properly the inclination of the two planes AS B, ASC only when the perpendicular B O falls on the same side of SA with SC; for if it fell on the other side, the angle of the two planes would be obtuse, and joined to the angle A of the triangle O A B it would make two right angles. But, in the same case, the angle of the two planes DTE, DTF would also be obtuse, and joined to the angle D of the triangle D P E it would make two right angles; and the angle A being thus always equal to the angle D, it would follow in the same manner that the inclination of the two planes ASB, ASC must be equal to that of the two planes DTE, DTF.
433. Scholium 2. If two triedral angles are formed by three plane angles respectively equal to each other, and if at the same time the equal or homologous angles are similarly situated, the two angles are equal. For, by the proposition, the planes which contain the equal angles of the triedral angles are equally inclined to each other.
434. Scholium 3. When the equal plane angles forming the two triedral angles are not similarly situated, these angles are equal in all their constituent parts, but, not admitting of superposition, are said to be equal by symmetry, and are called symmetrical angles.
435. A POLYEDRON is a solid, or volume, bounded by planes.
The bounding planes are called the faces of the polyedron; and the lines of intersection of the faces are called the edges of the polyedron.
436. A PRISM is a polyedron having two of its faces equal and parallel polygons, and the other faces parallelograms.
The equal and parallel polygons are called the bases of the prism, and the parallelograms its lateral faces. The lateral faces taken together constitute the lateral or convex surface of the prism.
Thus the polyedron ABCDE-K is a prism, having for its bases the equal and parallel polygons ABCDE, FGHIK, and for its lateral faces the parallelograms ABGF, BCH G, &c.
The principal edges of a prism are those which join the corresponding angles of the bases; as A F, B G, &c.
437. The altitude of a prism is a perpendicular drawn from any point in one base to the plane of the other.
438. A RIGHT PRISM is one whose principal edges are perpendicular to the planes of its bases. Each of the
edges is then equal to the altitude of the prism. Every other prism is oblique, and has each edge greater than the altitude.
439. A prism is triangular, quadrangular, pentangular, hexangular, &c., according as its base is a triangle, a quadrilateral, a pentagon, a hexagon, &c.
440. A PARALLELOPIPEDON is a prism whose bases are parallelograms; as the prism ABCD-H.
The parallelopipedon is rectangular when all its faces are rectangles; as the parallelopipedon ABCD-H.
441. A CUBE, or REGULAR HEXAEDRON, is a rectangular parallelopipedon having all its faces equal squares; as the paral- A lelopipedon A B C D - H.
442. A PYRAMID is a polyedron of which one of the faces is any polygon, and all the others are triangles meeting at a common point.
The polygon is called the base of the pyramid, the triangles its lateral faces, and the point at which the triangles meet its vertex. The lateral faces taken together constitute the lateral or convex surface of the pyramid.
Thus the polyedron A B C D E-S is a pyramid, having for its base the polygon A B C D E, for its lateral faces the triangles AS B, BSC, CSD, &c., and for its vertex the point S.