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mains AD<AC. The two sides AS, SD, are equal to the two AS, SC; the third side AD is less than the third side AC; therefore the angle ASD ASC (Book I. Prop. IX. Sch.). Adding BSD=BSC, we shall have ASD+BSD or ASB ASC+BSC.
PROPOSITION XX. THEOREM.
The sum of the plane angles which form a solid angle is always less than four right angles.
Cut the solid angle S by any plane ABCDE; from O, a point in that plane, draw to the several angles the straight lines AO, OB, OC, OD, OE.
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. A formed about the point O. But at the point B the sum of the angles ABO, OBC, equal to ABC, is less than the sum of the angles ABS, SBC (Prop. XIX.); in the same manner at the point C we have BCO+OCD<BCS+SCD; and so with all the angles of the polygon ABCDE: whence it follows, that the sum of all the angles at the bases of the triangles whose vertex is in O, is less than the sum of the angles at the bases of the triangles whose vertex is in S; hence 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 (Book I. Prop. IV. Sch.); therefore the sum of the plane angles, which form the solid angle S, is less than four right angles.
Scholium. This demonstration is founded on the supposition that the solid angle is convex, or that the plane of no one surface produced can ever meet the solid angle; if it were otherwise, the sum of the plane angles would no longer be limited, and might be of any magnitude.
PROPOSITION XXI. THEOREM.
If two solid angles are contained by three plane angles which are equal to each other, each to each, the planes of 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.
Having taken 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; draw AB, BC; next take TE=SB; draw EP perpendicular to the plane DTF; from the point P draw PD, PF, perpendicular respectively to TD, TF; lastly, draw DE, EF.
The triangle SAB is right angled at A, and the triangle TDE at D (Prop. VI.): and since the angle ASB=DTE we have SBA=TED. Likewise SB=TE; therefore the triangle SAB is equal to the triangle TDE; therefore 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 (Book I. Prop. XVII.); and consequently, the angle OAB-PDE. The angle OAB 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.
It must, however, be observed, that the angle A of the right angled triangle AOB is properly the inclination of the two planes ASB, ASC, only when the perpendicular BO 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 the obtuse angle together with the angle A of the triangle OAB would make two right angles. But in the same case, the angle of the two planes TDE, TDF, would also be obtuse, and the obtuse angle together with the angle D of the triangle DPE, would make two right angles; and the angle A being thus always equal to the angle at 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 TDE, TDF.
Scholium. If two solid angles are contained by 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 applied the one to the other. 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 perdendiculars 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, or, what is the same, if the perpendiculars OB, PE, instead of lying in the same direction with regard to the planes ASC, DTF, lay in opposite directions, then it would be impossible to make these solid angles coincide with one another. It would not, however, on this account, be less true, as our Theorem states, that the planes containing the equal angles must still be equally inclined to each other; so that the two solid angles would be equal in all their constituent parts, without, however, admitting of superposition. 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.
The same remark is applicable to solid angles, which are formed by more than three plane angles: thus a solid angle, formed by the plane angles A, B, C, D, E, and another solid angle, formed by the same angles in an inverse order A, E, D, C, B, may be such that the planes which contain the equal angles are equally inclined to each other. Those two solid angles, are likewise equal, without being capable of superposition, and are 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 may 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.
1. THE name solid polyedron, or simple polyedron, is given to every solid terminated by planes or plane faces; which planes, it is evident, will themselves be terminated by straight lines.
2. The common intersection of two adjacent faces of a polyedron is called the side, or edge of the polyedron.
3. The prism is a solid bounded by several parallelograms, which are terminated at both ends by equal and parallel polygons.
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 ABCDE-K, the solid so formed, will be a prism..
4. The equal and parallel polygons ABCDE, FGHIK, are called the bases of the prism; the parallelograms taken together constitute the lateral or convex surface of the prism; the equal straight lines AF, BG, CH, &c. are called the sides, or edges of the prism.
5. 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.
6. 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 every other case the prism is oblique, and the altitude less than the side.
7. A prism is triangular, quadrangular, pentagonal, hexagonal, &c. when the base is a triangle, a quadrilateral, a pentagon, a hexagon, &c.
8. A prism whose base is a parallelogram, and which has all its faces parallelograms, is named a parallelopipedon.
The parallelopipedon is rectangular when all its faces are rectangles.
9. Among rectangular parallelopipedons, we distinguish the cube, or regular hexaedron, bounded by six equal squares.
10. A pyramid is a solid formed by several triangular planes proceeding from the same point S, and terminating in the different sides of the same polygon ABCDE.
The polygon ABCDE is called the base of the pyramid, the point S the vertex; and the triangles ASB, BSC, CSD, &c. form its convex or lateral surface.
11. If from the pyramid S-ABCDE, the pyramid S-abcde be cut off by a plane parallel to the base, the remaining solid ABCDE-d, is called a truncated pyramid, or the frustum of a pyramid.
12. The altitude of a pyramid is the perpendicular let fall from the vertex upon the plane of the base, produced if necessary.
13. A pyramid is triangular, quadrangular, &c. according as its base is a triangle, a quadrilateral, &c.
14. A pyramid is regular, when its base is a regular polygon, and when, at the same time, the perpendicular let fall from the vertex on the plane of the base passes through the centre of the base. That perpendicular is then called the axis of the pyramid.
15. Any line, as SF, drawn from the vertex S of a regular pyramid, perpendicular to either side of the polygon which forms its base, is called the slant height of the pyramid.
16. The diagonal of a polyedron is a straight line joining the vertices of two solid angles which are not adjacent to each other.