b 4.1. Suppl. equal to the angle EAB, the base DB is equall to the base BE. Because BD, DC are greater than CB, and one of them BD has been proved equal to BE, a part of CB, the c 20. 1. other DC is greater than the remaining part EC. Because DA is equal to AE, and AC common, but the base DC d 25. 1. greater than the base EC, the angle DAC is greater than the angle EAC. By the construction the angle DAB is equal to the angle BAE. Wherefore the angles DAB, DAC are together greater than the angles BAE, EAC, that is, than the angle BAC. But BAC is not less than either of the angles DAB, DAC; therefore BAC with either of those angles are greater than the other. Wherefore, if a solid angle, &c. Q. E. D. PROP. XXI. THEOR. A THE planc angles which contain any solid angle are together less than four right angles. Let A be a solid angle contained by any number of planie angles BAC, CAD, DAE, EAF, FAB; these together are less than four right angles. Let the planes which contain the solid angle at A be cut by another plane, and let the section of them by that plane be the rectilineal figure BCDEF. Because the solid angle at B is contained by three plane angles CBA, ABF, FBE, of which any a 20.2 Sup. two are greater than the thirda, the angles. CBA, ABF are greater С F common vertex A) are greater than the third angle at the same point, which is one of the angles E of the figure BCDEF. Therefore all the angles at the bases of the triangles are together greater than all the angles of the figure. Now all the angles of the triangles are together equal to twice as many right an gles 'as there are trianglesb, or sides in the figure BCDEF; Book II. and all the angles of the figure, together with four right angles, are likewise equal to twice as many right angles as there b 32. 1. are sides in the figure. Therefore all the angles of the tri- 1. Cor. 32. 1 angles are equal to all the angles of the rectilineal figure, together with four right angles. But all the angles at the bases of the triangles are greater than all the angles of the rectilineal, as has been proved. Wherefore the remaining angles of the triangles, or those at the vertex, which contain the solid angle at A, are less than four right angles. Therefore, the plane angles, &c. Q. E. D. Otherwise. Let the sum of all the angles at the bases of the triangles =S, and the sum of all the angles of the rectilineal figure BCDEF=Z, and the sum of the plane angles at A=X, and R=a right angle. Then S+X=twice as many right angles as there are triangles or sides of the rectilineal figure BCDEF, and Z+4R =twice as many right angles as there are sides of the same figure; therefore S + X=Z + 4R. But, of the three plane angles which contain a solid angle, any two' are greater than the third, therefore S>Z; consequently X<4R, that is, the sum of the plane angles which contain the solid angle at A is less than four right angles. Q. E. D. SCHOLIUM. It is evident that, when any of the angles of the figure BCDEF is exterior, like the А angle at D in the annexed figure, the reasoning in the above proposition does not hold, because the solid angles at the base are not all contained by plane angles, of which two belong to the inclined planes of the triangles, and the third is an B C interior angle of the rectilineal figure, or base. Therefore it cannot be concluded that S is necessarily greater than Z. This proposition, therefore, is subject to a limitation, which is farther explained in the notes on this book. ELEMENTS OF GEOMETRY. SUPPLEMENT. BOOK III. DEFINITIONS. I. Book III. II. Similar solid figures are such as are contained by the same See N. number of similar planes similarly situated, and having like inclinations to one another. III. stituted between one plane and a point above it in which IV. two that are opposite are equal, similar, and parallel to Suppl. V. VI. VII. semicircle about a diameter, which remains unmoved. : VIII. the semicircle revolves. ! IX. The diameter of a sphere is any straight line which passes through the centre, and is terminated both ways by the superficies of the sphere. XI. right angled triangle about one of the sides containing the XII. triangle revolves. XIII. cent to the right angle, which revolves. XIV. right angled parallelogram about one of its sides, which re. |