b 4. 1. c 20. 1. d 25. 1. Suppl. equal to the angle EAB, the base DB is equal 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 other DC is greater than the remaining part EC. Because DA is equal to AE, and AC common, but the base DC greater than the base EC, the angle DAC is greaterd 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 an gle, &c. Q. E. D. / PROP. XXI. THEOR. THE plane angles which contain any solid angle are together less than four right angles. Let A be a solid angle contained by any number of plane 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 A CBA, ABF, FBC, of which any a 20.2 Sup. two are greater than the thirda, the angles CBA, ABF are greater than the angle FBC. For the same reason the two plane angles B at each of the points C, D, E, F (viz. the angles which are at the bases of the triangles having the common vertex A) are greater than the third angle at the same point, which is one of the angles of the figure BCDEF. There C fore all the angles at the bases of the triangles are together 1 : 32. 1. gles as there are triangles, 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 figures. Therefore all the angles of the tri- c 1. Cor. 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./ 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 A 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 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. SOLID is that which has length, breadth, and thickness. 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. A pyramid is a solid figure contained by planes that are constituted between one plane and a point above it in which they meet. IV. A prism is a solid figure contained by plane figures, of which two that are opposite are equal, similar, and parallel to each other; and the others are parallelograms. A parallelopiped is a solid figure contained by six quadrilateral figures, whereof every opposite two are parallel. / VI. (A cube is a solid figure contained by six equal squares. VII. A sphere is a solid figure described by the revolution of a semicircle about a diameter, which remains unmoved. VIII. The axis of a sphere is the fixed straight line about which the semicircle revolves. ! IX. The centre of a sphere is the same with that of the semicircle. Χ. 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. A cone is a solid figure described by the revolution of a right angled triangle about one of the sides containing the right angle, which side remains fixed. XII. The axis of a cone is the fixed straight line about which the triangle revolves. XIII. The base of a cone is the circle described by the side adjacent to the right angle, which.revolves. XIV. A cylinder is a solid figure described by the revolution of a right angled parallelogram about one of its sides, which re mains fixed. |