ELEMENTS OF GEOMETRY. SUPPLEMENT. BOOK III. OF THE COMPARISON OF SOLIDS. DEFINITIONS. A I. SOLID is that which has length, breadth, and thick- Book III. ness. II. Similar solid figures are such as are contained by the See N. same numbers 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 betwixt one plane and a point above it in which they meet. A prism is a solid figure contained by plane figures, of which two that are opposite are equal, similar, and pa rallel to one another; and the others are parallelograms. V. 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. VIIL 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. X. 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. XIIL The base of a cone is the circle described by that side, containing 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 remains fixed, XV. The axis of a cylinder is the fixed straight line about which the parallelogram revolves. XVI. The basis of a cylinder are the circles described by the two revolving opposite sides of the parallelogram. XVII. Similar cones and cylinders are those which have their axes, and the diameters of their bases proportionals. Book III. Supplement N. PROP. I. THEOR. If two solids be contained by the same number of equal and similar planes, similarly situated, and if the inclination of any two contiguous planes in the one solid be the same with the inclination of the two equal, and similarly si tuated planes in the other, the solids themselves are equal and similar. Let AG and KQ be two solids contained by the same number of equal and similar planes, similarly situated, so that the plane AC is similar and equal to the plane KM, the plane AF to the plane KP; BG to LQ, GD to QN, DE to NO, and FH to PR. Let also the inclination of the plane AF to the plane AC be the same with that of the plane KP to the plane KM, and so of the rest; the solid KQ is equal and similar to the solid AG. Let the solid KQ be applied to the solid AG, so that the bases KM and AC, which are equal and similar, may a 8. Ax. 1. coincide, the point N coinciding with the point D, K with A, L with B, and so on. And because the plane KM coincides with the plane AC, and, by hypothesis, the inclination of KR to KM is the same with the inclination of AH to AC, the plane KR will be upon the plane AH, and will coincide with it, because they are similar and equal, and because their equal sides KN and AD coincide. And in the same manner it is shewn, that the other planes of the solid KQ coincide with the other planes of the solid AG, each with each: wherefore the solids KQ and AG do wholly coincide, and are equal and similar to one another. Therefore, &c. Q. E. D. Book III, PROP. II. THEOR. If a solid be contained by six planes, two and two of which are parallel, the opposite planes are similar and equal parallelograms. Let the solid CDGH be contained by the parallel planes AC, GF; BG, CE; FB, AE: its opposite planes are similar and equal parallelograms. : B Sup. Because the two parallel planes BG, CE, are cut by the plane AC, their common sections AB, CD are parallel. Again, because the two parallel planes BF, AE a 14. 2. are cut by the plane AC, their common sections AD, BC are parallel and AB is parallel to CD; therefore AC is a parallelogram. In like manner, it may be proved that each of the figures CE, FG, GB, BF, AE is a parallelogram: join AH, DF; and because AB is parallel to DC, and BH to CF: the two straight lines AB, BH, which meet one another, are parallel to DC and CF, which meet one another; wherefore, though the first two are not in the same A D H G E plane with the other two, they contain equal angles b ; b 9. ?. Sup. the angle ABH is therefore equal to the angle DCF. And because AB, BH, are equal to DC, CF, and the angle ABH equal to the angle DCF; therefore the base Aй is equal to the base DF, and the triangle ABH to e 4. 1. the triangle DCF: For the same reason, the triangle AGH is equal to the triangle DEF; and therefore the parallelogram BG is equal and similar to the parallelogram CE. In the same manner, it may be proved, that the parallelogram AC is equal and similar to the parallelogram GF, and the parallelogram AE to BF. Therefore, if a solid, &c. Q. E. D. |