XII. A pyramid is a solid figure contained by planes that are constituted betwixt one plane and one point above it in which they meet. XIII. A prism is a solid figure contained by plane figures, of which two that are opposite are equal, similar, and parallel to one another; and the others parallelograms. XIV. A sphere is a solid figure described by the revolution of a semicircle about its diameter, which remains unmoved. XV. The axis of a sphere is the fixed straight line about which the semicircle revolves. XVI. The center of a sphere is the same with that of the semicircle. XVII. The diameter of a sphere is any straight line which passes through the center, and is terminated both ways by the superficies of the sphere. XVIII. A cone is a solid figure described by the revolution of a rightangled triangle about one of the sides containing the right angle, which side remains fixed. If the fixed side be equal to the other side containing the right angle, the cone is called a right-angled cone; if it be less than the other side, an obtuse-angled; and if greater, an acute-angled cone. XIX. The axis of a cone is the fixed straight line about which the triangle revolves. XX. The base of a cone is the circle described by that side containing the right angle, which revolves. XXI. A cylinder is a solid figure described by the revolution of a rightangled parallelogram about one of its sides which remains fixed. XXII. The axis of a cylinder is the fixed straight line about which the parallelogram revolves. XXIII. The bases of a cylinder are the circles described by the two revolving opposite sides of the parallelogram. XXIV. Similar cones and cylinders are those which have their axes and the diameters of their bases proportionals. XXV. A cube is a solid figure contained by six equal squares. XXVI. A tetrahedron is a solid figure contained by four equal and equilateral triangles. XXVII. An octrahedron is a solid figure contained by eight equal and equilateral triangles. XXVIII. A dodecahedron is a solid figure contained by twelve equal pentagons which are equilateral and equiangular. XXIX. An icosahedron is a solid figure contained by twenty equal and equilateral triangles. Def. A. A parallelopiped is a solid figure contained by six quadrilateral figures, whereof every opposite two are parallel. PROPOSITION I. THEOREM. One part of a straight line cannot be in a plane, and another part above it. If it be possible, let AB, part of the straight line ABC, be in the plane, and the part BC above it: A B D and since the straight line AB is in the plane, it can be produced in that plane: let it be produced to D; and let any plane pass through the straight line AD, and be turned about it, until it pass through the point C: and because the points B, C are in this plane, the straight line BC is in it: (I. def. 7.) therefore there are two straight lines ABC, ABD in the same plane that have a common segment AB; (I. 11. Cor.) which is impossible. Therefore, one part, &c. Q.E.D. PROPOSITION II. THEOREM. Two straight lines which cut one another are in one plane, and three straight lines which meet one another are in one plane. Let two straight lines AB, CD cut one another in E; and three straight lines EC, CB, BE, which meet one another, shall be in one plane. Let any plane pass through the straight line EB, and let the plane be turned about EB, produced if necessary, until it pass through the point C. Then, because the points E, Care in this plane, the straight line EC is in it: (1. def. 7.) for the same reason, the straight line BC is in the same: therefore the three straight lines EC, CB, BE are in one plane; If two planes cut one another, their common section is a straight line. Let two planes AB, BC cut one another, and let the line DB be their common section. If it be not, from the point D to B, draw, in the plane AB, the straight line DEB, (post. 1.) and in the plane BC, the straight line DFB : then two straight lines DEB, DFB have the same extremities, and therefore include a space betwixt them; which is impossible: (I. ax. 10.) therefore BD, the common section of the planes AB, BC, cannot but be a straight line. Wherefore, if two planes, &c. Q.E.D. PROPOSITION IV. THEOREM. If a straight line stand at right angles to each of two straight lines in the point of their intersection, it shall also be at right angles to the plane which passes through them, that is, to the plane in which they are. Let the straight line EF stand at right angles to each of the straight lines AB, CD, in E the point of their intersection. Then EF shall also be at right angles to the plane passing through AB, CD. Take the straight lines AE, EB, CE, ED all equal to one another; and through E draw, in the plane in which are AB, CD, any straight line GEH, and join AD, CB; then from any point F, in EF, draw FA, FG, FD, FC, FH, FB. And because the two straight lines AE, ED are equal to the two BE, EC, each to each, and that they contain equal angles AED, BEC, (I. 15.) and the angle DAE to the angle EBC: and the angle AEG is equal to the angle BEH: (1. 15.) therefore the triangles AEG, BEH have two angles of the one equal to two angles of the other, each to each, and the sides AE, EB, adjacent to the equal angles, equal to one another: wherefore they have their other sides equal: (1. 26.) and because AE is equal to EB, and FE common and at right angles to them, the base 4F is equal to the base FB; (1. 4.) for the same reason, CF is equal to FD: and because AD is equal to BC, and AF to FB, the two sides FA, AD are equal to the two FB, BC, each to each; and the base DF was proved equal to the base FC; therefore the angle FAD is equal to the angle FBC: (1. 8.) again, it was proved that GA is equal to BH, and also AF to FB; therefore FA and AG are equal to FB and BH, each to each; and the angle FAG has been proved equal to the angle FBH; therefore the base GF is equal to the base FH: (1. 4.) again, because it was proved that GE is equal to EH, and EF'is common; therefore GE, EF are equal to HE, EF, each to each; and the base GF is equal to the base FH; therefore the angle GEF is equal to the angle HEF; (1. 8.) and consequently each of these angles is a right angle. (1. def. 10.) Therefore FE makes right angles with GH, that is, with any straight line drawn through E in the plane passing through AB, CD. In like manner, it may be proved, that FE makes right angles with every straight line which meets it in that plane. But a straight line is at right angles to a plane when it makes right angles with every straight line which meets it in that plane : (XI. def. 3.) therefore EF is at right angles to the plane in which are AB, CD. Wherefore, if a straight line, &c. Q. E. D. PROPOSITION V. THEOREM. If three straight lines meet all in one point, and a straight line stands at right angles to each of them in that point; these three straight lines are in one and the same plane. Let the straight line AB stand at right angles to each of the straight lines BC, BD, BE, in B the point where they meet. Then BC, BD, BE shall be in one and the same plane. If not, let, if it be possible, BD and BE be in one plane, and let a plane pass through AB, BC, the common section of which, with the plane in which BD and BE are, is a straight line; (XI. 3.) let this be BF: therefore the three straight lines AB, BC, BF are all in one plane, viz. that which passes through AB, BC. And because AB stands at right angles to each of the straight lines BD, BE, it is also at right angles to the plane passing through them: (XI. 4.) and therefore makes right angles with every straight line meeting it in that plane (XI. def. 3.) but BF, which is in that plane, meets it; therefore the angle ABF is a right angle: but the angle ABC, by the hypothesis, is also a right angle; therefore the angle ABF is equal to the angle ABC, and they are both in the same plane, which is impossible; (I. ax. 9.) therefore the straight line BC is not above the plane in which are BD and BE: wherefore the three straight lines BC, BD, BE are in one and the same plane. Therefore, if three straight lines, &c. Q.E.D. PROPOSITION VI. THEOREM. If two straight lines be at right angles to the same plane, they shall be parallel to one another. Let the straight lines AB, CD be at right angles to the same plane. |