BOOK XI. DEFINITIONS. 1. A SOLID is that which has length, breadth, and thickness. 2. That which bounds a solid is a superficies. 3. A straight line is perpendicular, or at right angles, to a plane, when it makes right angles with every straight line meeting it in that plane. 4. A plane is perpendicular to a plane, when the straight lines drawn in one of the planes perpendicular to the common section of the two planes, are perpendicular to the other plane. 5. The inclination of a straight line to a plane is the acute angle contained by that straight line, and another drawn from the point at which the first line meets the plane to the point at which a perpendicular to the plane drawn from any point of the first line above the plane, meets the same plane. 6. The inclination of a plane to a plane is the acute angle contained by two straight lines drawn from any the same point of their common section at right angles to it, one in one plane, and the other in the other plane. 7. Two planes are said to have the same or a like inclination to one another, which two other planes have, when the said angles of inclination are equal to one another. 8. Parallel planes are such as do not meet one another though produced. 9. A solid angle is that which is made by more than two plane angles, which are not in the same plane, meeting at one point. 10. Equal and similar solid figures are such as are contained by similar planes equal in number and magnitude. [See the Notes.] 11. Similar solid figures are such as have all their solid angles equal, each to each, and are contained by the same number of similar planes. 12. A pyramid is a solid figure contained by planes which are constructed between one plane and one point above it at which they meet. 13. 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 are parallelograms. 14. A sphere is a solid figure described by the revolution of a semicircle about its diameter, which remains fixed. 15. The axis of a sphere is the fixed straight line about which the semicircle revolves. 16. The centre of a sphere is the same with that of the semicircle. 17. 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. 18. 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. 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 cone; and if greater, an acute-angled cone. 19. The axis of a cone is the fixed straight line about which the triangle revolves. 20. The base of a cone is the circle described by that side containing the right angle which revolves. 21. A cylinder is a solid figure described by the revolution of a right-angled parallelogram about one of its sides which remains fixed. 22. The axis of a cylinder is the fixed straight line about which the parallelogram revolves. 23. The bases of a cylinder are the circles described by the two revolving opposite sides of the parallelogram. 24. Similar cones and cylinders are those which bave their axes and the diameters of their bases proportionals. 25. A cube is a solid figure contained by six equal squares. 26. A tetrahedron is a solid figure contained by four equal and equilateral triangles. 27. An octahedron is a solid figure contained by eight equal and equilateral triangles. 28. A dodecahedron is a solid figure contained by twelve equal pentagons which are equilateral and equiangular. 29. An icosahedron is a solid figure contained by twenty equal and equilateral triangles. A. A parallelepiped is a solid figure contained by six quadrilateral figures, of which every opposite two are parallel PROPOSITION 1. THEOREM. One part of a straight line cannot be in a plane, and another part without it. If it be possible, let AB, part of the straight line ABC, be in a plane, and the part BC without it. A B Then 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 D straight line ĀD,and be turned about until it pass through the point C. Then, because the points B and C are in this plane, the straight line BC is in it. [I. Definition 7. Therefore there are two straight lines ABC, ABD in the same plane, that have a common segment AB; but this is impossible. [I. 11, Corollary. Wherefore, one part of a straight line &c. Q.E.D. PROPOSITION 2. 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 the two straight lines AB, CD cut one another at E: AB and CD shall be in one plane ; and the three straight lines EC, CB, BE which meet one another, shall be in one plane. Let any plane pass through the А. 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 and C are in this plane, the straight line EC is in it; [I. Definition 7. for the same reason, the straight line BC is in the same plane; and, by hypothesis, EB is in it. Therefore the three straight lines EC, CB, BE are in one plane. But AB and CD are in the plane in which EB and EC are; (XI. 1. therefore AB and CD are in one plane. Wherefore, two straight lines &c. Q.E.T. B PROPOSITION 3. THEOREM. If two planes cut one another their common section is a straight line. Let two planes AB, BC cut one another, and let BD be their common section: BD shall be a straight line. If it be not, from B to D, draw in the plane AB the straight line BED, and in the plane BC the straight line BFD. [Postulate 1. Then the two straight lines BED, BFD have the same extremities, and therefore include a space between them; but this is impossible. [Axiom 10. Therefore BD, the common section of the planes AB and BC cannot but be a straight line. Wherefore, if two planes &c. Q.E.D. PROPOSITION 4. THEOREM. If a straight line stand at right angles to each of troo straight lines at 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, at E, the point of their intersection: EÉ 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; join AD, CB; through E draw in the plane in which are AB, CD, any straight line cutting AD'at G, and CB at H; and from any point Fin EF draw FA, FG, FD, FC, FH, FB, |