BOOK VI. angle DBE be added to each of them, the angle ABD is equal to EBC. A B E D Because the angle ABD is equal to EBC, and the angle BDA to the angle BCE, for they are in the same b21. 3. fegment BCDA; the triangles ABD, €4.6. BCE are equiangular: Wherefore, as BD to DA, fo is BC to CE; and confequently the rectangle BD, СЕ d16. 6. is equal d to the rectangle AD, BC. Again, because the angle ABE is equal to the angle DBC, and the angle BAE to the angle b BDC; the triangle ABE is equiangular to BCD: as, therefore, BA to AE, fo is BD to DC; wherefore the rectangle BA, DC is equal to the rectangle BD, AE: But the rectangle BC, AD has been fhown equal to the rectangle BD, CE; therefore the whole rectangle AC, BD is equal to the rectangle AB, DC, together with the rectangle AD, BC. Wherefore, &c. Q. E. D. e 1.2. COR. If AD be equal to DC, or B E C Consequently AB, BC toge PROP. E. THEOR. I circle, much that the rectangle contained by the F two points be taken in the femidiameter of a fegments between them and the centre is equal to the fquare of the femidiameter: ftraight lines drawn from these points to any point of the circumference fhall have the fame ratio that the fegments of the diameter between them and the circumference have to one another. Let ABC be a circle, of which AC is the diameter, and D the centre; and let E, F be two points in AC, on the fame fide of the centre, so that the rectangle ED, DF is equal to the fquare of AD; and draw EB, FB to any point B of the circumference: EB is to FB, as EA to AF. Join AB, BD; and because the rectangle ED, DF is equal Book. VI. to the fquare of AD or DB, FD is to DB, as a DB to DE; that is, the fides of the tri angles FBD, EBD, about a 17.6. b their common angle D, are C EAB, ABE are equal to d FBD: of which EAB is equal to ABD, because BD is equal d 5. 1. to DA; therefore the remaining angle ABE is equal to the remaining angle ABF; that is, the angle FBE is bifected by BA: Wherefore, as FB to BE, fo is FA to AE. Therefore, e 3. 6. &c. Q. E. D. COR. Hence, AB bifects the angle FBE. And if BC be joined, and FB produced to G: because the angle ABC in a femicircle is a right angle f, it is half the fum of the angles f 31.3. FBE and EBG ; of which the angle ABE is the half of g 13. 1. FBE; therefore the remaining angle EBC is the half of the remaining angle EBG: Therefore BC bifects the exterior angle EBG. THE ELEMENTS OF EUCLID. BOOK XI. DEFINITIONS. I. SOLID is that which hath length, breadth, and thicknels. II. That which bounds a folid is a fuperficies. III. A ftraight line is perpendicular, or at right angles to a plane, when it makes right angles with every straight line meeting it in that plane. IV. A plane is perpendicular to a plane, when the ftraight lines drawn in one of the planes perpendicularly to the common fection of the two planes, are perpendicular to the other plane. V. The inclination of a ftraight line to a plane is the acute angle contained by that straight line, and another drawn from the point in which the firft line meets the plane, to the point in which a perpendicular to the plane drawn from any point of the first line above the plane, meets the fame plane. VI. See N. The inclination of a plane to a plane is the angle contained by two straight lines drawn from any the fame point of their common 171 common fection at right angles to it, one upon one plane, and Book XI. the other upon the other plane. VII. Two planes are faid to have the fame, or a like inclination to one another, which two other planes have, when the faid angles of inclination are equal to one another. Parallel planes are fuch as do not meet one another though produced. IX. Similar folid figures are fuch as are contained by the fame number of fimilar planes having the fame inclination to one another. X. Omitted. A folid angle is that which is made by the meeting of more than two plane angles, which are not in the fame plane, in one point, the inclinations of all the planes being inwards. A. A parallelopiped is a folid figure contained by fix quadrilateral figures, whereof every oppofite two are parallel. XII. A pyramid is a folid figure contained by planes that are conftituted betwixt one plane and one point above it in which they meet. XIII. A prifm is a folid figure contained by plane figures of which two that are oppofite are equal, fimilar, and parallel to one another; and the others parallelograms. XIV. A sphere is a folid figure defcribed by the revolution of a femicircle about its diameter, which remains unmoved. XV. The axis of a sphere is the fixed straight line about which the femicircle revolves. XVI. The centre of a fphere is the fame with that of the femicircle. XVII. The diameter of a sphere is any ftraight line which paffes through the centre, and is terminated both ways by the superficies of the fphere. XVIII. A cone is a folid figure described by the revolution of a right angled triangle about one of the fides containing the right angle, which fide remains fixed. See N. See N. See N. The axis of a cone is the fixed ftraight line about which the triangle revolves. XX. The base of a cone is the circle defcribed by that fide containing the right angle, which revolves. XXI. A cylinder is a folid figure described by the revolution of a right angled parallelogram about one of its fides, which remains fixed. XXII. The axis of a cylinder is the fixed ftraight line about which the parallelogram revolves. XXIII. The bafes of a cylinder are the circles defcribed by the two revolving oppofite fides of the parallelogram. XXIV. Similar cones and cylinders are those which have their axes and the diameters of their bafes proportionals. Ο PROP. I. THEOR. NE part of a firaight line cannot be in a plane, and another part above it. If it be poffible, let AB, part of the flraight line ABC, be in the plane, and the part BC above it: And fince the straight line AB is in the plane, it can be produced in that plane: Let the point C; and becaufe the points B, C, are in this plane, a 7.Def. 1. the ftraight line BC is in it: Therefore there are two straight lines ABC, ABD in the fame plane that have a common fegbio.Ax.. ment AB; which is impoffible. Therefore one part, &c. Q. E. D. PROP. II. THEOR. WO ftraight lines which cut one another are in one plane, and three ftraight lines which meet one another are in one plane. Let |