C 4. 6. Book VI. angle DBE be added to each of them, the angle ABD is equal to EBC. Because the angle ABD is equal B С b 21. 3. segment BCDA6; the triangles ABD, BCE are equiangular: Wherefore", confequently the rectangle BD, CE d:6.6. is equal to the rectangle AD, BC. Again, because the angle ABE is А. 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, lo cis BD to DC; wherefore the rectangle BA, DC is equal to the rectangle BD, AE: But the rectangle BC, AD has been shown equal to the rectangle BD, CE; therefore the B e 1.2. whole rectangle AC, BD is equal to the rectangle AB, DC, together С fore, &c. Q. E. D. Cor. If AD be equal to DC, or the angle ABD equal to DBC, the E rectangle AC, BD is equal to the A rectangles AD, BC and AD, AB; that is, to the rectangle contained by AD and the sum of AB and BC. Consequently AB, BC together are to BD, as AC to AD, PROP. E. THEOR. I F two points be taken in the semidiameter of a circle, such that the rectangle contained by the segments between them and the centre is equal to the square of the semidiameter : straight lines drawn from these points to any point of the circumference Thall have the same ratio that the segments 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 same side of the centre, so that the rectangle ED, DF is equal to the {quare of AD; and draw EB, FB to any point B of the circumference : EB is to FB, as EA to AF. € 32. I. Join AB, BD; and because the rectangle ED, DF is equal Book. VI. to the square of AD or DB, FD is to DB, as - DB to DE; that is, the sides of the tri a 17.6. angles FBD, EBD, about G their common angle D, are proportionals; therefore the triangles are equiangular b 6.6. С and have the angle FBDF equal to BED: But BED is equal o to the two EAB, ABE; therefore the two EAB, ABE are equal to FBD: of which EAB is equal d to ABD, because BD is equal d 5. 2. 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 eis FA to AE. Therefore, e 3.6. &c. Q. E. D. Cor. Hence, AB bisects the angle FBE. And if BC be joined, and FB produced to G: because the angle ABC in a semicircle is a right angle f, it is half the sum of the angles [ 31. 3. FBE and EBG 8; 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 bisects the exterior angle EBG. А I. II. III. when it makes right angles with every straight line meeting IV. drawn in one of the planes perpendicularly to the common V. contained by that straight line, and another drawn from the VI. See N. The inclination of a plane to a plane is the angle contained by two straight lines drain from any the same point of their common common section at right angles to it, one upon one plane, and Book XI. the other upon the other plane. VII. one another, which two other planes have, when the said VIII. Parallel planes are such as do not meet one another though produced. IX. ber of fimilar planes having the same inclination to one an- See N. See N. two plane angles, which are not in the same plane, in one point, the inclinations of all the planes being inwards. A. XII. tuted betwixt one plane and one point above it in which they meet. XIII two that are opposite are equal, fimilar, and parallel to one XIV. A sphere is a solid figure described by the revolution of a semicircle about its diameter, which remains unmoved. XV. XVI. XVII. through the centre, and is terminated both ways by the super- XVIII. angled triangle about one of the fides containing the right XIX. Y 2 Book XI. XIX. The axis of a cone is the fixed straight line about which the triangle revolves. XX. XXI. angled parallelogram about one of its fides, which remains XXII. The axis of a cylinder is the fixed straight line about which the parallelogram revolves. XXIII. XXIV. the diameters of their bases proportionals. PROP. I. THEOR. VE part of a straight line cannot be in a plane, and another part above it. If it be pofible, let AB, part of the straight line ABC, be in the plane, and the part BC above it : And since the straight line AB is in the plane, it can Le produced in that plane : Let it be produced to D: And let any plane pass through the straight line AD, and be turned A about it until it pass through the point C; and because the points B, C, are in this plane, a 7. Def. 1. the straight line BC is in it ?: Therefore there are two straight lines ABC, ABD in the same plane that have a common segb10Ax.I. ment AB; which is impoffible b. Therefore one part, &c. Q. E. D. B PROP. II. THEOR. TWO firaight lines which cut one another are in one plane, and three straight lines which meet one another are in one plane. Let |