Book VI. Soc N. a. 5.4. IF a PROP. B. THEO R. F an angle of a triangle be bisected by a straight line, which likewife cuts the bafe; the rectangle contained by the fides of the triangle is equal to the rectangle contained by the fegments of the bafe, together with the fquare of the ftraight line bifecting the angle. Let ABC be a triangle, and let the angle BAC be bifected by the ftraight line AD; the rectangle BA, AC is equal to the rectangle BD, DC together with the fquare of AD. 2 A C D Defcribe the circle ACB about the triangle, and produce AD to the circumference in E, and join EC. then because the angle BAD is equal to the angle CAE, and the b. 21. 3. angle ABD to the angle ↳ AEC, for they are in the fame fegment; the triangles ABD, AEC are equiangular to one another. therefore as BA to AD, fo is EA to AC, and confequently the rectangle BA, AC is equal to the rectangle EA, AD, that is to the rectangle ED, DA together with the fquare of C. 4. 6. d. 16. 6. C. 3.2. f. 35. 3. Sce N. B E f AD. but the rectangle ED, DA is equal to the rectangle ! BD, IF PROP. C. THEOR. F from an angle of a triangle a ftraight line be drawn perpendicular to the base; the rectangle contained by the fides of the triangle is equal to the rectangle contained by the perpendicular and the diameter of the circle defcribed about the triangle. Let ABC be a triangle, and AD the perpendicular from the angle A to the bafe BC; the rectangle BA, AC is equal to the rectangle contained by AD and the diameter of the circle defcribed about the triangle. Defcribe the circle ACB about the triangle, and draw its diameter AE, and join EC. becaufe the right angle BDA is equal to the angle ECA in a femicircle, and the angle ABD to the angle AEC in the fame fegment; the triangles ABD, AEC are equiangular. therefore as BA to AD, fo is EA to AC, and confequently the rectangle BA, AC is equal to the rectangle EA, AD. If therefore from an angle, &c. e. 16. 6. O. E. D. TH PROP. D. THEO R. 'HE rectangle contained by the diagonals of a quadrilateral infcribed in a circle, is equal to both the rectangles contained by its oppofite fides. Let ABCD be any quadrilateral infcribed in a circle, and join AC, BD; the rectangle contained by AC, BD is equal to the two rectangles contained by AB, CD and by AD, BC *. Make the angle ABE equal to the angle DBC; add to each of thefe the common angle EBD, then the angle ABD is equal to the angle EBC. and the angle BDA is equal to the angle BCE, because a. 21. 3. they are in the fame fegment; therefore the triangle ABD is equi angular to the triangle BCE. where a forebas BC is to CE, fo is BD to DA, B, and confequently the rectangle BC, C E BCD. as therefore BA to AE, fo is D A BD to DC; wherefore the rectangle. b. 4. 6. c. 16. 6. BA, DC is equal to the rectangle BD, AE. but the rectangle BC, AD has been fhewn 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. Therefore the rectangle, &c. Q_E. D. This is a Lemma of Cl. Ptolomaeus in page 9. of his meydan oúvražıs. Bock XI. THE ELEMENT S OF E U CL L I D. BOOK XI. A so DEFINITION S. I. SOLID is that which hath length, breadth, and thickness. 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 ftraight line meeting it in that plane. IV. A plane is perpendicular to a plane, when the straight lines drawn in one of the planes perpendicularly to the common section of the two planes, are perpendicular to the other plane. V. The inclination of a straight line to a plane is the acute angle contained by that straight line, and another drawn from the point in which the first line meets the plane, to the point in which a perpendicular to the plane drawn from any point of the firft line above the plane, meets the fame plane. VI. The inclination of a plane to a plane is the acute angle contained by two ftraight lines drawn from any the fame point of their common fection at right angles to it, one upon one plane, and 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. VIII. Parallel planes are fuch which do not meet one another tho' produced. IX. Book XI. A folid angle is that which is made by the meeting of more than see N. two plane angles, which are not in the fame plane, in one point. X. 'The tenth Definition is omitted for reasons given in the Notes.' See N. XI. Similar folid figures are fuch as have all their folid angles equal, See N. each to each, and which are contained by the fame number of fimilar planes. 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 Prism 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 femi circle about its diameter, which remains unmoved. XV. The axis of a sphere is the fixed ftraight line about which the femicircle revolves. XVI. The center of a sphere is the fame with that of the femicircle. XVII. The diameter of a sphere is any ftraight line which paffes thro the center, and is terminated both ways by the fuperficies of the sphere. 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, Book XI. If the fixed fide be equal to the other fide containing the right angle, the Cone is called a right angled Cone; if it be lefs than the other fide, an obtufe angled, and if greater, an acute angled Cone. XIX. The axis of a Cone is the fixed ftraight line about which the triangle revolves. XX. The bafe 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 opposite fides of the parallelogram. XXIV. Similar cones and cylinders are those which have their axes and the diameters of their bafes proportionals. XXV. A Cube is a folid figure contained by fix equal squares. XXVI. A Tetrahedron is a folid figure contained by four equal and equilateral triangles. XXVII. An Octahedron is a folid figure contained by eight equal and equilateral triangles. XXVIII. A Dodecahedron is a folid figure contained by twelve equal pentagons which are equilateral and equiangular. XXIX. An Icofahedron is a folid figure contained by twenty equal and equilateral triangles. DE F. A. A Parallelepiped is a folid figure contained by fix quadrilateral figures whereof every opposite two are parallel. |