and they are upon equal straight lines BC, CK: but similar segments of circles upon equal straight lines, are equal (24. 3.) to one another: therefore the segment BXC is equal to the segment COK: and the triangle BGC is equal to the triangle CGK; therefore the whole, the sector BGC, is equal to the whole, the sector CGK: for, the same reason, the sector KGL is equal to each of the sectors BGC, CGK: in the same manner, the sectors EHF, FHM, MHN may be proved equal to one another : therefore, what multiple soever the circumference BL is of the circumference BC, the same multiple is the sector BGL of the sector BGC: for the same reason, whatever multiple the circumference EN is of EF, the same multiple is the sector EHN of the sector EHF and if the circumference BL be equal to EN, the sector BGL is equal to the sector EHN and if the circumference BL be greater than EN, the sector BGL is greater than the sector EHÑ; and if less, less: since, then, there are four magnitudes, the two circumferences BC, EF, and the two sectors BGC, EHF, and of the circumference BC, and sector BGC, the circumference BL and sector BGL are any equal multiples whatever: and of the circumference EF, and sector EHF, the circumference EN and sector EHN are any equimultiples whatever; and that it has been proved, if the circumference BL be greater than EN, the sector BGL is greater than the sector EHN; and if equal, equal; and if less, less. Therefore (5. def. 5), as the circumference BC is to the circumference EF, so is the sector BGC to the sector EHF. Wherefore, in equal circles, &c. Q. E. D. PROP. B. THEOR. Ir an angle of a triangle be bisected by a straight line, which likewise cuts the base; the rectangle contained by the sides of the triangle is equal to the rectangle contained by the segments of the base, together with the square of the straight line bisecting the angle.* Let ABC be a triangle, and let the angle BAC be bisected by the straight line AD; the rectangle BA, AC is equal to the rectangle BD, DC, together with the square of AD. B A Describe the circle (5. 4.) 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 angle ABD to the angle (21. 3.) AEC, for they are in the same. segment the triangles ABD, AEC are equiangular to one another: therefore as BA to AD, so is (4. 6.) EA to AC, and consequently the rectangle BA, AC is equal (16. 6.) to the rectangle EA, AD, that is D E (3. 2.), to the rectangle ED, DA, together with the square of AD but the rectangle ED, DA is equal to the rectangle (35. 3.) BD, DC. Therefore the rectangle BA, AC is equal to the rectangle BD, DC, together with the square of AD. fore, if an angle, &c. Q. E. D. PROP. C. THEOR. Where IF from any angle of a triangle a straight line be drawn perpendicular to the base; the rectangle contained by the sides of the triangle is equal to the rectangle contained by the perpendicular and the diameter of the circle described about the triangle.* Let ABC be a triangle, and AD the perpendicular from the angle A to the base BC; the rectangle BA, AC is equal to the rectangle contained by AD and the diameter of the circle described about the triangle. • See Notes. Describe (5. 4.) the circle ACB about the triangle, and draw its diameter AE, and join EC: because the right angle BDA is equal (31. 3.) to the angle ECA in a semicircle, and B the angle ABD to the angle AEC in the same segment (21. 3.); the triangles ABD, AEC are equiangular : therefore as (4. 6.) BA to AD, so is EA to AC; and consequently the rectangle BA, AC is equal (16. 6.) to the rectangle EA, AD. If, therefore, from an angle, &c. Q. E. D. PROP. D. THEOR. THE rectangle contained by the diagonals of a quadrilateral inscribed in a circle, is equal to both the rectangles contained by its opposite sides.* Let ABCD be any quadrilateral inscribed 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.t Make the angle ABE equal to the angle DBC; add to each of these the common angle EBD, then the angle ABD is equal to the angle EBC; and the angle BDA is equal (21. 3.) to the angle BCE, because they are in the same segment; therefore the triangle ABD is equiangular to the triangle BCE; wherefore (4. 6.) as B BC is to CE, so is BD to DA; and consequently the rectangle BC, AD is equal (16. 6.) to the rectangle BD, CE: again, because the angle ABE is equal to the angle DBC, and the angle (21.3.) BAE to the angle BDC, the triangle ABE is equiangular to the triangle BCD: as therefore BA to AE, so is 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 whole rectangle AC, BD (1. 2.) is equal to the rectangle AB, DC, together with the rectangle AD, BC. Therefore the rectangle, &c. Q. E. D. • See Note. E D A †This is a Lemma of Cl. Ptolomæus, in page 9 of his μayaan curtağı. Bb THE ELEMENTS OF EUCLID. BOOK XI. DEFINITIONS. I. A SOLID is that which hath length, breadth, and thickness. II. That which bounds a solid is a superficies. III. 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. 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 first line above the plane, meets the same plane. VI. 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 upon one plane, and the other upon the other plane. VII. 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. VIII. Parallel planes are such which do not meet one another though produced. IX. A solid angle is that which is made by the meeting of more than two plane angles, which are not in the same plane, in one point.* X. 'The tenth definition is omitted for reasons given in the notes.** XI. Similar solid figures are such as have all their solid angles equal, each to each, and which are contained by the same number of similar planes.* 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 centre of a sphere is the same with that of a semicircle. XVII. 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. XVIII. 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. * See Notes. |