drawn to the opposite angles of the square described on the base, cut the diagonals of the square in E and F: prove that the line EF is parallel to the base. 98. Inscribe a square in a segment of a circle. 99. Inscribe a square in a sector of a circle, so that the angular points shall be one on each radius, and the other two in the circumference. 100. Inscribe a square in a given equilateral and equiangular pentagon. 101. Inscribe a parallelogram in a given triangle similar to a given parallelogram. 102. If any rectangle be inscribed in a given triangle, required the locus of the point of intersection of its diagonals. 103. Inscribe the greatest parallelogram in a given semicircle. 104. In a given rectangle inscribe another, whose sides shall bear to each other a given ratio. 105. In a given segment of a circle to inscribe a similar segment. 106. The square inscribed in a circle is to the square inscribed in the semicircle :: 5 : 2. 107. If a square be inscribed in a right-angled triangle of which one side coincides with the hypotenuse of the triangle, the extremities of that side divide the base into three segments that are continued proportionals. 108. The square inscribed in a semicircle is to the square inscribed in a quadrant of the same circle :: 8: 5. 109. Shew that if a triangle inscribed in a circle be isosceles, having each of its sides double the base, the squares described upon the radius of the circle and one of the sides of the triangle, shall be to each other in the ratio of 4: 15. 110. APB is a quadrant, SPT a straight line touching it at P, PM perpendicular to CA; prove that triangle SCT: triangle ACB:: triangle ACB: triangle CMP. 111. If through any point in the arc of a quadrant whose radius is R, two circles be drawn touching the bounding radii of the quadrant, and r, r' be the radii of these circles: shew that rr' = R. 112. If R be the radius of the circle inscribed in a right-angled triangle ABC, right-angled at A; and a perpendicular be let fall from A on the hypotenuse BC, and if r, r' be the radii of the circles inscribed in the triangles ADB, ACD: prove that r2 + r'2 = R2. XIV. 113. If in a given equilateral and equiangular hexagon another be inscribed, to determine its ratio to the given one. 114. A regular hexagon inscribed in a circle is a mean proportional between an inscribed and circumscribed equilateral triangle. 115. The area of the inscribed pentagon, is to the area of the circumscribing pentagon, as the square on the radius of the circle inscribed within the greater pentagon, is to the square on the radius of the circle circumscribing it. 116. The diameter of a circle is a mean proportional between the sides of an equilateral triangle and hexagon which are described about that circle. 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 perpendicular 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 as do not meet one another though produced. IX. A solid angle is that which is made by the meeting, in one point, of more than two plane angles, which are not in the same plane. X. Equal and similar solid figures are such as are contained by similar planes equal in number and magnitude. XI. 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. P 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 center of a sphere is the same with that of the semicircle. XVII. The diameter of a sphere is any straight line which passes through the center, and is terminated both ways by the superficies of the sphere. XVIII. A cone is a solid figure described by the revolution of a rightangled 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; and if greater, an acute-angled cone. XIX. The axis of a cone is the fixed straight line about which the triangle revolves. XX. The base of a cone is the circle described by that side containing the right angle, which revolves. XXI. A cylinder is a solid figure described by the revolution of a rightangled parallelogram about one of its sides which remains fixed. XXII. The axis of a cylinder is the fixed straight line about which the parallelogram revolves. XXIII. The bases of a cylinder are the circles described by the two revolving opposite sides of the parallelogram. XXIV. Similar cones and cylinders are those which have their axes and the diameters of their bases proportionals. XXV. A cube is a solid figure contained by six equal squares. XXVI. A tetrahedron is a solid figure contained by four equal and equilateral triangles. XXVII. An octrahedron is a solid figure contained by eight equal and equilateral triangles. XXVIII. A dodecahedron is a solid figure contained by twelve equal pentagons which are equilateral and equiangular. XXIX. An icosahedron is a solid figure contained by twenty equal and equilateral triangles. Def. A. A parallelopiped is a solid figure contained by six quadrilateral figures, whereof every opposite two are parallel. PROPOSITION I. THEOREM. One part of a straight line cannot be in a plane, and another part above it. If it be possible, let AB, part of the straight line ABC, be in the plane, and the part BC above it: C A B D and 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 straight line AD, and be turned about it, until it pass through the point C: and because the points B, C are in this plane, the straight line BC is in it: (I. def. 7.) therefore there are two straight lines ABC, ABD in the same plane that have a common segment AB; (I. 11. Cor.) which is impossible. Therefore, one part, &c. Q.E.D. PROPOSITION II. 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 two straight lines AB, CD cut one another in E; and three straight lines EC, CB, BE, which meet one another, shall be in one plane. Let any plane pass through the 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, C' are in this plane, the straight line EC is in it: (1. def. 7.) for the same reason, the straight line BC is in the same: therefore the three straight lines EC, CB, BE are in one plane; but in the plane in which EC, EB are, in the same are CD, AB: (XI. 1.) therefore, AB, CD are in one plane. Wherefore two straight lines, &c. Q.E.D. PROPOSITION III. THEOREM. If two planes cut one another, their common section is a straight line. Let two planes AB, BC cut one another, and let the line DB be their common section. If it be not, from the point D to B, draw, in the plane AB, the straight line DEB, (post. 1.) and in the plane BC, the straight line DFB: then two straight lines DEB, DFB have the same extremities, and therefore include a space betwixt them; which is impossible: (1. ax. 10.) therefore BD, the common section of the planes AB, BC, cannot but be a straight line. Wherefore, if two planes, &c. Q.E.D. |