extremities with the other extremities of the base will cut off equal segments from the sides, each of them a mean proportional between the other two segments, 82. AB, AC are the sides of a regular pentagon and decagon inscribed in a circle whose centre is O; if OD be drawn to AB, bisecting the angle AOC, shew that the triangles ACB, ACD, as also the triangles AOB, DOB, are similar, and hence that AB2- AC?=(rad). 83. Two circles touch in C; if any point D be taken without them, such that the radii AC, BC subtend equal angles at D, and DE, DF be tangents to the circles, then ĎE.DF=DC2, 84. Bisect a triangle by a line drawn parallel to one of its sides. 85. One side of a right-angled triangle is double of the other : shew that the perpendicular from vertex on hypothenuse will cut off parts in the ratio of 1:4. 86. If two triangles are to one another as their bases, they have the same altitude; and triangles and parallelograms of unequal altitudes are to one another in the ratio compounded of the ratios of their bases and altitudes. 87. Describe a rhombus, equal to a given rectilineal figure, and having an angle equal to a given rectilineal angle. 88. If two triangles have one angle of the one equal or supplementary to one angle of the other, they are to one another in the ratio compounded of the ratios of the sides containing them. 89. Describe a square, having given the difference between the diagonal and a side. 90. ABCD is a quadrilateral; if any line be drawn cutting AB, CD, in a, d, AD, BC, in b, c, and AC, BD, in e, f, then ab: cd :: af x be: cf x de. 91. Prove that the ratio of the diagonal of a square to the side is incommensurable, and thence shew that its numerical value is nearly 1.4142. 92. Through a given point describe a circle touching a given line and a given circle. 93. Describe a circle touching two given circles and a given line. 94. Through a given point describe a circle touching two given circles. 95. Describe a circle touching three given circles. 96. In any triangle the intersections of perpendiculars from the angles on the sides, of lines from the angles bisecting the sides, and of perpendiculars bisecting the sides, are in one line, and their distances from one another as 1, 2, 3. 97. Let six points be taken in a plane, A, C, E, in one line, and B, D, F, in another ; then shew that the intersections of AB and DE, BC and EF, CD and FA, are in one line. 98. If perpendiculars be dropped from the baseangles of a triangle on the line bisecting the vertical angle, a circle passing through the points of intersection and the foot of the perpendicular from the vertical angle will bisect the base; and the area of the triangle will be equal to the rectangle contained by either perpendicular and the segment of the bisecting line between the angle and the other perpendicular. 99. AC, BE are parallel lines, F, G, H, &c. a series of equidistant points in AC; draw a line through B, cutting EF, EG, EH, &c. in f, g, h, &c., and shew that Bf, Bg, Bh, &c. are in harmonic proportion. 100. If pairs of common tangents be drawn to each two of three given circles, the intersections of each pair will lie in one line. BOOK XI. DEFINITIONS. 1. 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, the other upon the other plane. vil. 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 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 paral. lelograms. 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 the 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. If the fixed side be equal to the other side containing the right angle, the cone is called a right-angled cone, if less, 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 right-angled 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. xxy. 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 octahedron 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. |