2. The locus of points, such that the distances of each one of them from two fixed points are in a constant ratio, is a sphere. 3. Find the locus of the centres of sections made in a sphere by planes which pass through a given point. 4. Find the locus of the centres of sections made in a given sphere by planes which contain a given straight line. 5. Find the locus of the points of contact of tangents drawn from the same point to a given sphere. 6. Find the locus of points of contact of planes tangent to a sphere and parallel to a given straight line. 7. Find the locus of the poles of great circles making a given angle with a given great circle. 8. Find the locus of all the points on the surface of a sphere, each one of which is equidistant from two given points on the surface. 9. Find the locus of all the points on the surface of a sphere, each one of which is equidistant from two great circles of the sphere. 10. The locus of the vertices of a spherical triangle, whose vertical angle is equal to the sum of the other two, is a small circle of the sphere find its pole and polar distance. II. Find the locus of the points from which a given straight line appears under the same angle. 12. Find the locus of the centres of spheres of a given radius tangent to a given plane. 13. Find the locus of centres of spheres of a given radius tangent externally to a given sphere. 14. What is the locus of the centres of spheres tangent to a given plane at a given point? to a given sphere at a given point? 15. Find the locus of the centres of spheres of a given radius and passing through two given points. 16. Find the locus of the centres of spheres which have three points in common. 17. Find the locus of the centres of spheres of given radius which are tangent to two given intersecting planes. 18. Find the locus of centres of spheres tangent to three given planes. 19. Find the locus of the centres of spheres of given radius tangent to two given spheres. PROBLEMS. 1. Bisect a given arc of a great circle. 2. Bisect the angle contained by two given arcs of great circles. 3. Circumscribe a circle about a given spherical triangle. 4. Inscribe a circle in a given spherical triangle. Scholium The pole of the circle inscribed in a spherical triangle is also the pole of the circle circumscribed about the polar triangle; and the radii of these circles are complements of each other. 5. At a point on an arc of a great circle draw a second arc making a given angle with the first. 6. From a point without an arc of a great circle draw a second arc, making a given angle with the first. 7. Construct a spherical triangle, given an angle and two adjacent sides. Application Given the latitudes and difference of longitudes of two places on the earth, construct the distance between them, i. e., the arc of great circle which joins them (regarding the earth as a sphere). 8. Construct a spherical triangle, given one side and the adjacent angles. 9. Construct a spherical triangle, given three sides. Application: Given the latitudes of two places on the earth and the distance between them, construct the difference of longitude. 10. Construct a spherical triangle, knowing the three angles. NOTE. The above problems may be solved graphically on a globe with a pair of spherical dividers or some simple substitute for these. In the absence of a globe the solutions can be indicated merely. 11. Compute the area of the spherical triangle in terms of the trirectangular triangle, given its angles 61°, 109°, and 127°. 12. Compute the area of the spherical triangle in terms of the trirectangular triangle, its angles being 52° 36′, 72° 15', 87° 40′. 13. Compute the angles of a spherical triangle, knowing that they are to each other as the numbers 4, 6, and 7, the area of the triangle being one-fourth of the tri-rectangular triangle. 14. Given the radius of the inscribed sphere in each case, compute the edges and volume of the regular tetraedron, hexaedron, and octaedron. 15. Given the side of a cube, find the diameter of the circumscribing sphere by a plane construction. 16. A tangent to the earth's surface (regarded as a sphere) from the top of a vertical pole ten feet high touches the earth at a distance of (about) four miles from the pole. Find the radius of the earth in miles. Indicate the solutions of the following problems: 17. Construct a sphere of given radius which shall pass through three given points. 18. Construct a sphere of given radius which shall pass through two given points and be tangent to a given plane. 19. Construct a sphere of given radius which shall pass through a given point and be tangent to two given planes which intersect each other. 20. Construct a sphere of given radius which shall be tangent to three given planes. 21. Construct a sphere of given radius which shall be tangent to a given sphere and pass through two given points. 22. Construct a sphere of given radius which shall pass through a given point and be tangent to two given spheres. 23. Construct a sphere of given radius which shall pass through a given point and be tangent to a given plane and a given sphere. BOOK VIII. MEASURES OF THE THREE ROUND BODIES. DEFINITIONS. 1. A cylinder of revolution is the solid generated by the revolution of a rectangle, ABCD, conceived to turn about the immovable side AB. In this movement the sides AD, BC, remaining always perpendicular to AB, describe the equal circles DHP, CGQ, which are called the bases of the cylinder, and the side CD describes its convex surface. The immovable line AB is styled the axis of the cylinder. P A E N M F G K L Every section, KLM, made in the cylinder perpen- B dicular to the axis, is a circle equal to each of the bases: for, whilst the rectangle ABCD turns about AB, the line IK, perpendicular to AB, describes a circular plane equal to the base, and this plane is nothing else than the section made perpendicular to the axis at the point I. Every section, PQGH, made through the axis, is a rectangle, double the generating rectangle ABCD. NOTE. The cylinder of revolution is often called a right cylinder with circular base. 2. A cone of revolution is the solid generated by the revolution of the right-angled triangle SAB, conceived to turn about the immovable side SA. In this movement the side AB describes a circle, BDCE, which is called the base of the cone, and the hypothenuse, SB, describes its convex surface. The point S is called the vertex of the cone, SA the axis, or the altitude, and SB the side, or slant height. Every section, HKFI, made perpendicular to the axis, is a circle; every section, SDE, made through the axis, is an isosceles triangle, double the generating triangle SAB. NOTE.-The cone of revolution is often called a right cone with circular base. 3. If from the cone SCDB we cut off the cone SFKH, by a section parallel to the base, the remaining solid, CBHF, is called a truncated cone, or frustum of a cone. S It may be conceived as described by the revolution of the trapezoid ABHG, whose angles, A and G, are right, about the side AG. The immovable line AG is called the axis, or the altitude of the frustum, the circles BDC, HFK are its bases, and BH is its side. 4. Two cylinders, or two cones, are similar when they are generated by similar rectangles, or by similar triangles respectively, (i. e.) when their axes are to each other as the diameters of their bases. 5. If, in the circle ACD, which forms the base of a cylinder, a polygon, ABCDE, be inscribed, F and on the base, ABCDE, a right prism be erected, equal in altitude to the cylinder, the prism is said to be inscribed in the cylinder, or the cylinder to be circumscribed about the prism. It is evident that the edges, AF, BG, CH, etc., of the prism, being perpendicular to the plane of the base, are included in the convex surface of the cylinder; hence, the prism and the cylinder touch each other along these edges. 6. Similarly, if ABCD is a polygon circumscribed about the base of a cylinder, and on the base, ABCD, a right prism be erected, equal in altitude to the cylinder, the prism is said to be circumscribed about the cylinder, or the cylinder inscribed in the prism. F Let M, N, etc., be the points of contact of the sides AB, BC, etc., and at the points M, N, etc., A let MX, NY, etc., be drawn perpendicular to the plane of the base; it is evident that these perpendiculars will be at the same time in the sur N S B E H R Ꮓ K A H |