EXERCISES. 1. Parallel sections of a prism are equal polygons. 2. The diagonals of a parallelopiped bisect one another. 3. Two square prisms are of equal altitude; a side of the base of one is twice a side of the base of the other; are they similar? 4. The cell of a honey-bee is a hexagonal prism; can a number of equal cells exactly fit into each other? What is the lateral surface of a cell, each of whose sides is 1 inch long and of an inch wide? What is the lateral surface of a square cell of the same length and same volume? 5. The sum of the squares of the four diagonals of a parallelopiped is equal to the sum of the squares of its twelve edges. 6. The volume of a right truncated* triangular prism is equal to the product of its base and one third the sum of its lateral edges. Divide the solid into three pyramids, as in the case of the frustum of a pyramid. 7. The lateral surface of a regular pyramid is greater than its base. 8. If a cylinder be cut by a plane through an element, the section is a parallelogram. 9. If a cone be cut by a plane through an element, the section is a triangle. 10. If a cone or cylinder be cut by a plane parallel to the base, the section is a circle. * A truncated prism is one cut off by a plane oblique to the base. 11. Two similar cylinders are on circles 2 and 3 feet in diameter; what is their relative magnitude? What is the relative magnitude of similar cones on the same bases? If the altitude of one cone be 10 feet, what is the altitude of the other? 12. Suppose that two men be similar solids, one 5, the other 6 feet high; what are the relative amounts of cloth required to cover them? What are their relative weights? 13. Two similar bottles have the diameters of their bases in the proportion of 1 to 2; what is the ratio of their surfaces? of their contents? 14. A cone is cut in two by a plane parallel to the base through the middle point of the altitude; what portion of the volume is cut off toward the vertex? 15. A cylinder is bent around so as to form a circular ring; if we know the inner diameter and thickness of the ring, deduce a rule for finding its volume. 16. To find the ratio of the volumes of two cylinders whose convex areas are equal. This and the following examples may best be worked out algebraically by using the formulæ for volumes and areas. 17. To find the ratio of the convex areas of two cylinders whose volumes are equal. 18. To find the ratio of the volumes of two cylinders generated by successively revolving a rectangle about its two adjacent sides. 19. What is the convex surface and the volume of a cone, the section of which, through the axis, is an equilateral triangle, whose side is unity? What is the value of this side, for the total surface to be a square metre? 20. The square of the diagonal of a rectangular parallelopiped is equal to the sum of the squares of its sides. BOOK IX. SPHERES. DEFINITIONS. 1. A sphere is a solid generated by the revolution of a semicircle about its diameter. Corollary. The centre of the semicircle is also the centre of the sphere; hence, any point on the surface is equally distant from the centre. 2. The distance from the centre to the surface is called a radius, and a line through the centre terminating both ways in the surface is called a diameter. Corollary.-Every diameter being equal to two radii, all diameters of a sphere are equal. 3. It will be proved that every section of a sphere by a plane is a circle. If the plane pass through the centre of the sphere, the circle is called a great circle; if not, a small circle. Corollary. The diameter of a great circle is also the diameter of the sphere; and the intersection of the planes of two great circles is a diameter of the sphere, for both planes pass through the centre. 4. A line through the centre of a circle of a sphere perpendicular to its plane is called its axis: and the intersections of 17* 197 this line with the spherical surface are the poles of the circle. 5. A spherical triangle is the space enclosed on the surface of a sphere by the arcs of three great circles, each arc being less than a semi-circumference. 6. A spherical polygon is the space enclosed on the surface of a sphere by arcs of any number of great circles. 7. A spherical angle is the angle between two tangents to the arcs of the great circles which form the angle, at the point where they meet. 8. A spherical sector is a solid formed by the revolution of a circular sector about any radius of the circle. 9. A spherical segment is the portion of a sphere cut off by a plane. 10. A spherical zone is the surface of a sphere between two parallel planes. Its altitude is the perpendicular disance between the planes. 11. A lune is the portion of the surface of a sphere between the semi-circumferences of two great circles. Proposition 1. Theorem.-Every section of a sphere made by a plane is a circle. Let ABC be any section of a sphere made by a plane; it is a circle. Take D, the centre of the sphere, and draw (VII. 10) DF perpendicular to the plane of ABC; also join D with any three points of the curve ABC, as G, B, C. From E, where DF meets the plane, draw EC, EB, EG. The angles DEC, DEB, DEG are right angles; also DB, DC, DG are equal, because they are radii of the sphere, and DE is common to the triangles DEB, DEC, DEG. Therefore (I. 42, Cor. 3) EC, EB, EG are F E G equal; and the same is true of all lines drawn from E to the circumference ABC. Therefore ABC is a circle, and E is its centre. Corollary 1.—The pole of a circle is equally distant from all points in it. For if CF, GF be joined, we may prove in the triangles CEF, GEF that the chords CF, GF are equal. Hence the arcs CF, GF are equal, and the same may be proved of any arcs from F to ABC. Corollary 2.-Any circle made by a plane parallel to ABC would have the same axis and pole. Proposition 2. Theorem.-A pole of a great circle is at the distance of a quadrant from any point of the circle; and, conversely, if arcs of two great circles between any point on the surface of a sphere and another great circle be quadrants, that point is the pole of the great circle. |