ly to their bases, these sections are equivalent (47). If now the planes forming these sections be supposed to move, remaining always parallel to the bases, and each keeping the same distance from the vertex as the other, these sections, always being equivalent to each other, will move over equal volumes; therefore, as the altitudes are equal, the pyramids must be equivalent. THEOREM XII. 49. A triangular pyramid is one third of a triangular prism of the same base and altitude. Let C-DEF be a triangular pyramid and ABC-DEF be a triangular prism on the same base DEF; then C-DEF is one third of ABC-DEF. Taking away the pyramid C-DEF there remains the quadrangular pyramid whose vertex is C and base the parallelogram ABED. Through the points A, C, E pass a plane; it A D F B E will divide the pyramid C-A BED into two triangular pyramids, which are equivalent to each other (48), since their bases are halves of the parallelogram ABED, and they have the same altitude, the perpendicular from their vertex C to the base A BED. But the pyramid C-A B E, that is, E-A B C, is equivalent to the pyramid C-DEF, as they have equal bases A B C and D E F, and the same altitude (48). Therefore the three pyramids are equivalent and the given pyramid is one third of the prism. 50. Corollary. The volume of a triangular pyramid is equal to one third the product of its base by its altitude. THEOREM XIII. 51. The volume of any pyramid is equal to one third of the product of its base by its altitude. Let A-BCDEF be any pyramid; its volume is equal to one third the product of its base B C D E F by its altitude ▲ N. Planes passing through the vertex A and the diagonals of the base BD, BE, will divide the pyramid into triangular pyramids whose bases together compose the base of B the given pyramid and which have as their common altitude A N, the altitude of the given pyramid. The volume of the given. F A D E pyramid is equal to the sum of the volumes of the several triangular pyramids, which is equal to one third of the sum of their bases multiplied by their common altitude; that is, is equal to one third of the product of the base BCDFE by the altitude A N. 52. Cor. 1. As a cone is a right pyramid (33), this demonstration includes the cone. A cone, therefore, is one third of a cylinder, or of any pyramid, of equivalent base and the same altitude. If R: radius of the base, the altitude, and V the volume of a cone, V = π R2 A. = 53. Cor. 2. The ratio of similar pyramids to one another is the same as that of similar prisms; that is, as the cubes of homologous lines. THE SPHERE. DEFINITIONS. 54. A Sphere is a solid bounded by a curved surface, of which every point is equally distant from a point within called A sphere can be described by the revolution of a semicircle about its diameter, which remains fixed. the centre. 55. The Radius of a sphere is the straight line from the centre to any point of the surface. 56. The Diameter of a sphere is a straight line passing through the centre and terminating at either end at the surface. 57. Corollary. All the radii of a sphere are equal; all the diameters are equal, and each is double the radius. THEOREM XIV. 58. Every section of a sphere made by a plane is a circle. Let ABD be a section made by a plane cutting the sphere whose centre is C; then is A B D a circle. Draw CE perpendicular to the plane, and to the points A, D, F, where the plane cuts the surface of the sphere, draw CA, CD, CF. As CA, CD, CF are radii of the sphere they are B D equal, and are therefore equally distant from the foot of the perpendicular CE (IV. 7). Therefore EA, ED, EF are equal, and the section ABD is a circle whose centre is E. 59. Corollary. If the section passes through the centre of the sphere, its radius will be the radius of the sphere. 60. Definition. A section made by a plane passing through the centre of a sphere is called a great circle. A section made by a plane not passing through the centre is called a small circle. THEOREM XV. 61. The surface of a sphere is equal to the product of its diameter by the circumference of a great circle. A P Let ABCDEF be the semicircle by whose revolution about the diameter AF, the sphere K may be described; then the surface of the sphere is equal to the diameter A F multiplied by the circumference of the circle whose radius is GA, or = AFX circ. GA. L G H M D Let ABCDEF be a regular semi-decagon inscribed in the semicircle. Draw GO per- N pendicular to one of its sides, as B C. F Draw BK, OP, CL, DM, EN perpendicular to the diameter A F, and B H perpendicular to CL. The surface described by BC is the convex surface of the frustum of a cone, and is equal to BCX circ. PO (45). But the triangles BCH and POG are similar (II. 21); therefore or (III. 28) BC: BH or KL GO: PO BC: KL circ. GO: circ. PO = .. BCX circ. P 0 = KLX circ. GO That is, the surface described by BC is equal to the altitude KL multiplied by circ. GO, or the radius of the circle inscribed in the polygon. In like manner it can be proved that the surfaces described by AB, CD, DE, and EF are respectively equal to their altitudes A K, LM, MN, and NF multiplied by circ. GO. Therefore the entire surface described by the semi-polygon will be equal to (AK+KL+LM+MN+NF) circ. GO AFX circ. GO This demonstration is true, whatever the number of sides of the semi-polygon; it is true, therefore, if the number of sides is infinite, in which case the semi-polygon would coincide with the semicircle; and the surface described by the semi-polygon would be the surface of the sphere, and the radius of the in scribed polygon would be the radius of the sphere. Therefore we have the surface of the sphere equal to AFX circ. GA 62. Corollary. Let S the surface of the sphere, C= the circumference, R = the radius, D = the diameter, then we have (III. 30) C = 2π R, or π D Therefore S = 2π RX 2 R = 4 π R2, or π D2 п That is, the surface of a sphere is equal to the square of its diameter multiplied by 3.14159. THEOREM XVI. 63. The volume of a sphere is the product of its surface by one third of its radius. A sphere may be conceived to be composed of an infinite number of pyramids whose vertices are at the centre of the sphere, and whose bases, being infinitely small planes, coincide with the surface of the sphere. The altitude of each of these pyramids is the radius of the sphere, and the sum of the surfaces of their bases is the surface of the sphere. The volume of each pyramid is the product of the area of its base by one third of its altitude, that is, of the radius of the sphere (51); and the volume of all the pyramids, that is, of the sphere, is, therefore, the product of the surface of the sphere by one third of its radius. That is, the volume of a sphere is the cube of the diameter multiplied by .5235, 65. Cor. 2. As in these equations and are constant, the volumes of spheres vary as the cubes of their radii, or as the cubes of their diameters. |