F CD is the diagonal, the triangle ACD is equal to the triangle CDF. Therefore the pyramid, whose base is the triangle ACD, and vertex the point E, is equivalent to the pyramid whose base is the triangle CDF, and vertex the point E. But the latter pyramid is equivalent to the pyramid E-ABC, for they have equal A bases, viz., the triangles ABC, DEF, and the same altitude, viz., the altitude of the prism ABC-DEF. Therefore the three pyramids E-ABC, E-ACD, E-CDF, are equivalent to each other, and they compose the whole prism ABC-DEF; hence the pyramid E-ABC is the third part of the prism which has the same base and the same altitude. Cor. The solidity of a triangular pyramid is measured by the product of its base by one third of its altitude. PROPOSITION XVII. THEOREM. The solidity of every pyramid is measured by the product of its base by one third of its altitude. Let A-BCDEF be any pyramid, whose base is the polygon BCDEF, and altitude AH; then will the solidity of the pyramid be measured by BCDEFAH. Divide the polygon BCDEF into triangles by the diagonals CF, DF; and let planes pass through these lines and the vertex A; they will divide the polygonal pyramid A-BCDEF into triangular pyramids, all having the same altitude AH. But each of these pyramids is measured by the product of its base by one third of its altitude (Prop. XVI., Cor.); hence the sum of the triangular pyramids, or the polygonal pyramid A-BCDEF, will be measured by the sum of the triangles BCF, CDF, DEF, or the polygon BCDEF, multiplied by one third of AH. Therefore every pyramid is measured by the product of its base by one third of its altitude. Cor. 1. Every pyramid is one third of a prism having the same base and altitude. Cor. 2. Pyramids of the same altitude are to each other as their bases; pyramids of the same base are to each other as their altitudes; and pyramids generally are to each other as the products of their bases by their altitudes. Cor. 3. Similar pyramids are to each other as the cubes of their homologous edges. Scholium. The solidity of any polyedron may be found by dividing it into pyramids, by planes passing through one of its vertices. PROPOSITION XVIII. THEOREM. A frustum of a pyramid is equivalent to the sum of three pyramids, having the same altitude as the frustum, and whose bases are the lower base of the frustum, its upper base, and a mean proportional between them. Case first. When the base of the frustum is a triangle. Let ABC-DEF be a frustum of a triangular pyramid. If a plane be made to pass through the points A, C, E, it will cut off the pyramid E-ABC, whose altitude is the altitude of the frustum, and its base is ABC, the lower base of the frustum. Pass another plane through the points C, D, E; it will cut off the pyramid C-DEF, whose altitude is that of the frustum, and its base is DEF, the upper base of the frustum. To find the magnitude of the remaining pyramid E-ACD, draw EG parallel to AD; join CG, DG. Then, because the two triangles AGC, DEF have the angles at A and D equal to each other, we have (Prop. XXIII., B. IV.) AGC: DEF:: AGXAC: DEXDF, :: AC: DF, because AG is equal to DE. Also (Prop. VI., Cor. 1, B. IV.), ACB ACG:: AB: AG or DE. But, because the triangles ABC, DEF are similar (Prop. XIII.), we have AB: DE:: AC: DF. Therefore (Prop. IV., B. II.), ACB: ACG:: ACG: DEF; that is, the triangle ACG is a mean proportional between ACB and DEF, the two bases of the frustum. Now the pyramid E-ACD is equivalent to the pyramid G-ACD, because it has the same base and the same altitude; for EG is parallel to AD, and, consequently, parallel to the K plane ACD. But the pyramid G-ACD has the same altitude as the frustum, and its base ACG is a mean proportional between the two bases of the frustum. Case second. When the base of the frustum is any polygon. Let BCDEF-bcdef be a frustum of any pyramid. K Let G-HIK be a triangular pyramid having the same altitude and an equivalent base with the pyramid A-BCDEF, and from it let a frustum HIK-hik be cut B off, having the same altitude. with the frustum BCDEFbcdef. The entire pyramids are equivalent (Prop. XV.); and the small pyramids A-bcdef, G-hik are also equivalent, for their altitudes are equal, and their bases are equivalent (Prop. XIII., Cor. 2). Hence the two frustums are equivalent, and they have the same altitude, with equivalent bases. But the frustum HIK-hik has been proved to be equivalent to the sum of three pyramids, each having the same altitude as the frustum, and whose bases are the lower base of the frustum, its upper base, and a mean proportional between them. Hence the same must be true of the frustum of any pyramid. Therefore, a frustum of a pyramid, &c. PROPOSITION XIX. THEOREM. There can be but five regular polyedrons. Since the faces of a regular polyedron are regular polygons, they must consist of equilateral triangles, of squares, of regular pentagons, or polygons of a greater number of sides. First. If the faces are equilateral triangles, each solid angle of the polyedron may be contained by three of these tri angles, forming the tetraedron; or by four, forming the octaedron; or by five, forming the icosaedron. No other regular polyedron can be formed with equilateral triangles; for six angles of these triangles amount to four right angles, and can not form a solid angle (Prop. XVIII., B. VII.). Secondly. If the faces are squares, their angles may be united three and three, forming the hexaedron, or cube. Four angles of squares amount to four right angles, and can not form a solid angle. Thirdly. If the faces are regular pentagons, their angles may be united three and three, forming the regular dodecaedron. Four angles of a regular pentagon, are greater than four right angles, and can not form a solid angle. Fourthly. A regular polyedron can not be formed with regular hexagons, for three angles of a regular hexagon amount to four right angles. Three angles of a regular heptagon amount to more than four right angles; and the same is true of any polygon having a greater number of sides. Hence there can be but five regular polyedrons; three formed with equilateral triangles, one with squares, and one with pentagons. BOOK IX. SPHERICAL GEOMETRY. Definitions. 1. A sphere is a solid bounded by a curved surface, all the points of which are equally distant from a point within, called the center. The sphere may be conceived to be described by the revolution of a semicircle ADB, about its diameter AB, which remains unmoved. 2. The radius of a sphere, is a straight line drawn from the center to any point of the surface. The diameter, or axis, is a line passing through the center, and terminated each way by the surface. All the radii of a sphere are equal; all the diameters are also equal, and each double of the radius. 3. It will be shown (Prop. I.), that every section of a sphere made by a plane is a circle. A great circle is a section made by a plane which passes through the center of the sphere. Any other section made by a plane is called a small circle. 4. A plane touches a sphere, when it meets the sphere, but, being produced, does not cut it. 5. The pole of a circle of a sphere, is a point in the surface equally distant from every point in the circumference of this circle. It will be shown (Prop. V.), that every circle, whether great or small, has two poles. 6. A spherical triangle is a part of the surface of a sphere, bounded by three arcs of great circles, each of which is less than a semicircumference. These arcs are called the sides of the triangle; and the angles which their planes make with each other, are the angles of the triangle. 7. A spherical triangle is called right-angled, isosceles, or equilateral, in the same cases as a plane triangle. |