THEOREMS. 764. Every section of a prism by a plane parallel to the lateral edges is a parallelogram. 765. The lateral areas of right prisms having equal altitudes are as the perimeters of their bases. 766. The diagonals of a rectangular parallelopiped are equal. 767. The square of a diagonal of a rectangular parallelopiped is equal to the sum of the squares of the three diagonals meeting in any vertex. 768. The volume of a triangular prism is equal to one half the product of any lateral face by its distance from the opposite edge. 769. The volume of any prism is equal to the product of its right section by an edge. 770. The four diagonals of a parallelopiped bisect each other. 771. If the four diagonals of a four-sided prism pass through a common point, the prism is a parallelopiped. 772. Any straight line passing through the center of a parallelopiped and terminated by two faces, is bisected at the center. N.B.-The center of a parallelopiped is the point of intersection of its diagonals. 773. Any plane passing through the center of a parallelopiped divides it into two equal solids. 774. If any two nonparallel diagonal planes of a prism are perpendicular to the base, the prism is a right prism. 775. The lateral surface of any pyramid is greater than its base. 776. The mid points of the edges of a regular tetrahedron are at the vertices of a regular octahedron. 777. The section of a triangular pyramid made by a plane passed parallel to two opposite edges is a parallelogram. 778. The section of a regular tetrahedron made by a plane passed parallel to two opposite edges is a rectangle. 779. The altitude of a regular tetrahedron is equal to the sum of the perpendiculars to the faces from any point within the figure. BOOK IX. THE THREE ROUND BODIES. Of the solids that are bounded by curved surfaces, only three are treated of in Elementary Geometry, viz., the cylinder, the cone, and the sphere, usually referred to as the three round bodies. CYLINDERS. AB C 566. A cylindrical surface is a curved surface generated by a straight line that moves so as continually to touch a given curve, while remaining parallel to its original position. Thus if the straight line 4a moves so as to remain always parallel to its first position 4a, while continually touching the curve ABCD, the surface ABCD-abcd thus generated is a cylindrical surface. The moving line is called the generatrix; the curve touched, the directrix; and any straight line Bb, that represents the generatrix in any of its positions, an element of the surface. d Since the generatrix is of indefinite length, a cylindrical surface may be regarded as extending indefinitely in two directions. As the directrix, moreover, may be a curve of any kind, close or not, the surface generated may present a corresponding variety of form. In elementary geometry, for obvious reasons, the directrix is usually assumed to be a circle. 567. A cylinder is a solid bounded by a cylindrical surface whose directrix is a closed curve, and by two parallel planes. These planes are called the bases of the cylinder; the curved surface, the lateral surface; and the perpendicular distance between the bases, the altitude. From the definition it is evident that the curved surface of a cylinder must have as directrix a closed curve; since, otherwise, besides the two parallel bases, at least one other plane face would be needed in order to form a solid. 568. A right cylinder has its elements perpendicular to its base, as AB; an oblique cylinder has its elements oblique to its base, as A'B'. 569. A circular cylinder is one that has a circle for each base. As only circular cylinders are treated of in this book, the term B cylinder is to be understood as signifying circular cylinder. 570. A right cylinder with a circular base is called a cylinder of revolution, because it may be generated by the revolution of a rectangle about one of its sides as axis. This side is then called the axis of the cylinder, and the radius of the base, the radius of the cylinder. B' EXERCISE 780. Only one straight line can be drawn through a given point on a cylindrical surface. 781. A straight line that joins two points on a cylindrical surface must coincide with an element of that surface. 782. If a plane contains one and only one straight line in common with the curved surface of a cylinder, the plane touches but does not intersect that surface. A tangent DEFINITIONS. Such a plane is said to be tangent to the cylinder, and the common element is called the element of contact. line touches but does not intersect the cylinder. PROPOSITION I. THEOREM. 571. Every section of a cylinder made by a plane. passing through an element is a parallelogram. B b Given: Ac, a section of cylinder AB-c by a plane through element Aa; To Prove: Ac is a parallelogram. For if through C a line be drawn || to Aa, that line will be an element of the surface; it will also be a line in plane Ac; .. the line will coincide with Cc, the intersection of plane Ac and the lateral surface; .. Cc is to Aa, and ac is to AC, (since they lie in parallel planes ;) .. Ac is a parallelogram. (566) (424) (451) Q.E.D. 572. COR. Every section of a right cylinder passing through an element is a rectangle. SCHOLIUM. It will be noticed that the properties established in Prop. I. and Prop. II., being independent of the form of the base, hold true, not only in regard to circular cylinders, but also to cylinders in general. A similar remark applies also to Prop. III. concerning the cone. PROPOSITION II. THEOREM. 573. The bases of a cylinder are equal. b Given: ABC, abc, bases of cylinder Ac; ABC equals abc. Through any element Aa, pass planes forming the sections Ab, Ac; and join BC, bc. If, then, the upper base be applied to the lower, so that ab + AB, then abc ▲ A B C, and c C; M that is, any point c in the perimeter of the upper base will coincide with a corresponding point in the perimeter of the lower base; ..the bases coincide and are equal. Q.E.D. 574. COR. 1. Any two parallel sections, S, S', cutting a cylindrical surface MN, are equal. 575. COR. 2. All sections of a circular cylinder parallel to the base are equal. S S |