401. A Right Parallelopiped is one whose lateral edges are perpendicular to its bases; that is, the lateral faces are rectangles. 402. A Rectangular Parallelopiped is a right parallelopiped whose bases are rectangles; that is, all the faces are rectangles. Such a solid is sometimes called a cuboid. It is contained between three pairs of parallel planes. The Dimensions of a rectangular parallelopiped are the three edges which meet at any vertex. 403. A Cube is a rectangular parallelopiped whose six faces are all squares, and edges consequently equal. 404. Similar Polyedrons are those which are bounded by the same number of similar polygons, similarly placed. Parts which are similarly placed, whether faces, edges, or angles, are called Homologous. 405. A Cylindrical Surface is a curved surface traced by a straight line, so moving as to intersect a given curve and always be parallel to a given straight line not in the curve. Thus if the line EF moves so as to continually intersect the curve DC, and always be parallel to GH, the surface AC is a cylindrical surface. E A B GM F C D H N 406. The moving line EF is the Generatrix, the fixed curve DC the Directrix, and EF in any of its positions is an Element of the surface. 407. A General Cylinder is a solid bounded by a cylindrical surface and two parallel planes called Bases. 408. The Lateral Surface is the curved surface. 408 a. A plane which contains an element of the cylinder and does not cut the surface is called a tangent plane, and the element contained by the tangent plane is the element of contact. 409. The Altitude of a cylinder is the perpendicular distance between the bases or the planes of the bases. 410. The Right Cylinder is the cylinder whose element is perpendicular to its base. If the base is distorted so as to be no longer ос regular, the cylinder is still a right though not a regular cylinder. 411. A Circular Cylinder is one whose directrix is a circle. NOTE. Hereafter the term cylinder is used for circular cylinder. 412. A right cylinder may be conceived as formed by the revolution of a rectangle about one of its sides. Similar cylinders of revolution are generated by similar rectangles. 413. Since the base of a cylinder is a circle of an infinite number of sides, the cylinder itself may be regarded as a prism of an infinite number of faces; that is, a cylinder is only a prism under this condition of infinite faces. 414. Hence the cylinder will have the properties of a prism, and all demonstrations for prisms will include cylinders when so stated in the theorem or in the corollary. PROPOSITION I. THEOREM. 415. The lateral area of a prism is equal to the product of the perimeter of a right section by a lateral edge. Let AD' be a prism, and FGHIK a right section. To prove that the lateral area = AA' (FG+ GH + HI, etc.). Since a right section is perpendicular to the lateral edges, FG, GH, HI, etc., are altitudes of the parallelograms which form the faces of the prism. Hence, and area of A'ABB' = AA' × FG (by 251), area of B'BCC' = BB' × GH, etc. But (by 399) the lateral edges are equal; that is, AA' = BB' = CC', etc. Therefore the total lateral surface will be AA' × FG + AA' × GH + AA' × HI + etc., or lateral surface AD = AA' (FG + GH + HI+ etc). Q.E.D. 416. COR. 1. The lateral area of a right prism is equal to the product of the perimeter of its base by its altitude. 417. COR. 2. The lateral area of a cylinder is equal to the perimeter of a right section of a cylinder multiplied by an element. PROPOSITION II. THEOREM. 418. An oblique prism is equivalent to a right prism having for its base a right section of the oblique prism and for its altitude a lateral edge of the oblique prism. Let ABCDE-I be an oblique prism, and A'B'C'D'E'G' a right section of it. = = Produce A'F to F", making A'F' AF, likewise B'G' BG, C'H' = CH, D'I' = DI, E'K' = EK; then will F'-I' be a plane (by 329) parallel to A'-D', which is a right section; hence A'B'C'D'E'-I' will be a right prism. = To prove that prism AI prism A'I'. The prisms F-I' and A-D' are equivalent, since the faces FGG'F" and ABB'A' are equal by construction, likewise G'GHH' and B'BCC', and so with each pair of faces. The diedral angles are equal, being formed by a continuation of the same faces; that is, LA'A=LFF', ▲ B'B=ZG'G, etc. Therefore the space occupied by A-D' could be exactly filled by F-I', or vice versa; that is, the prisms are equivalent. Hence if from the entire prism A-I' we subtract the prism A-D', we have left the right prism A'-I', and if from the same prism we subtract the equal prism F-I', we have left the oblique prism A-I. Therefore prism A-I = right prism A'-I'. Q.E.D. PROPOSITION III. THEOREM. 419. Two rectangular parallelopipeds having equal bases are to each other as their altitudes. Let P and Q be two rectangular parallelopipeds having equal bases, and let their altitudes AA' and BB' be commensurable. Let AC be a common measure of AA' and BB', and suppose it to be contained 4 times in AA' and 3 times in BB'. Then, AA' 4 (1) At the several points of division of AA' and BB' pass planes perpendicular to these lines. Then the parallelopiped P will be divided into 4 equal parts, of which the parallelopiped Q will contain 3. When the altitudes are incommensurable, the demonstration follows the method pursued in section 173. |