21. Any plane through the point of intersection of the diagonals of a parallelopiped divides the parallelopiped into two equivalent solids. 22. The sum of two opposite lateral edges of a truncated parallelopiped is equal to the sum of the other two lateral edges. 23. The middle points of the edges of a regular tetrahedron are the vertices of a regular octahedron. 24. What is the edge of a cube whose entire surface is one square foot? 25. In a regular pyramid the sum of the squares of the lateral edges is equal to the sum of the squares of the base edges increased by n times the square of the slant height. [n = no. of sides of base.] 26. A section of a tetrahedron made by a plane parallel to two nonintersecting edges is a parallelogram. 27. If the diagonals of a quadrangular prism pass through a common point, the figure is a parallelopiped. 28. Given the lengths of the diagonals of the three faces about a trihedral angle of a rectangular parallelopiped to determine the edges. 29. The plane that bisects a dihedral angle of a tetrahedron divides the opposite face into two segments that are proportional to the areas of the adjacent faces. 30. The straight lines joining the middle points of the opposite edges of a tetrahedron all pass through the center of gravity of the tetrahedron. 31. If from any point within a regular tetrahedron perpendiculars be drawn to the faces, their sum is equal to an altitude of the tetrahedron. 32. On three given lines in space that intersect in a common point, as edges, construct a parallelopiped. 33. If a pyramid is cut by three parallel planes so that the distance of one of the planes from the vertex is a mean proportional between the distances of the other two planes from the vertex, then is the section formed by that plane a mean proportional between the other two sections. 34. Divide a tetrahedron into four equivalent tetrahedrons. BOOK VIII 994. DEFINITIONS. A cylindrical surface is a surface generated by a moving straight line that constantly intersects a fixed curve, and in all its positions is paral lel to a fixed straight line not in the plane of the given curve. The moving line is called the generatrix; the fixed curve, the directrix. The generatrix in any of its positions is called an element of the cylindrical surface. If the directrix is a closed convex1 curve, the solid bounded by a cylindrical surface and two parallel plane surfaces is called a cylinder. The cylindrical surface is called its lateral surface, and the parallel plane surfaces are its bases. The altitude of a cylinder is the perpendicular distance between its bases. The elements of a cylinder are equal. (?) A right cylinder is a cylinder whose elements are perpendicular to its bases, and the elements of an oblique cylinder are oblique to its bases. A circular cylinder is a cylinder whose bases are circles. A cylinder of revolution is a right circular cylinder, and is generated by revolving a rectangle about one of its sides as an axis. A section of a cylinder is the figure formed by its intersection with a plane passing through it. 1 A curve is convex if a straight line can intersect it in only two points. PROPOSITION I. THEOREM 995. The bases of a cylinder are equal. E B Let ABC and DEF be the bases of the cylinder AE. Proof. Let A, B, and C be any three points in the perimeter of the lower base. From these points draw the elements AD, BE, and CF. Draw AB, BC, CA, DE, EF, and FD. Prove ABC and DEF equal. The base ABC may be placed on the base DEF so that the points 4, B, and C shall coincide with D, E, and F. (?) But 4, B, and C are any points in the perimeter of the lower base. Therefore all points in the perimeter of the lower base will coincide with corresponding points of the upper base, and the bases are equal. Q.E.D. 996. COROLLARY I. A section of a cylinder made by a plane parallel to the base, is equal to the base. 997. COROLLARY II. Sections of a cylinder made by parallel planes that cut all the elements are equal. 998. EXERCISE. Show that a right section of an oblique circular cylinder is not a circle. PROPOSITION II. THEOREM 999. Any section of a cylinder made by a plane passing through an element is a parallelogram. D A B Let ABCD be a section of the cylinder AC made by a plane passing through the element AB. To Prove ABCD a parallelogram. Proof. Suppose a line drawn through D || to AB. This line lies in the plane ABCD. (?) This line is an element of AC. (?) This line is the intersection of the plane ABCD with the lateral surface. It is DC. DC is || AB, and BC || to AD. (?) .. ABCD is a parallelogram. Q.E.D. 1000. COROLLARY. Any section of a right cylinder made by a plane passing through an element is a rectangle. 1001. EXERCISE. Through an element of a right circular cylinder pass a plane cutting off a section whose area is a maximum. 1002. EXERCISE. Through a point on the lateral surface of a cylinder only one straight line can be drawn lying on the surface. 1003. DEFINITION. The axis of a circular cylinder is the straight line joining the centers of its bases. 1004. EXERCISE. The axis of a circular cylinder is parallel to the elements of the lateral surface. 1005. EXERCISE. The axis of a circular cylinder passes through the centers of all sections of the cylinder that are parallel to the bases. B 1006. DEFINITION. A plane tangent to a cylinder is a plane that contains one element of the cylinder and no point of the surface without that element. 1007. EXERCISE. A plane passed through an element of a cylinder and tangent to the base is tangent to the cylinder. 1008. DEFINITIONS. A prism is inscribed in a cylinder if its bases are inscribed in the bases of the cylinder and its lateral edges are elements of the cylinder. A prism is circumscribed about a cylinder if its bases are circumscribed about the bases of the cylinder and its lateral edges are parallel to the elements of the cylinder. Similar cylinders of revolution are cylinders that are generated by similar rectangles revolved about homologous sides as axes. 1009. AXIOM. A plane surface is less than any other surface having the same boundaries. |