where the law is, that, in the successive values of E, the number to be added to V is a unit less than the number of faces. The last line expresses the relation for the open surface of F-1 faces, that is, for the open surface which wants but one face to make the closed surface of F faces. But the number of edges and the number of vertices of this open surface are the same as in the closed surface. Therefore, in a closed surface of F faces, we have This theorem was discovered by Euler, and is called Euler's Theorem on Polyedrons. PROPOSITION XXXIII.-THEOREM. 98. The sum of all the angles of the faces of any polyedron is equall to four right angles taken as many times as the polyedron has vertices less two. Let E denote the number of edges, V the number of vertices, F the number of faces, and S the sum of all the angles of the faces, of any polyedron. If we consider both the interior angles of a polygon and the exterior ones formed by producing its sides as in (I. 101), the sum of all the angles both interior and exterior is 2R × n, where R denotes a right angle, and n is the number of sides of the polygon. If, then, E denotes the number of edges of the polyedron, 2E denotes the whole number of sides of all its faces considered as independent polygons, and the sum S of the interior angles of all the F faces plus the sum of their exterior angles is 2R × 2E. But the sum of the exterior angles of one polygon is 4R, and the sum of the exterior angles of the F polygons is 4R X F; that is, or, reducing, S+ 4RX F = 2R × 2E, S= 4RX (E-F). But by Euler's Theorem. EFV-2; hence, 8=4RX (V-2). BOOK VIII. THE THREE ROUND BODIES. Of the various solids bounded by curved surfaces, but three are treated of in Elementary Geometry-namely, the cylinder, the cone, and the sphere, which are called the THREE ROUND BODIES. THE CYLINDER. 2. Definition. A cylindrical surface is a curved surface generated by a moving straight line which continually touches a given curve, and in all of its positions is parallel to a given fixed straight line not in the plane of the curve. Thus, if the straight line Aa moves so as continually to touch the given curve ABCD, and so that in any of its positions, as Bb, Cc, Dd, etc., it is parallel to a given fixed straight line Mm, the surface ABCDdeba is a cylindrical surface. If the moving line is of indefinite length, a surface of indefinite extent is generated. M m a В с be The moving line is called the generatrix; the curve which it touches is called the directrix. Any straight line in the surface, as Bb, which represents one of the positions of the generatrix, is called an element of the surface. In this general definition of a cylindrical surface, the directrix may be any curve whatever. Hereafter we shall assume it to be a closed curve, and usually a circle, as this is the only curve whose properties are treated of in elementary geometry. 3. Definition. The solid Ad bounded by a cylindrical surface and two parallel planes, ABD and abd, is called a cylinder; its plane surfaces, ABD, abd, are called its bases; the curved surface is sometimes called its lateral surface; and the perpendicular distance between its bases is its altitude. A cylinder whose base is a circle is called a circular cylinder. 4. Definition. A right cylinder is one whose elements are perpendicular to its base. a C B 5. Definition. A right cylinder with a circular base, as AB Ca, is called a cylinder of revolution, because it may be generated by the revolution of a rectangle AOoa about one of its sides, Oo, as an axis; the side Aa generating the curved surface, and the sides OA and oa generating the bases. Oo is the axis of the cylinder. The radius of the base is called the adius of the cylinder. The fixed side PROPOSITION I.-THEOREM. 6. Every section of a cylinder made by a plane passing through an element is a parallelogram. Let Bb be an element of the cylinder Ac; then, the section BbdD, made by a plane passed through Bb, is a parallelogram. A B d a C 1st. The line Dd in which the cutting plane intersects the curved surface a second time is an element. For, if through any point D of this intersection a straight line is drawn parallel to Bb, this line by the definition of a cylindrical surface, is an element of the surface, and it must also lie in the plane Bd; therefore, this element, being common to both surfaces, is their intersection. 2d. The lines BD and bd are parallel (VI. 25), and the elements Bb and Dd are parallel; therefore, Bd is a parallelogram. 7. Corollary. Every section of a right cylinder made by a plane perpendicular to its base is a rectangle. PROPOSITION II.-THEOREM. 8. The bases of a cylinder are equal. Let BD be the straight line joining any two points of the perimeter of the lower base, and let a plane passing through BD and the element Bb cut the upper base in the line bd; then, BD bd (6). B = ab and AD = ad Let A be any third point in the perimeter of the lower base, and Aa the corresponding element. Join AB, AD, ab, ad. Then AB (6); and the triangles ABD, abd, are equal. Therefore, if the upper base be applied to the lower base with the line bd in coincidence with its equal BD, the triangles will coincide and the point a will fall upon A; that is, any point a of the upper base will fall on the perimeter of the lower base, and consequently the perimeters will coincide throughout. Therefore, the bases are equal. 9. Corollary I. Any two parallel sections MPN, mpn, of a cylindrical surface Ab, are equal. For, these sections are the bases of the cylinder Mn. 10. Corollary II. All the sections of a circular cylinder parallel to its bases are equal circles; and the straight line joining the centres of the bases passes through the centres of all the parallel sections. This line is called the axis of the cylinder. 11. Definition. A tangent plane to a cylinder is a plane which passes through an element of the curved surface without cutting this surface. The element through which it passes is called the element of contact. |