96. Scholium. The student may derive some aid in comprehending the preceding discussion of the regular polyedrons by constructing models of them, which he can do in a very simple manner, and at the same time with great accuracy, as follows.

Draw on card-board the following diagrams; cut them out entire, and at the lines separating adjacent polygons cut the card-board half through; the figures will then readily bend into the form of the respective surfaces, and can be retained in that form by glueing the edges.

97. In any polyedron, the number of its edges increased by two is equal to the number of its vertices increased by the number of its faces.

Let E denote the number of edges of any polyedron, V the number of its vertices, and F the number of its faces; then we are to prove that

E+2V+F.

In the first place, we observe that if we remove a face, as ABCDE, from any convex polyedron GH, we leave an open surface, terminated by a broken line which

was the contour of the face removed; and in this open surface the number of edges and the number of vertices remain the same as in the original surface.

B

H

Now let us form this open surface by putting together its faces successively, and let us examine the law of connection between the number of edges E, the number of vertices V, and the number of faces, at each successive step. Beginning with one face we have E=V. Annexing a second face, by applying one of its edges to an edge of the first, we form a surface having one edge and two vertices in common with the first; therefore, whatever the number of sides of the new face, the whole number of edges is now one more than the whole number of vertices; that is,

Annexing a third face, adjacent to each of the former, the new surface will have two edges and three vertices in common with the preceding surface; therefore the increase in the number of edges is again one more than the increase in the number of vertices; and we have

At different stages of this process the number of common edges to two successive open surfaces may vary, but in all cases it is apparent that the addition of a new face increases E by one more unit than it increases V; and hence we have the following series of results:

where the law is, that, in the successive values of E, the number to be added to Vis 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

or

E=V+F-2,

E+2=V+ F,

as was to be proved.

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 equal 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 2Rn, 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 × F; that is,

ɔr, reducing,

S+ 4RX F= 2R × 2E,

S4RX (E-F).

But by Euler's Theorem FFV-2; hence,

S= 4RX (V-2).

BOOK VIll.

THE THREE ROUND BODIES.

Or 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 ABCDdcba is a cylindrical surface. If the moving line is of indefinite length, a surface of indefinite extent is generated.

α

b c

D

M

B C

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 nay 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.