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SUPPLEMENTARY PROPOSITIONS. PROPOSITION XXV. THEOREM. (EULER’s.) 594. In any polyhedron 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 polyhedron;

V the number of its vertices, F the number of its
faces.
We are to prove E + 2 = V + F.

Beginning with one face ABCDE, we have E= V.

Annex a second face SAB by applying one of its edges to an edge of the first face.

There is formed a surface having one edge A B, and two vertices A and D B common to both faces.

.. whatever the number of the

sides of the new face, the whole number of edges is now one more than the whole number of vertices.

. for 2 faces E = V + 1. Annex a third face, SBC, adjacent to each of the former. The new surface will have two edges S B and BC,

and three vertices S, B and C, in common with the preceding surface.

... the increase in the number of edges is again one more than the increase in the number of vertices.

According to the same law, for an incomplete surface of F-1 faces

E = V + F – 2. When we add the last face SE A, necessary to complete the surface, • its edges SE, SA and A E, and its vertices S, E and A will be in common with the preceding surface. .. in a polyhedron of F faces E= V + F 2. .. E + 2 = V + F.

Q. E. D.

PROPOSITION XXVI. THEOREM. 595. The sum of all the angles of the faces of any polyhedron is equal to four right angles taken as many times as the polyhedron 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 polyhedron.

We are to prove

S= 4 rt. A X (V — 2).

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Since E denotes the number of the edges of the polyhedron,

2 E will denote the whole number of sides of all its faces, considered as sides of independent polygons.

AR And since the sum of all the interior and exterior of each polygon is equal to 2 rt. s taken as many times as it has sides,

the sum of the interior and exterior of all the faces is equal to 2 rt. 6 X 2 E.

And since the sum of the exterior of each face is 4 rt. 2,

the sum of the exterior { of aļl the faces is equal to 4 rt. 4 X F.

S + 4 rt. A X F= 2 rt. A x 2 E.

$ 159

That is,

Since

§ 594

S= 4 rt. X (E F).

E + 2 = V + F,

E-F=V – 2,
..S= 4 rt. 4 X (V – 2).

Q. E. D.

ON THE CYLINDER.

596. Dec. A Cylindrical surface is a curved surface generated by a moving straight line which continually touches a given curve and in all its positions is parallel to a given fixed straight line not in the plane of the curve.

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Thus, the surface A B C D, generated by the moving line A D continually touching the curve A B C and always parallel to a given straight line M, is a cylindrical surface.

597. Def. The moving line is called the Generatrix ; the curve which directs the motion of the generatrix is called the Directrix ; the generatrix in any position is called an Element of the surface.

The generatrix may be indefinite in extent, and the directrix a closed or an open curve. In elementary geometry the directrix is considered a circle.

598. DEF. A Cylinder is a solid bounded by a cylindrical surface and two parallel planes.

599. DEF. The Bases of a cylinder are its plane surfaces.

600. Def. The Lateral surface of a cylinder is its cylindrical surface.

601. Def. The Axis of a cylinder is the straight line joining the centres of its bases.

602. DEF. The Altitude of a cylinder is the perpendicular distance between the planes of its bases.

603. DEF. A Section of a cylinder is a plane figure whose boundary is the intersection of its plane with the surface of the cylinder.

604. DEF. A Right section of a cylinder is a section perpendicular to the elements.

605. DEF. A Radius of a cylinder is the radius of the base.

606. DEF. A Right cylinder is a cylinder whose elements are perpendicular to its bases. Any element of a right cylinder is equal to its altitude.

607. Def. An Oblique cylinder is a cylinder whose elements are oblique to its bases. Any element of an oblique cylinder is greater than its altitude.

608. DEF. A Cylinder of Revolution is a cylinder generated by the revolution of a rectangle about one side as an axis.

.609. DEF. Similar cylinders of revolution are cylinders generated by similar rectangles revolving about homologous sides.

610. DEF. A Tangent line to a cylinder is a straight line which touches the surface of the cylinder, but does not intersect it.

611. DEF. A Tangent plane to a cylinder is a plane which embraces an element of the cylinder, The element embraced by the tangent plane is called the Element of Contact.

612. DEF. A prism is inscribed in a cylinder when its lateral edges are elements of the cylinder and its bases are inscribed in the bases of the cylinder.

613. DEF. A prism is circumscribed about a cylinder when its lateral faces are tangent to the cylinder and its bases are circumscribed about the bases of the cylinder.

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PROPOSITION XXVII. THEOREM. 614. Every section of a cylinder made by a plane passing through an element is a parallelogram.

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Let A B C D be a section of the cylinder A C, made by

a plane passing through A D.
We are to prove the section A B C D a parallelogram.

The line BC, in which the cutting plane intersects the curved surface a second time, is an element ; I for, if through the point B a line be drawn || to AD,

it will be an element of the surface.

It will also lie in the plane AC,

(for its extremities lie in the plane). This element, lying in both the cylindrical surface and plane surface, is their intersection. Now

A D is ll to BC,
(being elements of the cylinder),

A B is || to DC,
(the intersections of two || planes by a third plane are \ lines).

.. the section A B C D is a 0. § 125

and

§ 465

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615. COROLLARY. Every section of a righi cylinder embracing an element is a rectangle.

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