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ON THE CYLINDER.

596. DEF. 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 CD, generated by the moving line AD continually touching the curve ABC 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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614. Every section of a cylinder made by a plane passing through an element is a parallelogram.

A

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;

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 A C,

(for its extremities lie in the plane).

This element, lying in both the cylindrical surface and plane surface, is their intersection.

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(the intersections of two || planes by a third plane are | lines).

.. the section ABCD is a .

§ 465

§ 125

Q. E. D.

615. COROLLARY. Every section of a right cylinder embrac

ing an element is a rectangle.

PROPOSITION XXVIII. THEOREM.

616. The bases of a cylinder are equal.

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Let ABE and DCG be the bases of the cylinder A C.

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Any sections AC and AG, embracing A D, an element of

the cylinder, are §.

§ 614

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Apply the upper base to the lower base, so that DC will coincide with A B.

Then A GDC will coincide with ▲ EA B, and point G will fall upon point E.

That is, any point G in the perimeter of the upper base will coincide with the point in the same element in the lower base. ..the bases coincide, and are equal.

Q. E. D.

617. COROLLARY 1. Any two parallel sections A B C and A'B'C', cutting all the elements of a cylinder E F, are equal. For these sections are the bases of the cylinder A C'.

618. COR. 2. Any section of a cylinder parallel to the base is equal to the base.

AC' is not ac

PROPOSITION XXIX. THEOREM.

619. The lateral area of a cylinder is equal to the product of the perimeter of a right section of the cylinder by an element of the surface.

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B'

B

Let ABCDE be the base, and A A' any element of the cylinder A C'; and let the curve abcde be any right section of its surface.

Denote the perimeter of the right section by P,

and the lateral surface of the cylinder by S.
SPXA A'.

We are to prove

Inscribe in the cylinder a prism having the regular polygon ABCDE as its base.

The right section a b c d e of this prism will be a regular polygon inscribed in the right section abcde of the cylinder. § 604 Denote the lateral area of the prism by 8,

Then

and the perimeter of its right section by p.

s= pX A A',

§ 524

(the lateral area of a prism is equal to the product of the perimeter of a right section by a lateral edge).

Now let the number of lateral faces of the inscribed prism

be indefinitely increased,

the new edges continually bisecting the arcs in the right section.

Then p approaches P as its limit,

and s approaches S as its limit.

But, however great the number of faces,

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