BOOK VIII.

THE CYLINDER.

DEFINITIONS.

542.

A cylindrical surface is a curved surface formed by a moving straight line which constantly touches a given curve and is at all times parallel to a fixed straight line not in the plane. The moving straight line is called the generatrix. The given curved line is called the directrix.

543.

x

y

Any straight line in the cylindrical surface is called an element of the surface, as x y above.

There may be any number of cylindrical surfaces.

544.

A cylinder is a solid bounded by a cylindrical surface and two parallel planes. The plane surfaces are called the bases and the curved surface the lateral or convex surface of the cylinder.

(How do the elements compare in length?)

545.

A line joining the centers of the bases is called the axis of the cylinder.

546.

A right section of a cylinder is the intersection of the cylinder and a plane perpendicular to an element.

547.

The altitude of a cylinder is the perpendicular distance between its bases.

548.

An oblique cylinder is one whose bases form oblique angles with the elements.

549.

A right cylinder is one whose elements are perpendicular to its bases.

550.

A circular cylinder is one whose bases are circles.

551.

In how many ways may a right circular cylinder be generated?

552.

A right circular cylinder generated by the revolution of a rectangle about one of its sides as an axis is called a cylinder of revolution.

553.

Similar cylinders of revolution are generated by the revolution of similar rectangles about homologous sides.

554.

An axial section is formed by passing a plane through the axis.

555.

A plane is tangent to a cylinder when it passes through an element of the cylinder, but does not cut the surface.

556.

A tangent line to a cylinder is a line which touches the cylindrical surface at a point, but does not intersect the surface.

557.

Remark: The generatrix is supposed to be indefinite in extent; hence the surface generated is also of indefinite extent.

558.

PROPOSITION I.

Can you show that any section of a cylinder made by passing a plane through an element is a parallelogram?

B

Let the plane A C contain an element A B. Is A B C D a parallelogram?

Sug. Is point D common to the plane and the surface of the cylinder? Draw a line through D || to A B. Where will it lie?

Pupil complete proof and write the proposition, calling it Prop. I.

559.

Cor. Every section of a right cylinder made by passing a plane through an element is a Proof.

560.

PROPOSITION II.

Can you show that the bases of a cylinder are

equal?

H

B

Sug. 1. Suppose A B' to be any cylinder and F, G, any two points in the perimeter of the upper base. Pass the plane FD containing the line F G and the element F C. Compare F G and C D.

Sug. 2. Take H as any other point in the perimeter of the upper base and H E an element through H. Pass planes through H E, F C and H E, G D. Compare F H, CE and G H, D E.

Sug. 3. Compare As. Superpose the upper base on the lower base so F G will fall on C D and F G H wil fall on AC DE. What follows?

Make the generalization. Write it and call it Prop. II.