BOOK IX

CYLINDERS AND CONES

584. A surface, generated by a moving straight line which always remains parallel to its original position and continually touches a given curved line, is called a Cylindrical Surface.

The moving straight line is called the generatrix, and the given curved line is called the directrix.

The generatrix in any position is called

an element of the surface.

585. A solid bounded by a cylindrical surface and two parallel planes which cut all its elements is called a Cylinder.

The plane surfaces are called the bases and the cylindrical surface is called the lateral, or convex surface of the cylinder.

All elements of a cylinder are equal. § 464.

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

586. A section of a cylinder made by a plane perpendicular to its elements is called a Right Section.

587. A cylinder whose elements are perpendicular to its base is called a Right Cylinder.

588. A cylinder whose elements are not perpendicular to its base is called an Oblique Cylinder.

589. A cylinder whose bases are circles is a Circular Cylinder. The straight line joining the centers of the bases of a circular cylinder is called the axis of the cylinder.

590. A right circular cylinder is called a Cylinder of Revolution, because it may be generated by the revolution of a rectangle about one of its sides.

Cylinders of revolution generated by similar rectangles revolving about homologous sides are similar.

591. A plane which contains an element of a cylinder and does not cut the surface is a Tangent Plane to the cylinder.

The element is called the element of contact.

592. Any straight line that lies in a tangent plane and cuts the element of contact is a Tangent Line to the cylinder.

593. When the bases of a prism are inscribed in the bases of a cylinder and its lateral edges are elements of the cylinder, the prism is said to be inscribed in the cylinder.

594. When the bases of a prism are circumscribed about the bases of a cylinder and its lateral edges are parallel to the elements of the cylinder, the prism is said to be circumscribed about the cylinder.

Proposition I

595. 1. Form a cylinder and cut it by any plane through an element of its surface (§ 519 N.). What plane figure is the section made by the cutting plane?

2. If the cylinder is a right cylinder, what plane figure does such a plane make?

Theorem. Any section of a cylinder made by a plane passing through an element is a parallelogram.

Data: Any section of the cylinder EF, as ABCD, made by a plane passing through AB, an element of the surface.

To prove ABCD a parallelogram.

Proof. The plane passing through the element AB cuts the circumference of the base in a second point, as D. Draw DC AB. Then, § 63, DC is in the plane BAD; and, § 584,

E

DC is an element of the cylinder.

Hence, DC, being common to the plane and the lateral surface of the cylinder, is their intersection.

596. Cor. Any section of a right cylinder made by a plane passing through an element is a rectangle.

Proposition II

597. 1. Form a cylinder. How do its bases compare?

2. Cut the cylinder by parallel planes which cut all its elements. How do the sections thus made compare with each other?

3. How does a section made by a plane parallel to the base compare with the base?

Theorem. The bases of a cylinder are equal.

Data: Any cylinder, as MG, whose bases are HG and MN.

Proof. Take any three points in the perimeter of the upper base, as D, E, F, and from them draw the elements of the surface DA, EB, FC, respectively.

Draw AB, BC, AC, DE, EF, and DF.

§§ 585, 584, AD, BE, and CF are equal and parallel;

.. § 150, AE, AF, and BF are parallelograms;

and

hence,

AB DE, AC = DF, BC = EF;

AABC:
= A DEF.

Why?

Apply the upper base to the lower base so that DE shall fall upon AB.

But F is any point in the perimeter of the upper base, therefore, every point in the perimeter of the upper base will fall upon the perimeter of the lower base.

Hence, § 36,

HG MN.

Therefore, etc.

Q.E.D.

598. Cor. I. The sections of a cylinder made by parallel planes cutting all its elements are equal.

599. Cor. II. A section of a cylinder made by a plane parallel to the base is equal to the base,

Proposition III

600. To what is the lateral surface of any prism equivalent? (§ 519) If the number of its lateral faces is indefinitely increased, what solid does the prism approach as its limit? How, then, does the lateral surface of any cylinder compare with the rectangle formed by an element and the perimeter of a right section?

Theorem. The lateral surface of a cylinder is equivalent to the rectangle formed by an element of the surface and the perimeter of a right section.

Data: Any cylinder, as FK; any right section of it, as ABCDE; and any element of its surface, as FG.

Denote the lateral surface of FK by S, and the perimeter of its right section by P.

Proof. Inscribe in the cylinder a prism; F

denote its lateral surface by s' and the

perimeter of its right section by P'.

Then, § 593, each lateral edge is an element of the cylinder, and, § 585, the elements are all equal;

Now, if the number of lateral faces of the inscribed prism is indefinitely increased,

601. Cor. The lateral surface of a cylinder of revolution is equivalent to the rectangle formed by its altitude and the circumference of its base.

Arithmetical Rules: To be framed by the student.

$ 339

602. Formulæ : Let A denote the lateral area, T the total area, H the altitude, and R the radius of the base of a cylinder of revolution.

Then, $395, and, § 398,

A=2 TRX H,

T=2 TRX H+2 TR2 = 2 TR (H+R).

Proposition IV

603. Compute the areas of any two similar cylinders of revolution, as those whose altitudes are 4" and 2" and whose radii are 2" and 1", respectively. How does the ratio of their lateral areas, or of their total areas, compare with the ratio of the squares of their altitudes, or with the ratio of the squares of their radii?

Theorem. The lateral areas, or the total areas, of similar cylinders of revolution are to each other as the squares of their altitudes, or as the squares of their radii.

Data: Any two similar cylinders of revolution, whose altitudes are H and H', and radii R and R', respectively.

Denote their lateral areas by A and A', and their total areas by T and T', respectively.

H

H

R

604. Cor. The lateral areas, or the total areas, of similar cylinders of revolution are to each other as the squares of any of their like dimensions.