BOOK IX.

MEASUREMENT OF THE THREE ROUND BODIES.

THE CYLINDER.

1. DEFINITION. The area of the convex, or lateral, surface of a cylinder is called its lateral area.

2. Definition. A prism is inscribed in a cylinder when its base is inscribed in the base of the cylinder and its lateral edges are elements of the cylinder. It follows that the upper base of the prism is inscribed in the upper base of the cylinder.

B C

To inscribe, then, a prism of any given number of lateral faces in a cylinder, we have merely to inscribe in the base a polygon of the given number of sides, and through the vertices of the polygon to draw elements of the cylinder. Planes passed through adjacent elements will form the lateral faces of the prism which is obviously wholly contained in the cylinder.

3. Definition. A prism is circumscribed about a cylinder when its base is circumscribed about the base of the cylinder and its lateral edges are parallel to elements of the cylinder.

It follows that its lateral faces

are tangent to the lateral faces of

the cylinder (VIII., 11); for any

D

D

B

face, as AB', contains the element bb', since it contains the

parallel line AA' and the point b (VI., Proposition II.), and, by VIII., Proposition I., it cannot cut the surface of the cylinder again unless AB cuts the base again; and that its upper base is circumscribed about the upper base of the cylinder. The cylinder is obviously wholly contained in the prism.

5. Definition. Similar cylinders of revolution are those which are generated by similar rectangles revolving about homologous sides.

PROPOSITION I.-THEOREM.

6. If a prism whose base is a regular polygon be inscribed in or circumscribed about a given cylinder, its volume will approach the volume of the cylinder as its limit, and its lateral surface will approach the lateral surface of the cylinder as its limit as the number of sides of its base is indefinitely increased.

For, if we could make the base of the prism exactly coincide with the base of the cylinder, the prism and the cylinder would coincide throughout, and their volumes would be equal and their lateral surfaces equal.

But, by increasing the number of sides of the base of the prism,

D

D

A

b

B

B

we can make it come as near as we please to coinciding with the base of the cylinder (V., Proposition VII.); we can then make

the prism and cylinder fail of coincidence by as small an amount as we choose. Consequently, by increasing at pleasure the number of sides of the base of the circumscribed or inscribed prism, we can make the difference between the volumes of prism and cylinder, and between the lateral surfaces

a

D

al

B

B

of prism and cylinder, as small as we choose, but cannot make it absolutely zero.

7. Scholium. The proposition just proved is true when the base of the prism is not a regular polygon; but it is only for the case of the regular polygon that a rigorous proof has been given in Book V.

PROPOSITION II.-THEOREM.

8. 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.

Let ABCDEF be the base and AA' any element of a cylinder, and let the curve abcdef be any right section of the surface. Denote the perimeter of the right section by P, the element AA' by E, and the lateral area of the cylinder by S.

Inscribe in the cylinder a prism ABCDEFA' of any arbitrarily chosen

B

F

E

с

number n of faces. The right section, abcdef, of this prism will be a polygon inscribed in the right section of the cylinder formed by the same plane (4). Denote the lateral area of the prism by s, and the perimeter of its right section by p; then, the lateral edge of the prism being equal to E, we have (VII., Proposition II.)

s=p E,

no matter what the value of n. If n is indefinitely increased, s approaches the limit S (Proposition I.), and p × E, the limit PX E. Therefore, by III., Theorem of Limits,

S = PXE.

9. COROLLARY I. The lateral area of a cylinder of revolution is equal to the product of the circumference of its base by its altitude.

This may be formulated,

S= 2πR.H,

if R is the radius of the base and H the altitude.

10. COROLLARY II. The lateral areas of similar cylinders of revolution are to each other as the squares of their altitudes, or as the squares of the radii of their bases.

PROPOSITION III.-THEOREM.

11. The volume of a cylinder is equal to the product of its base by its altitude.

Let the volume of the cylinder be denoted by V, its base by B, and its altitude by H. Let the volume of an inscribed prism be denoted by V', and its base by B'; its altitude will also be H, and we shall have (VII., Proposition XII., Corollary)

V' = B' × H,

B

no matter what the number of faces of the prism.

H

If the number of faces of the prism is indefinitely increased, V' has the limit V, and B'X H the limit BX H. Therefore

V=BX H.

12. COROLLARY I. For a cylinder of revolution this proposition may be formulated, VR. H. (V., Proposition IX., Corollary.)

13. COROLLARY II. The volumes of similar cylinders of revotion are to each other as the cubes of their altitudes, or as the the surface ir radii.

the right secti

AA' by E, and th

cylinder by S.

Inscribe in the

ABCDEFA' of any

THE CONE.

9 area of the convex, or lateral, surface ts lateral area.