cones are proportional to the cubes of their altitudes or to the cubes of the radii of their bases.

3. What is the volume of a cone whose altitude is 12 feet, and the circumference of whose base is 10 feet?

4. A cask, consisting of two equal conic frustums united at their larger bases, has its bung diameter 28 inches, the head diameter 20 inches, and length 40 inches; required the capacity of the cask in gallons.

5. The volume of a cone is 8.83575 cubic feet, the altitude 15 feet; required the diameter of the base.

6. The convex surface of a frustum of a right cone is 378 square feet, the slant height 24 feet, and the area of the smaller base 28.2745; required the diameter of the greater base.

Note.-There are three curves, formed by the intersection of a plane with the surface of a cone, which are very celebrated by reason both of their application in the arts and sciences, and of the beauty of their forms and properties.

If a right cone with a circular base be cut by a plane, at an angle with the base less than that made by the generatrix, the section is an Ellipse, as A B.

If the cutting plane is parallel to the generatrix, the section is a Parabola, as BC.

If the cutting plane makes an angle with the base greater than that made by the generatrix, the section is a Hyperbola, as D E

FH.

These curves are known as Conic Sections; since they may be formed by passing a plane, in different positions, through a right circular cone.

The distinguishing property of the Ellipse is, that the sum of the distances of any point in the bounding curve from two fixed points is constant.

The distinguishing property of the Parabola is, that the distance of any point in the curve from a fixed point is equal to its distance from a fixed right line.

F.

The distinguishing property of the Hyperbola is, that the difference of the distances of any point in either branch of the curve from two fixed points, is constant.

The fixed points are the Foci of the curves.
The heavenly bodies all move in conic sections.

The path of each of the planets is an ellipse, of which the sun, as an attractive force, occupies one of the foci.

Most of the comets are found to move in curves which cannot be distinguished from parabolas; and several are supposed to move in hyperbolas.

These curves were first studied in the school of Plato, with no other motive however, than a simple love of intellectual activity, a matter of sublime speculation. He and his disciples had not thought of serving any practical end, and were reproached with the inutility of their inquiries. It was reserved for Galileo, and Kepler, and Newton, after two thousand years had rolled away, to give to these investigations a practical significance, by showing that these very curves with which Plato and the geometers of his school had amused themselves, were the paths of celestial movement: the first, to show that the path of a body projected obliquely in a vacuum is a parabola; the second, to discover that the planetary orbits are ellipses; the third, to demonstrate that a body which moves under the influence of a central attractive force, like that of the sun, whose intensity varies inversely as the square of the distance, must move in one of the conic sections, an Ellipse, a Parabola, or a Hyperbola.

2. The path traced by the semicircumference is the boundary or Surface of the sphere.

Cor. I. The centre, radius, and diameter of the sphere are the same as those of the generating semicircle or circle.

For the centre, radius, and diameter of the semicircle or circle are, it is evident, precisely the same for every position in the revolution.

Cor. II.-Every point on the surface of a sphere is equally distant from the centre.

For, in every position assumed by the revolving semicircle or circle, every point of the circumference is still equally distant from the centre C.

Cor. III. All radii and all diameters of the same sphere are equal.

For these radii and diameters are the radii and diameters of the generating semicircle or circle.

Cor. IV. Spheres having equal radii are equal.

Cor. V.-Two spheres are externally tangent if the distance between their centres is equal to the sum of their radii.

For this would be true of their generating circles in every possible position of the revolution.

Cor. VI. Two spheres are tangent internally, if the distance between their centres is equal to the difference of their radii.

Cor. VII. One sphere is wholly within another, if the distance between their centres is less than the difference of their radii.

Cor. VIII. Two spheres are concentric, if the distance between their centres is zero.

A

B

Cor. IX. Two spheres intersect, if the distance between their centres is less than the sum of their radii and greater than the difference.

For this is true of the gen

erating circles, C and C', in every position of their revolution about A B.

Cor. X.-Two spheres are external to each other, if the distance between their centres is greater than the sum of their radii.

THEOREM I.

Any plane section of a sphere is a circle.

E

Let A B F be any plane section of the B sphere whose centre is C; then A B F is a circle.

For, from the centre C, draw CE perpendicular to the plane of the section. Join C with different points in the boundary of

the section, as A, B, and F.

Now, A, B, and F are points on the surface of the sphere; hence, C A, C B, and C F are radii, and therefore are equal.

Hence, EA, EF, and EB are equal; for, if these distances were unequal, the radii, CA, CB, and CF, would be unequal (Ch. XII, Th. II).

Hence, A, B and F are equally distant from E.

Hence, the section A B F is a circle (Ch. IV, Th. VII; Ch.

XII, Th. II, Sch.).

Cor. I. The centre of every plane section of a sphere is the foot of the perpendicular from the centre of the sphere Af to the section; and, hence, lies in a diameter of the sphere.

As appears in the course of the preceding demonstration.

B

Q. E. D.

[This diameter of the sphere, perpen

P

dicular to the circle, is called the Axis of the circle; the extremities of the axis, as P and P', are called the Poles of the circle.]

Cor. II.-All circles of a sphere equally distant from the centre of the sphere are equal.

For, letting r denote the radius of the sphere, the radius of any section, and d the distance of the section from the centre of the sphere, we have from the right triangle A B C,

r'=√ p2 — d2.

Similarly, if " and d' denote the radius and distance of any other section, we should have

Hence, the radii being equal, the circles are equal.

Cor. III. A section is the largest possible, when it passes through the centre of the sphere.