If we put A to represent the altitude of the zone which forms the base of a sector, then the solidity of the sector will be represented by

2πRA× R=3πR3A.

Cor. 3. Every sphere is two thirds of the circumscribed cylinder.

For, since the base of the circumscribed cylinder is equal to a great circle, and its altitude to a diameter, the solidity of the cylinder is equal to a great circle, multiplied by the diameter (Prop. II.). But the solidity of a sphere is equal to four great circles, multiplied by one third of the radius; or one great circle, multiplied by of the radius, or of the diameter. Hence a sphere is two thirds of the circumscribed cylinder.

A spherical segment with one base, is equivalent to half of a cylinder having the same base and altitude, plus a sphere whose diameter is the altitude of the segment.

Let BD be the radius of the base of the segment, AD its altitude, and let the segment be generated by the revolution of the circular half segment AEBD about the axis AC. B Join CB, and from the center C draw CF perpendicular to AB.

The solid generated by the revolution of

E

the segment AEB, is equal to the difference of the solids generated by the sector ACBE, and the triangle ACB. Now, the solid generated by the sector ACBE is equal to

CBAD (Prop. VIII., Cor. 2).

And the solid generated by the triangle ACB, by Prop. VIII., is equal to CF, multiplied by the convex surface described by AB, which is 2πCF × AD (Prop. VII.), making for the solid generated by the triangle ACB,

TCFX AD.

Therefore the solid generated by the segment AEB, is equal TAD×(CB-CF2), πADX BF2;

to

or

because CB-CF" is equal to BF, and BF' is equal to one fourth of AB'.

Now the cone generated by the triangle ABD is equal to ADX BD (Prop. V., Cor. 2).

Therefore the spherical segment in question, which is the sum of the solids described by AEB and ABD, is equal to

that is,

πAD(2BD2+AB2);
TAD(3BD+ AD3),

because AB is equal to BD+AD2.

This expression may be separated into the two parts
ADX BD2,
TAD3.

and

The first part represents the solidity of a cylinder having the same base with the segment and half its altitude (Prop. II.); the other part represents a sphere, of which AD is the diameter (Prop. VIII., Cor. 2). segment, &c.

Therefore, a spherical

Cor. The solidity of the spherical segment of two bases, generated by the revolution of BCDE about the axis AD, may be found by subtracting that of the segment of one base generated by ABE, from that of the segment of one base generated by ACD.

CONIC SECTIONS.

THERE are three curves whose properties are extensively applied in Astronomy, and many other branches of science, which, being the sections of a cone made by a plane in dif ferent positions, are called the conic sections. These are The Parabola, The Ellipse, and The Hyperbola.

PARABOLA.

Definitions.

1. A parabola is a plane curve, every point of which is equally distant from a fixed point, and a given straight line. 2. The fixed point is called the focus of the parabola and the given straight line is called the directrix.

Thus, if F be a fixed point, and BC a B given line, and the point A move about F in such a manner, that its distance from F D is always equal to the perpendicular distance from BC, the point A will describe a parabola, of which F is the focus, and BC the directrix.

D

3. A diameter is a straight line drawn through any point of the curve perpendicular to the directrix. The vertex of the diameter is the point in which it cuts c the curve.

A

Thus, through any point of the curve, as A, draw a line DE perpendicular to the directrix BC; DE is a diameter of the parabola, and the point A is the vertex of this diameter.

4. The axis of the parabola is the diameter which passes through the focus; and the point in which it cuts the curve is called the principal vertex.

Thus, draw a diameter of the parabola, GH, through the

focus F; GH is the axis of the parabola, B and the point V, where the axis cuts the E curve, is called the principal vertex of the parabola, or simply the vertex.

It is evident from Def. 1, that the line FH is bisected in the point V.

H

5. A tangent is a straight line which E meets the curve, but, being produced, does not cut it.

C

6. An ordinate to a diameter, is a straight line drawn from any point of the curve to meet that diameter, and is parallel

to the tangent at its vertex.

Thus, let AC be a tangent to the parabola at B, the vertex of the diameter BD. From any point E of the curve, draw EGH parallel to AC; L then is EG an ordinate to the diameter BD.

It is proved in Prop. IX., that EG is equal to GH; hence the entire line EH is called a double ordinate.

7. An abscissa is the part of a diameter intercepted between its vertex and an ordinate.

Thus, BG is the abscissa of the diameter BD, corresponding to the ordinate EG.

8. A subtangent is that part of a diameter intercepted between a tangent and ordinate to the point of contact.

Thus, let EL, a tangent to the curve at E, meet the diameter BD in the point L; then LG is the subtangent of BD, corresponding to the point E.

9. The parameter of a diameter is the double ordinate which passes through the focus.

Thus, through the focus F, draw IK parallel to the tangent AC; then is IK the parameter of the diameter BD.

10. The parameter of the axis is called the principal parameter, or latus rectum.

11. A normal is a line drawn perpendicular to a tangent from the point of contact, and terminated by the axis. 12. A subnormal is the part of the axis intercepted between the normal, and the corresponding ordinate.

Thus, let AB be a tangent to the parabola at any point A. From A B draw AC perpendicular to AB; draw, also, the ordinate AD. Then AC is the normal, and DC is the subnormal corresponding to the point A

A

D C

PROPOSITION I. PROBLEM.

To describe a parabola.

Let BC be a ruler laid upon a plane, and let DEG be a square. Take a thread equal in length to EG, and attach B one extremity at G, and the other at some point as F. Then slide the side of the square DE along the ruler BC, and, at the same time, keep the thread continually tight by means of the pencil A; the pencil will describe one part of a parabola, of which F is the focus, and BC the directrix. For, in every position of the square,

and hence

AF+AG-AE+AG,
AE;

AF

D

E

that is, the point A is always equally distant from the focus F and directrix BC.

If the square be turned over, and moved on the other side of the point F, the other part of the same parabola may be described.

PROPOSITION II. THEOREM.

A tangent to the parabola bisects the angle formed at the point of contact, by a perpendicular to the directrix, and a line drawn to the focus.

Let A be any point of the parabola AV, from which draw the line AF to the focus, and AB perpendicular to the directrix, and draw AC bisecting the angle BAF; then will AC be a tangent to the curve at the point A.

For, if possible, let the line AC meet the curve in some other point as D. Join DF, DB, and BF; also, draw DE perpendicular to the directrix.

E
B

VF

D

ཎ་

Since, in the two triangles ACB, ACF, AF is equal to AB (Def. 1), AC is common to both triangles, and the angle CAB is, by supposition, equal to the angle CAF; therefore CB is equal to CF, and the angle ACB to the angle ACF.