But we have proved that the solid described by the triangle ABO, is equal to area BKAO; it is, therefore, equal to OHX surface described by AB.

Secondly. To find the value of the solid formed by the revolution of the triangle BCO.

Produce BC until it meets AG produced in L. It is evident, from the preceding demonstration, that the solid described by the triangle LCO is equal to

OMX surface described by LC;

and the solid described by the triangle LBO is equal to

C

N

D

OMX surface described by LB;

M

B

hence the solid described by the triangle BCO is equal to OMX surface described by BC.

In the same manner, it may be proved that the solid de scribed by the triangle CDO is equal to

ONX surface described by CD;

and so on for the other triangles. But the perpendiculars OH, OM, ON, &c., are all equal; hence the solid described by the polygon ABCDEFG, is equal to the surface described by the perimeter of the polygon, multiplied by OH.

Let, now, the number of sides of the polygon be indefinitely increased, the perpendicular OH will become the radius ŎA, the perimeter ACEG will become the semi-circumference ADG, and the solid described by the polygon becomes a sphere; hence the solidity of a sphere is equal to one third of the product of its surface by the radius.

Cor. 1. The solidity of a spherical sector is equal to the product of the zone which forms its base, by one third of its radius. For the solid described by the revolution of BCDO i equal to the surface described by BC+CD, multiplied b OM. But when the number of sides of the polygon is in definitely_increased, the perpendicular OM becomes the radius OB, the quadrilateral BCDO becomes the sector BDO, and the solid described by the revolution of BCDO becomes a spherical sector. Hence the solidity of a spherical sector is equal to the product of the zone which forms its base, by one third of its radius.

Cor. 2. Let R represent the radius of a sphere, D its diameter, S its surface, and V its solidity, then we shall have S=4πR or TD' (Prop. VII., Cor. 5). V=R×S=TR3 or 1πD3;

Also,

hence the solidities of spheres are to each other as the cubes of their radii

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=πR3A.

Cor. 3. Every sphere is two thirds of the circumscribea 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 cy.inder 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 1 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.

E

C

gen.

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

TCBX AD (Prop. VIII., Cor. 2).

And the solid generated by the triangle ACB, by Prop. VII., 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 TADX (CB-CF2), πADX BF2; ";

to

or

that is,

because CB-CF is equal to BF', and BF2 is equal to one "ourth of AB'.

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

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

that is,

πAD(2BD2+AB');
TAD(3BD'+ AD),

because AB is equal to BD'+AD'.

This expression may be separated into the two pirts
TAD × BD2,
¿πAD'.

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 C segment of one base generated by CD.

B

CONIC SECTIONS.

THERE are three curves whose properties are extensivelv 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.

B

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 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 H FH is bisected in the point V.

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.

A

G

F

C K

H

7. An abscissa is the part of a diameter intercepted be tween 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 R 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

F D C