SECTION IV.

OF CONIC SECTIONS, AND OF THE FIGURES ARISING FROM, OR DEPENDING ON, THEM.

I.

C

DEFINITIONS.

MONIC Sections are the plane figures formed by cutting a cone.

According to the different pofitions of the cutting plane there will arife five different figures or fections.

2. If the cutting plane pafs through the vertex, and any part of the bafe, the fection will be a triangle.

3. If the cone be cut parallel to the bafe, the fection will be a circle.

4. The fection is called an ellipfis, when the cone is cut obliquely through both fides.

D

5. The fection is a parabola, when the cone is cut parallel to one of its fides.

6. The fection is an hyperbola, when the cutting plane meets the oppofite cone continued above the vertex, where it will make another fection or hyperbola.

7. The vertices of any fection, are the points where the cutting plane meets the oppofite fides of the cone.

8. The tranfverfe axis is the line between the two vertices. And the middle point of the tranfverfe, is the center of the conic fection.

9. The conjugate axis, is a line drawn through the center, and perpendicular to the tranfverfe.

10. A diameter is any right line drawn through the center, and terminated on each fide by the curve; the interfections of the diameter and curve being the vertice

$ 4

vertices of the diameter.-Hence every diameter of the ellipfe and hyperbola have two vertices; but of the parabola, only one; unlefs we confider the other as infinitely diftant. And hence, alfo, all the diameters of a parabola are parallel to each other, and infinite.

11. If a tangent to the curve be drawn through the vertex of any diameter, and another diameter be drawn parallel to the tangent, thofe diameters are faid to be conjugates the one to the other.-The axes, or principal conjugate diameters, are perpendicular to each other.

12. An abfcifs is any part of a diameter, terminated at the vertex.

13. An ordinate to any diameter, is a line contained between the diameter and the curve, and is parallel to the conjugate diameter, or to the tangent at the vertex.-The ordinates to the axe are perpendicular to it. And in the ellipfe and hyperbola, every ordinate hath two abfciffes, in the parabola only one.

14. The parameter of any diameter, is a third proportional to that diameter and its conjugate.

15. The focus is the point of intersection of the axe and an ordinate, to it, which is equal to half the parameter of the axe.-The ellipfe and hyperbola have each two foci, the parabola only one.

16. A fpheroid, or ellipfoid, is a folid generated by the revolution of an ellipfe about one of its axes. It is a prolate one, when the revolution is made about the tranfverfe axis; and oblate, when about the conjugate.

17. A conoid is a folid formed by the revolution of a parabola, or hyperbola, about the axis. And is accordingly called parabolic, or hyperbolic. The parabolic conoid is alfo called a paraboloid; and the hyperbolic conoid, an hyperboloid.

If any folid, formed by the revolution of a conic fection about its axe, i. e. a fpheroid, paraboloid, or hyperboloid, be cut by a plane in any pofition; the fection will be fome conic fection, and all the parallel fections will be like and fimilar figures.

DEMONSTRATION.

Let ABC be the generating fection, or a fection of the given folid through its axe BD, and perpendicular to the propofed fection AFC, their common

interfection

B

interfection being AC; let GH be any other line meeting the generating fection in G and H, and cutting AC in E; and erect EF perpendicular to the plane ABC, and meeting the propofed plane in F.

A

Then, if AC and GI be conceived to be moved continually parallel to themfelves, will the rectangle AE X EC be to the rectangle GE X EH, always in a conftant ratio; but if GH be perpendicular to BD, the points G, F, H will be in the circumference of a circle whofe diameter is GH, fo that GE X EH will be EF; therefore AEX EC will be to EF, always in a conftant ratio; confequently AFC is a conic fection, and every fection parallel to AFC will be of the fame kind with, and fimilar to, it. 2. E. D.

Corol. I. The above conftant ratio, in which AE X EC is to EF2, is that of KI2 to IN2, the fquares of the diameters of the generating fection refpectively parallel to AC, GH; that is, the ratio of the fquare of the diameter parallel to the fection, to the fquare of the revolving axe of the generating plane.

This will appear by conceiving Ac and GH to be moved into the pofitions KL, MN, interfecting in 1, the center of the generating fection.

Corol.