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Corol. 2. And hence it appears that the axes AC and 2EF of the fection, fuppofing E now to be the middle of AC, will be to each other, as the diameter KL is to the diameter MN of the generating fection.

Corol. 3. If the fection of the folid be made fo as to return into itfelf, it will evidently be an ellipfe. Which always happens in the fpheroid, except when it is perpendicular to the axe; which pofition is also to be excepted in the other folids, the fection being always then a circle: in the paraboloid the fection is always an ellipfe, excepting when it is parallel to the axe and in the hyperboloid the fection is always an ellipfe, when its axe makes with the axe of the folid, an angle greater than that made by the faid axe of the folid and the afymptote of the generating hyperbola; the fection being an hyperbola in all other cafes, but when thofe angles are equal, and then it is a parabola.

Corol. 4. But if the fection be parallel to the fixed axe BD, it will be of the fame kind with, and fimilar to, the generating plane ABC; that is, the fection parallel to the axe, in a fpheroid, is an ellipfe fimilar to the generating ellipfe; in the paraboloid, the fection is a parabola fimilar to the generating parabola; and in an hyperboloid, it is an hyperbola fimilar to the generating hyperbola of the folid.

Corol. 5. In the fpheroid, the fection through the axe is the greatest of the parallel fections; but in the hyperboloid, it is the leaft; and in the paraboloid, all the fections parallel to the axe, are equal to one

another.

For the axe is the greatest parallel chord line in the ellipfe, but the leaft in the oppofite hyperbolas, and all the diameters are equal in a parabola.

I

Corol.

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Corol. 6. If the extremities of the diameters KL, MN, be joined by the line KN, and Ao be drawn parallel to KN, and meeting GEH in o, E being the middle of AC, or AE the femi-axe, and GH parallel to MN. Then Eo will be equal to EF, the other femi-axe of the fection.

For, by fimilar triangles, KI IN AE: EO.

Or upon GH as a diameter defcribe a circle meeting EQ, perpendicular to GH, in q; and it is evident that EQ will be equal to the femi-diameter EF.

Corol. 7. Draw AP parallel to the axe BD of the folid, and meeting the perpendicular GH in P. And it will be evident that, in the fpheroid, the femi-axe EF EO will be greater than EP; but in the hyperboloid, the femi-axe EF LO, of the elliptic fection, will be lefs than EP; and in the paraboloid, EF = Eo is always equal to EP.

SCHOL I U M.

The analogy of the fections of an hyperboloid to thofe of the cone, are very remarkable, all the three conic fections being formed by cutting an hyperboloid in the fame pofitions as the cone is cut.

Thus,

Thus, let an hyperbola and its afymptote be revolved together about the tranfverfe axe, the former defcribing an hyperboloid, and the latter a cone circumfcribing it; then let them be fuppofed to be both cut by a plane in any pofition, and the two fections will be like, fimilar, and concentric figures: that is, if the plane cut both the fides of each, the fections will be concentric, fimilar ellipfes; if the cutting plane be parallel to the afymptote, or to the fide of the cone, the fections will be parabolas; and in all other pofitions, the fections will be fimilar and concentric hyperbolas.

Aa,

That the fections are like figures, appears from the foregoing corollaries. That they are concentric, will be evident when we confider that cc is producing AC both ways to meet the afymptotes in a and c. And that they are fimilar, or have their tranfverfe and conjugate axes proportional to each other, will appear thus: Produce GH both ways to meet the afymptotes in g and b; and on the diameters GH, gb, defcribe the femi-circles GQH, gRb, meeting EQR, drawn perpendicular to GH, in q and R; EQ and ER being then evidently the femiconjugate axes, and EC, EC, the femi-tranfverfe axes of the fections. Now if GH and AC be conceived to be moved parallel to themfelves, AE X EC or CE will be to GE X EH or EQ, in a conftant ratio, or CE to EQ will be a conftant ratio; and fince CE is as Eg, and aɛ as Eh, аE X Ec or CE will be to ge × Eb or ER2, in a conftant ratio, or CE to ER will be a conftant ratio; but at an infinite diftance from the vertex, c and coincide, or EC EC, as alfo EG Eg, confequently EQ ER, and then CE to EQ will be

CE to ER; but as thefe ratios are conftant, if they be equal to each other in one place, they muft be always fo; and confequently CE: EC :: QE: ER.

And

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And this analogy of the fections will not feem ftrange, when we confider that a cone is a fpecies of the hyperboloid; or a triangle a fpecies of the hyperbola, whofe axes are infinitely little.

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If sI be the femi-diameter belonging to the double ordinate AEC of the generating plane, AEC being the diameter of the fection AFC, conceived to be moved continually parallel to itself; and if x denote any part of the diameter s1, intercepted by E the middle of Ac, and any given fixed point taken in s1; then will the fection AFC be always as a + bx + cxx; a, b, c, being conftant quantities; b in fome cafes affirmative, and in others negative; c being affirmative in the hyperbola, negative in the ellipfe, and nothing in the parabola; and a may always be fuppofed to denote the distance of the given fixed point from the vertex s.

DEMONSTRATION.

In any conic fection, ac2 is as a + bx + cxx; but all the parallel fections are like and fimilar figures, and fimilar plane figures are as the fquares of their like dimenfions; therefore the fection AFC is as AC2, that is, as a + bx + cxx. 2. E.D

Corollary. If the given fixed point, where x begins, coincide with the vertex s, then will a be equal to nothing, and the fection will be as bxcxx, or as xdxx, in the hyperbola and ellipfe, and as bx, or as x, in the parabola.

SEC

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D

RAW the tranfverfe TR, and its conjugate co, bifecting each other perpendicularly in

the center c.

With the radius Tc and center c, defcribe an arc cutting TR in the points F, f; which will be the two foci of the ellipfe.

Take any point P in the tranfverfe; then with the radii TP, PR, and centers F, f, defcribe two arcs interfecting in 1; fo will the point 1 be in the curve or circumference of the ellipfe.

And thus, by affuming feveral points P in the tranfverfe, there will be found as many points in the

The truth of this conftruction will appear, by obferving that the tranfverfe axe is equal to the fum of two lines drawn from the foci to meet in any point in the curve.

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