But all the AF's form an arithmetical progression, beginning at 0 or nothing, and having the greater term, and the sum of all the terms, each expressed by the whole axis AD.

And since the sum of all the terms of such progression is equal to AD, multiplied by AD, or AD2, half the product of the greatest term, and the number of terms; therefore AD is equal to the sum of the AF's, and consequently 4p×+AD2, or 2×pXAD2 is the sum of the revoloid. But by the properties of the parabola p: DC: : DC: AD,

or p

DC2

AD

; consequently 2XpX AD2, becomes 2×AD XDC for the solid content of the revoloid. But 4X AD XDC is equal to the prism HIBC, consequently the parabolic revoloid is equal to half of its circumscribing prism, and from a parity of reasoning, with regard to the paraboloid and cylinder, the paraboloid is equal to half its circumscribing cylinder.

Cor. Hence each of the ungulas of which the parabolic revoloid is composed is equal to half its circumscribing prism, and these ungulas are such as are cut from a parabolic cylinder or prism, by planes meeting in the vertical plane, passing through the vertices of its two parabolic bases.

PROPOSITION XIII. THEOREM.

The solidity of a frustum of a polar hemisphere of a parabolic revoloid is equal to a prism of equal altitude, and whose base is half the sum of the two bases of the frustum; and the solidity of a frustum of a polar hemisphere of a paraboloid is equal to that of a cylinder of equal altitude, and whose base is equal to half the sum of the two circular bases of the frustum.

For in the frustum BEGC last proposition. 2pX AD2= the solid ABC

and 2pXAF2 = the solid AEG. Therefore the difference 2p× (AD2 — AF2)

BEGC.

But AD-AF2 DFX (AD+AF)

=

the frustum

therefore, 2pXDFX (AD+AF) = the frustum BEGC. But, by the parabola pxAD = DC', and pXAF = FG3, therefore 2X DFX (DC+FG) = the frustum BEGC,

= =

that is frustum BEGC half the sum DC, FG, of the frustum multiplied by the altitude DF.

And for the same reasons as adduced in the the last proposition, this demonstration is equally applicable to the frustum of a revoloid or a paraboloid.

Cor. Hence any conjugate section of an ungula, being part of a polar hemisphere of a parabolic revoloid, is equal to a prism of equal altitude, and whose base is equal to half the sum of the two bases.

PROPOSITION XIV. THEOREM.

A conjugate parabolic revoloid is equal to of its circumscribing prism; and a conjugate parobolic spindle is equal to of its circumscribing cylinder.

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15

For this revoloid is composed of parabolic ungulas, such as are cut from the vertical side of a parabolical prism, which (Prop. XVIII. Cor. 2, B II.) are equal to their circumscribing prisms; hence a number of associated ungulas are equal to their associated prisms, and since the inscribed paraboloid bears the same proportion to its circumscribing cylinder, as the revoloid to its circumscribing prism; hence a conjugate paraboloid or a parabolic spindle is of its circumscribing prism.

Scholium. The frustum of a parabolic spindle may be resolved into three portions-first, a cylinder whose base is one of the bases of the frustum, and whose altitude is the length of the frustum-second, the angular portion of the ring remaining after taking away the cylinder, which is equivalent to a parabolic spindle formed by the revolution of the section of the ring on the chord or double ordinate-third, a parabolic prism, whose base is a section of the ring, and whose altititude is equal to the inner circumference of the ring.

The first is equal to its base multiplied by its altitude. The second is equal to of its circumscribing prism; the third is equal to of its circumscribing rectangular prism.

PROPOSITION XV. THeorem.

If any solid, formed by the rotation of a conic section about its axis, that is a spheroid, paraboloid, or hyperboloid, be cut by a plane in any position; the section will be some conic section, and all the parallel sections will be like and similar figures.

Let ABC be the generating section, or a section of the given solid through its axis BD, and perpendicular to the proposed section AFC, their common intersection being AC; and GH be any other line meeting the generating section in G and H, and cutting AC in E; and erect EF perpendicular to the plane ABC, and meeting the proposed plane in F.

Then, if AC and GH be conceived to be moved continually parallel to themselves, will the rectangle AE × EC be to the rectangle GE X EH, always in a constant ratio; but if GH be perpendicular to BD, the points G, F, H will be in the circumference of a circle whose diameter is GH, so that GE × EH will be EF; therefore AE x EC will be to EF, always in a constant ratio; consequently AFC is a conic section, and every section parallel to AFC will be of the same kind with it, and similar to it.

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Cor. 1. The above constant ratio, in which AE × EC is to EF2, is that of KI2 to IN2, the squares of the diameters of the generating section respectively parallel to AC, GH; that is, the ratio of the square of the diameter parallel to the section, to the square of the revolving axis of the generating plane.

This will appear by conceiving AC and GH to be moved into the positions KL, MN, intersecting in I, the centre of the geernating section.

Cor. 2. And hence it appears, that the axes AC and 2EF of the section, supposing 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 section.

Cor. 3. If the section of the solid be made so as to return into itself, it will evidently be an ellipse. Which always happens in the spheroid, except when it is perpendicular to the axis; which position is also to be excepted in the other solids, the section being always then a circle: in the paraboloid the section is always an ellipse, excepting when it is parallel to the axis; and in the hyperboloid the section is always an ellipse, when its axis makes with the axis of the solid, an angle greater than that made by the said axis of the solid and the asymptote of the generating hyperbola; the section being an hyperbola in all other cases, but when those angles are equal, then it is a parabola.

Cor. 4. But if the section be parallel to the fixed axis BD, it will be of the same kind with, and similar to, the generating plane ABC; that is, the section parallel to the axis, in a spheroid, is an ellipse similar to the generating ellipse; in the paraboloid, the section is a parabola similar to the generating parabola; and in an hyperboloid, it is an hyperbola similar to the generating hyperbola of the solid.

Cor. 5. In the spheroid, the section through the axis is the greatest of the parallel sections; but in the hyperboloid, it is the smallest ; and in the paraboloid, all the sections parallel to the axis, are equal to one another.-For, the axis is the greatest parallel chord line in the ellipse, but the least in the opposite hyperbolas, and all the diameters are equal in a parabola. Cor. 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 midde of AC, or AE the semi-axis, and GH parallel to MN. Then EO will be equal to EF, the other semi-axis of the section.

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

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

Cor. 7. Draw AP parallel to the axis BD of the solid, and meeting the perpendicular GH in P. Then it will be evident that, in the spheroid, the semi-axis EF = EO will be greater than EP; but in the hyperboloid, the semi-axis EF = EO, of the elliptic section, will be less than EP; and in the paraboloid, EF EO is always equal to EP.

Scholium. The analogy of the sections of an hyperboloid to those of the cone, are very remarkable, all the three conic sections being formed by cutting an hyperboloid in the same position as the cone is cut.

Thus, let an hyperbola and its asymptote be revolved together about the transverse axis, the former describing an hyperboloid, and the latter a cone circumscribing it; then let them be supposed to be both cut by a plane in any position and the two sections will be like, similar, and concentric figures; that is, if the plane cut both sides of each, the sections will be concentric, similar ellipses; if the cutting plane be parallel to the asymptote, or to the side of the cone, the sections will be parabolas; and in all other positions, the sections will be similar and concentric hyperbolas.

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That the sections are like figures, appears from the foregoing corollaries. That they are concentric, will be evident when we consider that Cc is Au, producing AC both ways to meet the asymptotes in a and c. And that they are similar, or have their transverse and conjugate axes proportional to each other, will appear thus: Produce GH both ways to meet the asymptotes in g and h; and on the diameters GH, gh, describe the semi-circles GQH, gRh, meeting EQR, drawn perpendicular to GH, in Q and R; EQ and ER being then evidently the semi-conjugate axes, and EC, Ec, the semi-transverse axes of the sections. Now if GH and AC be conceived to be moved parallel to themselves, AE x EC or CE2, will be to GE × EH or EQ2, in a constant ratio, or CE to EQ will be a constant ratio; and since cE is as Eg, and aE as Eh aE × Ec or cE2, will be to gE × Eh or ER2, in a constant ratio, or cE to ER will be a constant ratio; but at an infinite distance from the vertex, C and c coincide, or EC Ec, as also EG Eg, consequently EQ is then ER, and CE to EQ

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