gula is evidently proportional to the length of those lines, forming one of the dimensions of one of its sides.

Cor. 6. Since all the vertical sections of a right revoloid except those bisecting its sides at right angles, are ellipses, (Prop. I, Cor. 1.) it may be inferred that an elliptical revoloid is also equal to two-thirds of its circumscribing prism; and hence that an ellipsoid is also equal to two-thirds its circumscribing cylinder. For the ellipsoid evidently bears the same relation to the elliptical revoloid as the sphere does to the right revoloid, since the elliptical revoloid may, by multiplying the number of the sides of its prime, be shaded off into the ellip soid without changing its relation to its circumscribing prism, which in such case becomes a cylinder.

Scholium. Lest this latter corollary should not appear sufficiently satisfactory in view of the important principle enunci ated; and as it may seem to require a more rigorous demonstration, it will be made the subject of the following propo

sition.

PROPOSITION VII. THEOREM.

An elliptical revoloid is equal to two thirds its circumscribing prism; and an ellipsoid is equal to two-thirds of its circumscribing cylinder.

Let us imagine the annexed A figure ABCD, to be a conjugate section of a right quadrangular revoloid by a plane through its centre, and let abcd, be regarded as another conjugate section of the re- H voloid between the central section and the extremity of the axis, and let them be so projected that the axis I, perpendicular to the plane c of projection may be sup

posed to pass through their centres; and let the circles EFGH, and efgh be similar sections of a sphere of the same diameter and similarly posited.

Now let it be conceived that vertical sections or ungulas be taken from each side of the revoloid and sphere; such that KIL, kll, shall be sections of the ungula from the side

AIB, then will the sectoral portions of those triangles be respectively, sections through the spherical ungulas; now let the plane sections of the revoloid and sphere be reduced so as to close the spaces formed by the triangles so removed, but retaining their former figures. Thus let ABCD be reduced to opqr; abdc to stav, &c., then, because (Prop. XXIII., B. I., El. Geom.) if any number of squares are proportional, the sides of those squares are proportional, it follows that if P be the length of the side AB of the revoloidal section, and P' = the length of the sides or, and if p = the length of the side ab and p' the length of the side st; then will P' the square ABDC and Pthe same square reduced, or opqr; p' will=the square abdc and p" the same square reduced, or stuv. And we have shown that P2: P′′ : : p2 : p"

H

=

Hence also (Prop. XX, B. I, El. Geom.) P: P'::p:p'

That is, as the side AB is to the same side reduced, so is the side ab, to the same side reduced, and hence the sides, or diameters of those sections are reduced by the removal of the ungulas, in the ratio of those sides, or their diameters respectively, and because the circumferences of circles are as their diameters, and their areas as their squares, the diameters of the circular sections of the sphere, are each reduced in the ratio of these diameters; and the same will hold true with regard to any parallel sections of the revoloid or sphere.

Now the several radii El, el, &c. of the several sections may be regarded as ordinates to the vertical axis, I, of the revoloid and sphere; and the radii ml, wl, &c., may be regarded as the corresponding ordinates of the vertical sections of the solids so reduced. Now, since these latter ordinates are severally proportional to their corresponding ordinates of a vertical circular section of the revoloid and sphere, it follows that the same sections made by the same plane, through the solids so reduced are ellipses, for (Prop. XII, Cor. Ellipse) this is a property of an ellipse, when compared with a circle; hence the revoloid becomes an elliptical revoloid (Def. 10) and the sphere becomes an ellipsoid.

Now if a prism is supposed to circumscribe the revoloid before being reduced, and if a cylinder is supposed to circumscribe the sphere, they must in order to accommodate themselves to the elliptical revoloid and ellipsoid, be reduced in every conjugate section, equal in amount to the reduction of the central conjugate sections of the revoloid and sphere; and hence the prism will have been reduced in the same ratio as that of the revoloid; and the cylinder will in like manner, have been reduced in the same ratio as that of the sphere, so that if (Prop. VI) a right revoloid is equal to of its circumscribing

prism, and a sphere is equal to of its circumscribing cylin der, then also will an elliptical revoloid be equal to of its circumscribing prism, and an ellipsoid will be equal to of its circumscribing cylinder.

Cor. Since an elliptical revoloid is formed of ungulas, cut from an elliptical cylinder (Prop. I, Cor. 2) whose bases are severally the semi-base of the cylinder, it follows that such ungulas of an elliptical cylinder, are equal to of their respective circumscribing prisms.

Scholium. Since the solidity of a cylinder may be expressed by RH (Prop. II., Sch. B. III., El. S. Geom.,) that is since its solidity is as the square of its radius or diameter multiplied by its height, it follows that ellipsoids are proportional to each other as the square of their revolving axes mul tiplied by their fixed axes, and hence the same is true also of the elliptical revoloid, If RH = the solidity of a cylinder, 2 = RH the solidity of a spheroid

the solidity of a prism will be 4R*H,

and the revoloid will be R2H,

or if D = 2R then the revoloid is = D'H.

Cor. 2. As in Cor. 3, prop. VI, in relation to the segments of a right revoloid, or a sphere, so in relation to the segments of an elliptical revoloid, or spheroid, they are respectively equal to the difference between the corresponding sections of their circumscribing prisms or cylinders, and inscribed pyramids or cones. (See diagrams Prop. VI).

Cor. 3. Since a spheroid is equivalent to a sphere drawn out as in the case of a prolate, or contracted, as in the case of an oblate spheroid, and in such manner, as that every line or section through the spheroid, in the direction of the expansion or contraction, is drawn out or contracted, in the ratio of the increase or decrease of the axis in such direction, it follows that any segment of a spheroid, by a plane parallel to its axis of revolution, is to a corresponding segment of its inscribed sphere, if a prolate spheroid, or that of its circumscribing sphere if the spheroid is oblate, as the diameter of that sphere to the axis of the spheroid, or as in the conjugate axis of the seg ment's base to the transverse of the base, or the axis parallel to the axis of the spheroid.

And any segment by a plane perpendicular to the axis is also proportional to a corresponding segment of the sphere from which it may be conceived to be produced as the axis of the sphere to the axis of the spheroid, or as the height of a similar spherical segment to the height or altitude of the seg ment of the spheroid.

H

jad:

Scholium. Let ABCD be the complement of a cylinder from which is taken the ungula GEFH-a quadrant of a revoloid, and also a similar opposite ungula, cut by planes meeting in the diameter EF; and there may be taken two cones whose bases are the two bases of the complement, and whose vertices D Instaibio F to 2910pod are in the centre J, and the parts remaining will be equal to the remaining cylindrical surthe radius of the base or distance JF. (Prop. IV.,

face B. II.)

Let the complement be divided in the line EF, and let the segments be inverted so that the bases shall comprehend the line EF, and if the planes ABJ, DCJ cut off the segments AEBJ, DFCJ then there shall be left the pyramid AJ,BJ,CJ,DJ, whose base is equal to a central section of the cylinder along its axis; viz., ABCD, and its vertical height is equal to the radius of the cylin- D der; and as each side of the cy

linder is supposed to be cut alike, we shall have two of those pyramids, which together are equal to one-third of the prism circumscribing the revoloid.

It follows therefore that the two ungulas together with the two pyramids are equal to a full quadrangular revoloid.

Hence there remains four portions ABJFH to be determined, which when placed together, so that their several vertices J, shall coincide, their cylindric surfaces turned inward, their plane surfaces will be outward, forming a pyramid equal to

one of the former pyramids, minus a pyramidal portion PSQRJ, which is required to complete the pyramid. It will be perceived that every section pqrs of this latter solid, parallel to the base is a square, and the square of pq, the versed sine of the arc Jp, therefore this solid is equal to the squares of an infinite series of equi

distant versed sines drawn into their distance; or is to its circumscribed prism, erected on the same base PSQR as an infinite series of the squares of equidistant versed 's sines to a similar series of the

J

squares of radii, as will be more fully discussed in another place.

PROPOSITION VIII. THEOREM.

If the solidity of a sphere is equal to one or several cylindrical ungulas of the same cylinder, the surface of the sphere will also be equal to the cylindrical surface of such ungula or ungulas.

Let HK be a sphere AJB a cylindric ungula equal to the sphere, then will the surface of the sphere be equal to the cylindrical surface of the ungula.

For let an indefinite number of planes be passed through the two solids perpendicular to the axis J of the sphere, formed by the intersection of the planes LOJ, MNJ, and the sphere will be divided into an indefinite number of circles, from the great circle of the sphere down to the smallest about the axis, and the ungula will be divided in like manner into an indefinite number of similar triangles, with bases AB, cc, bb, &c., which was shown (Prop. IV) to be equal to the circumfer ence of the circles through the corresponding sections of the sphere; and because this is the case throughout, it follows