are in the centre J, and the

E

H

parts remaining will be equal to the remaining cylindrical surface X the radius of the base or distance JF. (Prop. IV.,

B. II.)

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 pars 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 equidistant versed sines drawn into

J

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

S

R

sines to a similar series of the

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 circumference of the circles through the corresponding sections of the sphere; and because this is the case throughout, it follows that the surface of the sphere, which may be represented by the sum of the several circumferences of the circles, is equal to the cylindric surface of the ungula, which may likewise be represented by the sum of the several bases of the triangles.

And because cylindric ungulas of the same base are proportional as their altitudes, (Prop. IX, Cor. 2, B. II) and because their cylindrical surfaces are proportional to their altitudes, it follows that if several ungulas, cut from a cylinder of a given diameter are equal to a sphere of the same diameter, the surface of the sphere will be equal to the sum of their cylindrical surfaces.

PROPOSITION IX. THEOREM.

The solidity of a sphere as well as a revoloid, is equal to the product of its surface by one-third of its radius.

For since the revoloid is made up of sections of the cylinder, whose several solidities are equal to their curve surface multiplied by one-third of the radius of the cylinder, whence they are conceived to be taken (Prop. III, B. II) which radius is equal to the vertical height of the several elimentary pyramids of which these sections are formed, and is also the radius of the revoloid, it follows that the solidity of the whole revoloid composed of all the sections, is equal to the whole curve surface of all the sections multiplied by one-third the radius.

Thus, if the revoloid consist of six facial sides a,b,c,d,e,f, the solidity of each of which is equal to its surface X + radius

or

r

or

3' surface a × r = solidity a, surface b× r = solidity b, &c., their surface a+b+c+d+e+fx r equal to the solidity of the whole revoloid, and as the number of sides of a revoloid may be increased indefinitely without altering the relation of its elimentary pyramids, it follows that the same relation exists between the solidity of the sphere, and its surface, as in the revoloid, viz., the solidity of a sphere, as also of a revoloid, is equal to the product of their respective surfaces by one-third of their respective radii.

Scholium. 1. Conceive also a polyedron, all of whose faces touch the sphere; this polyedron may be considerd as formed of pyramids, each having for its vertex the centre of the sphere, and for its base one of the polyedrons faces. Now, it is evident that all these pyramids will have the radius of the sphere for their common altitude; so that each pyramid will be equal to one face of the polyedron multiplied by one-third of the radius; hence the whole polyedron will be equal to its whole surface multiplied by a third of the radius of the inscribed sphere (Prop. XV. Cor. B. III. El. S. Geom.) It is therefore manifest that the solidities of polyedrons, as well as revoloids circumscribed about a sphere, are to each other as the surfaces of those polyedrons or revoloids respectively.

Now, also, as with a revoloid, the number of polyedron's faces may be inscribed till the polydron becomes identical with the sphere, and then its solidity is equal to the product of its surface with one-third of its radius; hence the sphere may be conceived to be made up of an indefinite number of indefinitely small pyramidals, whose bases when associated, form the surface of the sphere, and this surface has the same relation to the whole solid, or the sphere, as the base of each individual pyramidal has to the solidity of each.

Cor. 1. Hence the solidity of any sector of a sphere or of a revoloid is equal to its spherical or cylindrical surface multiplied by of the radius, for a sector consists of an association of regular pyramidals, the sum of whose bases form the curve surface of the sector.

Scholium. 2. Since the axis of the cylinder circumscribing a sphere is equal to its diameter, its solidity is equal to its whole surface, including the two ends multiplied by a third of the radius. For it may be conceived to be made up of two cones, whose bases are two ends of the cylinder, and vertical height = half the axis or length of the cylinder = radius; and the elementary pyramids of the curve surface, whose vertices termiinate in the centre of the cylinder with those of the cones, and hence, its solidity bears the same ratio to its surface, that the solidity of a sphere, a right revoloid, or polyedron, circumscribing a sphere do to their respective surfaces.

Cor. 2. Hence we have three orders of surfaces, which, taken as bases of pyramids, and multiplied by one-third of the distance of such base to the vertice of the pyramid, will determine the solidity of such pyramid; but, as observed in Schol. to Prop. III, B. III, El. S. Geom. in reference to cylindrical surfaces, the vertice of the pyramid with a spherical base, must be in the centre of the spherical curvature.

Cor. 3. Hence, as the solidity of a sphere is equal to twothirds that of its circumscribing cylinder, and as the surface of the sphere is equal to the curve surface of the cylinder; and as the solidities of each of these bodies are equal to the products of their respective surfaces, by one-third of their common radius, the surface of the two ends of the cylinder is equal to half the curve surface, for if a = the surface of the sphere, and x = the two bases of the cylinder, then will a + x = the whole surface of the cylinder, including the ends. Then axr or radius = the solidity of the sphere, and + ra + rx = the solidily of the cylinder.

But ra = (+ra++rx) x = ra + rx.
Transposing and dividing a = rx.

Hence a = 2x, therefore the area of the two ends is equal to half the area of the convex surface of the cylinder.

The same may be inferred from the ratio of the inscribed cones, to the remaining portion of the cylinders.

Scholium. 3. Since the surface of a sphere whose radius is R, is expressed by 4R2 (Prop. III, Cor. 2,) it follows that the surfaces of spheres are to each other as the squares of their radii; and since their solidities are as their surfaces multiplied by their radii, it follows that the solidities of spheres are to each other as the cubes of their radii or diameters, and the same is true also of revoloids. If the diameter is called D, we shall have R = + D, and R3 = +D*; hence the solidity of the sphere may likewise be expressed by × ¦ D3 = D.

PROPOSITION X. THEOREM.

Every segment of a sphere is measured by the half sum of its bases multiplied by its altitude, plus the solidity of a sphere whose diameter is this same altitude.

0

A

B

H
I

T

C

Let BH, DL, be the radii of the two bases of a segment, HL its altitude, the segment being generated by the revolution of the circular zone DLHB, about the axis AG passing D through the centre of curvature C; from Č draw CO perpendicular to the chord DB, draw also the radii CD, CB. The solid described by the section BCD is measured by , CB, LH (Prop. XXIII, Sch. 2, B. III., El. S. Geom.); but the solid described by E the isosceles triangle DCB, has for its measure. CO. LH, (Prop. XVII, Cor. B. III, El. S. Geom.,); hence the solid described by the segment BDO = « . LH .

N

R

G