PROPOSITION X. THEOREM.

If the triangle BAC and the rectangle BE, having the same base and the same altitude, turn simultaneously about the common base BC, the solid described by the revolution of the triangle will be a third of the cylinder described by the revolution of the rectangle.

On the axis, let fall the per- F pendicular AD: the cone described by the triangle ABD is

the third part of the cylinder de-
scribed by the rectangle AFBD;
(4.8.Cor.;) also the cone described B

E

D C

by the triangle ADC is the third part of the cylinder described by the rectangle ADCE: hence the sum of the two cones, or the solid described by ABC, is the third part of the two cylinders taken together, or of the cylinder described by the rectangle BCEF.

If the perpendicular AD falls F without the triangle; the solid described by ABC will, in that case, be the difference of the two cones described by ABD and ACD ; but B

E A

C D

at the same time, the cylinder described by BCEF will be the difference of the two cylinders described by AFBD and AECD. Hence the solid described by the revolution of the triangle, will still be a third part of the cylinder described by the revolution of the rectangle having the same base and the same altitude. Therefore,

If a triangle and a rectangle having the same base and the same altitude revolve together about a common axis, the solid described by the triangle will be a third of the cylinder described by the rectangle.

Scholium. The circle of which AD is radius, has for its measure × AD2; hence ж× AD3× BC measures the cylinder described by BCEF, and × AD3× BC measures the solid described by the triangle ABC.

The solidity of a sphere is equal to the product of its surface into a third of its radius.

Let us suppose the sphere to be filled with any number of regular inscribed pyramids standing on the surface of the sphere, and having their common A vertex in the centre of the sphere; and let ABDEFG, be a section of the sphere, and CAB, CBD, CDE, &c., be sections

E

H

of the inscribed pyramids made by a plane passing through the centre of the sphere and the axes of the pyramids. Bisect BD in H, and join CH: CH is the axis of a pyramid of which CBD is a section.

For, since the pyramids are regular, their axes are perpendicular to the plane of their bases; hence the plane AGH, which passes through these axes, must also be perpendicular to their bases; and since CB=CD, being radii of the same sphere, CH is perpendicular to BD; (11. 1. Sch. ;) therefore CH being let fall from the vertex C perpendicular to the plane of the base and passing through the centre of this base, must be the axis to the pyramid CBD; (Def. 14. 7 ;) and it is also the altitude of the pyramid. (Def. 15. 7.) In like manner it may be shown to be the altitude of all the inscribed pyramids CDE, CEF, &c. (8. 2.) Now if the arcs AB, BD, &c., be continually bisected, the number of sides of the polygon ABDEFG, being indefinitely increased,

will become indefinitely small, and will ultimately coincide with the circumference of the great circle ADF; (9.5. Cor. 1;) consequently the bases of the inscribed pyramids will lie wholly in the surface of the sphere, and the areas of their bases will be equal to the surface of the sphere; also CH the common altitude of the pyramids, will terminate in the surface, and be equal to the radius of the sphere.

Now since the inscribed pyramids exactly fill the sphere, the sum of their solidities must be equal to the solidity of the sphere. (Ax. 13.) But the solidity of a pyramid is equal to the product of its base by a third of its altitude; hence, the solidity of all the inscribed pyramids is equal to the sum of all their bases multiplied by a third of their altitude.

But the sum of all the bases is equal to the surface of the sphere and their altitude CH, is equal to the radius of the sphere. Therefore,

The solidity of the sphere must be equal to the product of its surface into a third of its radius.

Cor. The surfaces of spheres being as the squares of their radii, these surfaces multiplied by the squares of the radii, must be as the cubes of the latter. Hence,

The solidities of two spheres are as the cubes of their radii, or as the cubes of their diameters.

Scholium. Let R be the radius of a sphere; its surface will be expressed by 47R3, and its solidity by 4πR2 × R, or R. If the diameter is called D, we shall have RD, and R'D': hence the solidity of the sphere may likewise be expressed by

The surface of a sphere is to the whole surface of the circumscribed cylinder including its bases as 2 is to 3. The solidities of these two bodies are to each other in the same ratio.

Let MNPQ be a great circle of the sphere; ABCD the circumscri- D bed square if the semicircle PMQ and the half square PADQ are at the same time made to revolve about the M diameter PQ, the semicircle will generate the sphere, while the halfsquare will generate the cylinder circumscribed about that sphere.

A

B

P

The altitude AD of that cylinder is equal to the diameter PQ; the base of the cylinder is equal to the great circle, since its diameter AB is equal to MN; hence the convex surface of the cylinder is equal to the circumference of the great circle multiplied by its diameter. (1.8.) This measure is the same as that of the surface of the sphere. (9.8.) Hence,

The surface of the sphere is equal to the convex surface of the circumscribed cylinder.

But the surface of the sphere is equal to four great circles; hence the convex surface of the cylinder is also equal to four great circles and adding the two bases, each equal to a great circle, the total surface of the circumscribed cylinder will be equal to six great circles; hence the surface of the sphere is to the total surface of the circumscribed cylinder as 4 is to 6, or as 2 is to 3; which was the first part of our Proposition.

In the next place, since the base of the circumscribed

cylinder is equal to a great circle, and its altitude to the diameter, the solidity of the cylinder will be equal to a great circle multiplied by its diameter. (2. 8.) But the solidity of the sphere is equal to four great circles multiplied by a third of the radius; (11.8;) in other words, to one great circle multiplied by of the radius, or by of the diameter; hence the sphere is to the circumscribed cylinder as 2 to 3, and consequently the solidities of these two bodies are as their surfaces.

Scholium. Conceive a polyedron, all of whose faces touch the sphere; this polyedron may be considered as formed of pyramids, each having for its vertex the centre of the sphere, and for its base one of the polyedron's 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 a third of the radius: hence the whole polyedron will be equal to its surface multiplied by a third of the inscribed sphere's radius.

It is therefore manifest, that the solidities of polyedrons circumscribed about the sphere are to each other as the surfaces of those polyedrons. Thus the property, which we have shown to be true with regard to the circumscribed cylinder, is also true with regard to an infinite number of other bodies.

We might likewise have observed, that the surfaces of polygons, circumscribed about the circle, are to each other as their perimeters.