Inscribe in the semicircle ABCDE a regular semi-polygon, having any number of sides, and let OI be the radius of the circle inscribed in the polygon.

B

D

A

F

Ο

E

If the semicircle and semi-polygon be revolved about EA, the semicircle will C describe a sphere, and the semi-polygon a solid which has for its measure ŎI2× EA (Prop. XIII.); and this will be true whatever be the number of sides of the polygon. But if the number of sides of the polygon be indefinitely increased, the semi-polygon will become the semicircle, OI will become equal to OA, and the solid described by the semi-polygon will become the sphere: hence the solidity of the sphere is equal to OA3× EA, or by substituting 20A for EA, it becomes 7.OA3× OA, which is also equal to 4лOA2× OA. But 4л.OA? is equal to the surface of the sphere (Prop. X. Cor.): hence the solidity of a sphere is equal to its surface multiplied by a third of its radius.

Scholium 1. The solidity of every spherical sector is equal to the zone which forms its base, multiplied by a third of the radius. For, the solid described by any portion of the regular polygon, as the isosceles triangle OAB, is measured by OI2 × AF (Prop. XII. Cor.); and when the polygon becomes the circle, the portion OAB becomes the sector AOB, OI becomes equal to OA, and the solid described becomes a spherical sector. But its measure then becomes equal to 37.AO AF, which is equal to 2π.AO × AF XAO. But 27.AO is the circumference of a great circle of the sphere (Book V. Prop. XII. Cor. 2.), which being multiplied by AF gives the surface of the zone which forms the base of the sector (Prop. X. Sch. 1.): and the proof is equally applicable to the spherical sector described by the circular sector BOC: hence, the solidity of the spherical sector is equal to the zone which forms its base, multiplied by a third of the radius.

Scholium 2. Since the surface of a sphere whose radius is R. is expressed by 4лR2 (Prop. X. Cor.), 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 as the cubes of their diameters.

Scholium 3. Let R be the radius of a sphere; its surface will be expressed by 47R2, and its solidity by 47R2 × R, or R3. If the diameter is called D, we shall have R=D, and R3=1D3: hence the solidity of the sphere may likewise be expressed by

PROPOSITION XV. THEOREM.

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

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

A

Р

N

The altitude AD of the 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 (Prop. 1.). This measure is the same as that of the surface of the sphere (Prop. X.): 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 branch of the 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 (Prop. II.). But the solidity of the sphere is equal to four great circles multiplied by a third of the radius (Prop. XIV.); in other terms, 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 conse quently the solidities of these two bodies are as their surfacer

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 radius of the inscribed sphere.

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.

PROPOSITION XVI. PROBLEM.

If a circular segment be supposed to make a revolution about a diameter exterior to it, required the value of the solid which it describes.

Let the segment BMD revolve about AC. On the axis, let fall the perpendiculars BE, DF; from the centre C, draw CI D perpendicular to the chord BD; also draw the radii CB, CD.

A

The solid described by the sector BCD is measured by a CB2.EF (Prop. XIV. Sch. 1).

C

But the

solid described by the isosceles triangle DCB has for its measure.CI.EF (Prop. XII. Cor.); hence the solid described by the segment BMD=37.EF.(CB2-CI). Now, in the rightangled triangle CBI, we have CB2-CI-BI-BD2; hence the solid described by the segment BMD will have for its mea-' sure 7.EF.BD2, or 17.BD2.EF: that is one sixth of into the square of the chord, into the distance between the two perpendiculars let fall from the extremities of the arc on the

axis.

Scholium. The solid described by the segment BMD is to the sphere which has BD for its diameter, as 7.BD2 EF is to .BD3, or as EF to BD.

PROPOSITION XVII. 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.

Let BE, DF, be the radii of the two bases of the segment, EF its altitude, the segment being described by the revolution of the circular space BMDFE about the axis FE. The solid described by the segment BMD is equal to .BD2.EF D (Prop. XVI.); and the truncated cone described by the trapezoid BDFE is equal to 17.EF.(BE2+DF2+ BE.DF) (Prop. VI.);

hence the segment of the sphere, which is the sum of those two solids, must be equal to 7.EF.(2BE2+2DF2+2BE.DF+BD?) But, drawing BO parallel to EF, we shall have DO=DF-BE, hence DO2 DF2_2DF.BE+BE2 (Book IV. Prop. IX.); and consequently BD2=BO2+DO2=EF2+DF2—2DF. BE+BE2. Put this value in place of BD2 in the expression for the value of the segment, omitting the parts which destroy each other we shall obtain for the solidity of the segment,

EF.(3ВE2+3DF2 + EF2),

an expression which may be decomposed into two parts; the

one †Ã.EF.(3BE2+3DF2), or EF.(".BE2+

2

being the

half sum of the bases multiplied by the altitude; while the other .EF represents the sphere of which EF is the diameter (Prop. XIV. Sch.): hence every segment of a sphere, &c.

Cor. If either of the bases is nothing, the segment in question becomes a spherical segment with a single base; hence any spherical segment, with a single base, is equivalent to half the cylinder having the same base and the same altitude, plus the sphere of which this altitude is the diameter.

General Scholium.

Let R be the radius of a cylinder's base, H its altitude: the solidity of the cylinder will be лR2 x H, or "R2H.

Let R be the radius of a cone's base, H its altitude: the solidity of the cone will be R2 × H, or πR2H.

Let A and B be the radii of the bases of a truncated cone,

H its altitude: the solidity of the truncated cone will be 17.H. (A2+B2+AB).

Let R be the radius of a sphere; its solidity will be πR3. Let R be the radius of a spherical sector, H the altitude of the zone, which forms its base: the solidity of the sector will be 7R2H.

Let P and Q be the two bases of a spherical segment, H its P+Q.H+π.H3.

altitude: the solidity of the segment will be

2

If the spherical segment has but one base, the other being nothing, its solidity will be PH+‡πH3.

1. A spherical triangle is a portion of the surface of a sphere, bounded by three arcs of great circles.

These arcs are named the sides of the triangle, and are always supposed to be each less than a semi-circumference. The angles, which their planes form with each other, are the angles of the triangle.

2. A spherical triangle takes the name of right-angled, isosceles, equilateral, in the same cases as a rectilineal triangle. 3. A spherical polygon is a portion of the surface of a sphere terminated by several arcs of great circles.

4. A lune is that portion of the surface of a sphere, which is included between two great semi-circles meeting in a common dia.neter.

5. A spherical wedge or ungula is that portion of the solid sphere, which is included between the same great semi-circles, and has the lune for its base.

6. A spherical pyramid is a portion of the solid sphere, included between the planes of a solid angle whose vertex is the centre. The base of the pyramid is the spherical polygon intercepted by the same planes.

7. The pole of a circle of a sphere is a point in the surface equally distant from all the points in the circumference of this circle. It will be shown (Prop. V.) that every circle, great or small, has always two poles.