Of the Sphere.

rence ON: hence, the surface described by ED+DC, is equal to (HI+IP)× circumference ON, or equal to HPX circumference ON.

THEOREM XXIII.

The surface of a sphere is equal to the product of its diameter by the circumference of a great circle.

Let ABCDE be a semicircle. Inscribe in it any regular semi-polygon, and from the centre O draw OF perpendicular to one of the sides.

Let the semicircle and the semipolygon be revolved about the axis AE: the semicircumference ABCDE will describe the surface of a sphere (Def. 26); and the perimeter of the semi-polygon will describe a surface which has for its measure AEX cir

circle.

B

E

cumference OF (Th. xxii); 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, its perimeter will coincide with the circumference ABCDE, the perpendicular OF will become equal to OE, and the surface described by the perimeter of the semi-polygon will then be the same as that described by the semicircumference ABCDE. Hence, the surface of the sphere is equal to AEX circumference OE.

Cor. Since the area of a great circle is equal to the product of its circumference by half the radius, or by one-fourth of the diameter (Bk. IV. Th. xxvii), it follows that the surface of a sphere is equal to four of its great circles.

Of the Zone.

THEOREM XXIV.

The surface of a zone is equal to its altitude multiplied by the circumference of a great circle.

For, the surface described by any portion of the perimeter of the inscribed polygon, as BC+CD is equal to EHX circumference OF (Th. xxii. Cor). But when the number of sides of the polygon is indefinitely increased, BC+CD, becomes the arc BCD, OF becomes equal to OA, and the surface described by BC+CD, becomes the surface of the zone described by the arc BCD: hence, the surface of the zone is equal to EHX circumference OA.

A

Sch. 1. When the zone has but one base, as the zone described by the arc ABCD, its surface will still be equal to the altitude AE multiplied by the circumference of a great circle. て

Sch. 2. Two zones taken in the same sphere, or in equal spheres, are to each other as their altitudes; and any zone is to the surface of the sphere as the altitude of the zone is to the diameter of the sphere.

The solidity of a sphere is equal to one third of the product of the surface multiplied by the radius.

For, conceive a polyedron to be inscribed in the sphere.

Of the Sphere.

This polyedron may be considered as formed of pyramids, each having for its vertex the centre of the sphere, and for its hase one of the faces of the polyedron. Now, the solidity of each pyramid, will be equal to one third of the product of its base by its altitude (Th. xvii).

But if we suppose the faces of the polyedron to be continually diminished, and consequently, the number of the pyramids to be constantly increased, the polyedron will finally become the sphere, and the bases of all the pyramids will become the surface of the sphere. When this takes place, the solidities of the pyramids will still be equal to one third the product of the bases by the common altitude, which will then be equal to the radius of the sphere.

Hence, the solidity of a sphere is equal to one third of the product of the surface by the radius.

The surface of a sphere is equal to the convex surface of the circumscribing cylinder; and the solidity of the sphere is two thirds the solidity of the circumscribing cylinder.

Let MPNQ be a great circle of the sphere; ABCD the circumscribing 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 describe the sphere, while the half square will describe the cylinder circumscribed about

that sphere.

N

M

F

The altitude AD, of the cylinder, is equal to the diameter

Of the Sphere.

M

N

PQ; the base of the cylinder is equal to the great circle, since its diameter AB is equal MN; hence, the convex surface of the cylinder is equal to the circumference of the great circle multiplied by its diameter (Th. ii). This measure is the same as that of the surface of the sphere (Th. xxiii): hence, the surface of the sphere is equal to the convex surface of the circumscribing cylinder.

B

In the next place, since the base of the circumscribing 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 a diameter (Th. xiv. Cor). But the solidity of the sphere is equal to its surface multiplied by a third of its radius; and since the surface is equal to four great circles (Th. xxiii. Cor.), the solidity is equal to four great circles multiplied by a third of the radius; in other words, to one great circle multiplied by four-thirds of the radius, or by two-thirds of the diameter; hence, the sphere is two-thirds of the circumscribing cylinder.

Appendix.

APPENDIX

OF THE FIVE

REGULAR POLYEDRONS.

A regular polyedron, is one whose faces are all equal polygons, and whose polyedral angles are equal. There are five such solids.

1. The Tetraedron, or equilateral pyramid, is a solid bounded by four equal triangles.

2. The hexaedron or cube, is a solid, bounded by six equal squares.

+

3. The oclaedron, is a solid, bounded by eight equal equilateral triangles.