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in the same manner, by circumscribing a polygon about the circle BCD, it may be shown that the cone ABCD is not greater than the third part of the cylinder BFKG; therefore it is equal to the third part of that cylinder.

PROPOSITION XXI. THEOREM.

If a hemisphere and a cone have equal bases and altitudes, a series of cylinders may be inscribed in the hemisphere, and another series may be described about the cone, having all the same altitudes with one another, and such that their sum shall differ from the sum of the hemisphere, and the cone by a solid less than any given solid.

Let ADB be a semicircle, of which the centre is C, and let CD be at right angles to AB; let DB and DA be squares described on DC, draw DE, and let the figure thus constructed revolve about DC; then the sector BCD, which is the half of the semicircle ADB, will describe a hemisphere having C for its centre (II. Def. 10), and the triangle CDE will describe a cone, having its vertex at C, and having for its base the circle (II. Def. 12) described by DE, equal to that described by BC, which is the base of the hemisphere. Let W be any given solid. A series of cylinders may be inscribed in the hemisphere ADB, and another described about the cone ECL, so that their sum shall differ from the sum of the hemisphere and the cone, by a solid less than the solid W.

Upon the base of the hemisphere let a cylinder be constituted equal to W, and let its altitude be CX.

into such a number of L

equal parts, that each

A

of them shall be less than CX; let these be CH, HG, GF, and FD. Through the points F, G, H, draw FN, GO, HP, parallel

Divide CD

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to CB, meeting the circle in the points K, L, and M; and the straight line CE in the points Q, R, and S. From the points K, L, M, draw Kf, Lg, Mh, perpendicular to GO,

HP, and CB; and from Q, R, and S, draw Qq, Rr, Ss, perpendicular to the same lines. It is evident that the figure being thus constructed, if the whole revolve about CD, the rectangles Ff, Gg, Hh, will describe cylinders (II. Def. 13) that will be circumscribed by the hemisphere BDA; and that the rectangles DN, Fq, Gr, Hs, will also describe cylinders that will circumscribe the cone ICE. Now, it may be demonstrated, as was done of the prisms inscribed in a pyramid (II. 15), that the sum of all the cylinders described within the hemisphere, is exceeded by the hemisphere by a solid less than the cylinder generated by the rectangle HB, that is, by a solid less than W, for the cylinder generated by HB is less than W. In the same manner, it may be demonstrated that the sum of the cylinders circumscribing the cone ICE is greater than the cone by a solid less than the cylinder generated by the rectangle DN, that is, by a solid less than W. Therefore, since the sum of the cylinders inscribed in the hemisphere, together with a solid less than W, is equal to the hemisphere; and since the sum of the cylinders described about the cone is equal to the cone together with a solid less than W; adding equals to equals, the sum of all these cylinders, together with a solid less than W, is equal to the sum of the hemisphere and the cone together with a solid less than W. Therefore, the difference between the whole of the cylinders and the sum of the hemisphere and the cone, is equal to the difference of two solids, which are each of them less than W; but this difference must also be less than W; therefore the difference between the two series of cylinders, and the sum of the hemisphere and cone, is less than the given solid W.

Or thus: let I= the sum of the interior cylinders within the hemisphere; E = the sum of the exterior without the cone; and Y and Z two solids each less than W. Then I+Y=H, and E=C+Z; therefore, adding equals to equals, I+E+Y=H+C+Z. Hence the difference between (I+E) and (H + C) is equal to that between Z and Y; but as these two solids are each less than W, their difference is still less than W; and hence also the difference between (I+E) and (H+ C) is less than W.

PROPOSITION XXII. THEOREM.

The same things being supposed as in the last proposition, the sum of all the cylinders inscribed in the hemisphere, and described about the cone, is equal to a cylinder, having the same base and altitude with the hemisphere.

Let the figure DCB be constructed as before, and supposed to revolve about CD; the cylinders inscribed in the hemisphere, that is, the cylinders described by the revolution of the rectangles Hh, Gg, Ff, together with those described about the cone, that is, the cylinders described by the revolution of the rectangles Hs, Gr, Fq, and DN, are equal to the cylinder described by the revolution of the rectangle DB.

Let L be the point in which GO meets the circle ADB; then, because CGL is a right angle if CL be joined, the circles described with the distances CG and GL are equal to the circle described with the distance CL (Pl. Ge. Quad. 8, Cor. 2) or GO; now CG is equal to GR, because CD is equal to DE, and therefore also, the circles described with the distances GR and GL are together equal to the circle described with the distance GO, that is, the circles described by the revolution of GR and GL about the point G, are together equal to the circle described by the revolution of GO about the same point G; therefore also, the cylinders that stand upon the two first of these circles having the common altitude GH, are equal to the cylinder which stands on the remaining circle, and which has the same altitude GH. The cylinders described by the revolution of the rectangles Gg and Gr are therefore equal to the cylinder described by the rectangle GP. And as the same may be shown of all the rest, therefore the cylinders described by the rectangles Hh, Gg, Ff, and by the rectangles Hs, Gr, Fq, DN, are together equal to the cylinder described by DB, that is, to the cylinder having the same base and altitude with the hemisphere.

PROPOSITION XXIII. THEOREM.

Every sphere is two thirds of the circumscribing cylinder. Let the figure be constructed as in the two last proposi

tions, and if the hemisphere described by BDC be not equal

to two thirds of the I
cylinder described by

BD, let it be greater
by the solid W. Then,
as the cone described
by CDE is one third
of the cylinder (II.20)
described by BD, the
cone and the hemis- A

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phere together will exceed the cylinder by W. But that cylinder is equal to the sum of all the cylinders described by the rectangles Hh, Gg, Ff, Hs, Gr, Fq, DN (II. 22); therefore the hemisphere and the cone added together exceed the sum of all these cylinders by the given solid W which is absurd, for it has been shown that the hemisphere and the cone together differ from the sum of the cylinders by a solid less than W. The hemisphere is therefore equal to two thirds of the cylinder described by the rectangle BD; and therefore the whole sphere is equal to two thirds of the cylinder described by twice the rectangle BD, that is, to two thirds of the circumscribing cylinder.

SPHERICAL GEOMETRY.

DEFINITIONS.

1. A sphere is a solid conceived to be generated by the revolution of a semicircle about its diameter.

2. The centre of the semicircle is equally distant from every point on the surface of the sphere, and is therefore called the centre of the sphere.

3. Circles of the sphere, whose planes pass through the centre, are called great circles, and all others small circles. 4. A straight line, drawn through the centre of any circle of the sphere, perpendicular to its plane, and limited on both sides, by the surface of the sphere, is called the axis of that circle.

5. The poles of a circle of the sphere are the extremities of its axis.

6. By the distance of two points on the surface of the sphere is meant an arc of a great circle intercepted between them.

7. A spherical angle is that formed on the surface of the sphere by arcs of two great circles meeting at the angular point, and is measured by the inclination of the planes of the circles.

8. A spherical triangle is a figure formed on the surface of the sphere by arcs of three great circles, called its sides, each of which is less than a semicircle.

9. A quadrantal triangle is that of which one of the sides is a quadrant.

10. A lunary surface is a part of the surface of the sphere, contained by the halves of two great circles.

11. A segment of a sphere is a part cut off by a plane.

PROPOSITION I.

Every section of a sphere is a circle.

G

Let a plane cut the sphere AGH in any direction. If it pass through the centre, the section is evidently a circle. But if it do not pass through the centre, let ABCD be the section, and from E, the centre of the sphere, let A EF be drawn perpendicular to its plane; also, let FA, FB, FC, be drawn

in the plane, to meet the surface of the sphere. Then EA, EB, EC, being joined, the right-angled triangles EAF, EBF, ECF, have equal hypothenuses, EA, EB, EC, because they are radii

D

H

F

C

E

of the sphere, and one side EF common to all. Now (Pl. Ge. I. 47) EF2 + FB2 = EB2 = EC2 EF2 + FC2, and taking away the common part EF2, there remains FB2= FC2, or FB FC. It is similarly proved that FB FA. Con

=

sequently ABCD is a circle, whose centre is F.

COR. 1.-Any two great circles cut one another in al diameter of the sphere, and therefore mutually bisect each other.

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