D N Q R 日, P r с h B tions, and if the hemisphere described by BDC be not equal to two thirds of the cylinder described by BD, let it be greater by the solid W. Then, as the cone described fa by CDE is one third of the cylinder (II. 20) described by BD, the cone and the hemis- Ā 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. 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, theright-angled triangles EAF, EBF, ECF, have equal hypothenuses, EA, EB, EC, because they are radii of the sphere, and one side EF common to all. Now (Pl. Ge. I. 47) EF? + FB2 = EB2 – EC2 EF + FC, and taking away the common part EF, there remains FBP=FC, or FB=FC. It is similarly proved that FB = FA. Consequently 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. G D B H Cor. 2.-Only one great circle can pass through the same two points, in the surface of the sphere, that are not diametrically opposite. For the plane of the great circle must pass through these two points and through the centre of the sphere (Sp. Ge. Def. 3), and only one plane can do so (So. Ge. I. 2, Cor. 2). Cor. 3.-Any two sides of a spherical triangle being pro duced, intersect again at the distance of a semicircle. Cor. 4.—The two poles of any circle, its centre, and the centre of the sphere, are always in the same straight line, and that straight line is perpendicular to the plane of the circle. Cor. 5.-And, therefore, if a line or plane be perpendicu lar to a circle of the sphere, and pass through one of these points, it will pass through the other three. Or, if it pass through two of them, it will be perpendicular to the circle, and also pass through the remaining two. Cor. 6.--Hence, two great circles, whose planes are per pendicular, pass through each other's poles; and con versely. Cor. 7.-And, if one great circle pass through a pole of another, the latter will pass through the poles of the former. CoR. 8.-All parallel circles have the same axis and the same poles ; and conversely. Schol. It appears by the fourth and fifth corollaries, that, of these five conditions of passing through the two poles of a circle of the sphere, through the centre of the circle, through the centre of the sphere, and of being perpendicus lar to the plane of the circle-if a straight line or a plane fulfil any two, it will also satisfy the other three. PROPOSITION II. Each pole of any circle of the sphere is equally distant on the surface from every point in its circumference. Let ABCD be any circle of the sphere, whose centre is F, and axis GFH. Its poles G, H, are each of them equally distant from its circumference. For, let the sphere be cut by planes passing through G, H, D D C B В E H and let the sections--which will be great circles, because the line GH passes through the centre of the sphere-meet ABCD in the lines of common section FA, FB, FC. A Then GA, GB, GC, being joined, the right-angled triangles GFA, GFB, GFC, have the sides FA, FB, FC, equal, because they are radii of the same circle, and one side GF common to all; and the angles at F are right angles (Sp. Ge. I. Cor. 4); therefore (Pỉ. Ge. I. 4) the hypothenuses GA, GB, GC, and consequently the arcs which they subtend, are likewise equal. Cor. 1.--The pole of a great circle is at the distance of a quadrant from its circumference. Cor. 2.-Hence any plane passing through the centre of the sphere, divides it into two equal parts, which are therefore called hemispheres. Cor. 3.- If a point in the surface of the sphere be at the distance of a quadrant from other two points not diametrically opposite, it will be the pole of the great circle passing through them. For only one great circle can pass through these two points, and its pole is distant from them by a quadrant. (Sp. Ge. I. Cor. 2, and II. Cor. 1.) Cor. 4.-The radius of a small circle is the sine of its distance from either pole to the radius of the sphere, or the cosine of its distance from the parallel great circle. Cor. 5. Hence, those small circles, whose planes are equally distant from the centre, are equal, and conversely; and of two circles unequally distant, that which is nearer the centre is the greater; and conversely. Cor. 6.—Parallel circles intercept equal arcs on those great circles which pass through their poles. PROPOSITION III. The intercepted arc of a great circle, whose pole is the angular point, is the measure of a spherical angle. Let ABC be a spherical angle, of which the angular point M B N P B is the pole of the great circle ACD. Then is the intercepted arc AC the measure of ABC. For, let the tangents MB, NB, and the radii of the sphere EA, EB, EC, be drawn. The angle MBN is the same with the spherical angle ABC, for the tangents are perpendicular to BE (So. Ge. I. Def. 4, and Sp. Ge. Def. 7); but MBN is equal to AEC; be- A cause, since AB, BC, are quadrants, and BEA, BEC, right angles, MB, BN, are parallel to AE, EC. Wherefore the spherical angle ABC is equal to AEC, the measure of which is the arc AC to the radius of the sphere. Cor. 1.-The circumferences of two great circles cut each other at right angles, when their planes are perpendi cular; and conversely. Cor. 2.-At the point of intersection of two great circles, the opposite angles are equal, the two adjacent angles are together equal to two right angles, and each angle is equal to its opposite one, at the other point of inter section (Sp. Ge. Def. 7). Cor. 3.-The distance of the adjacent poles of two great circles is the measure of their inclination, or of the spherical angle. For, since AB, BC, pass through the poles of ACD, ACD passes through the poles of AB, BC (Sp. Ge. I. Cor. 7); let P be the pole of AB, and Q the adjacent pole of BC; then AP, CQ, are quadrants, and AC =PQ. Cor. 4.—The intercepted arc of any circle, whose pole is the angular point, is the measure of the spherical angle to the radius of that circle. Cor. 5.-Two great circles, which pass through the poles of parallel circles, intercept similar arcs. Cor. 6.-A spherical angle is equal to the inclination of two tangents, to the arcs containing it, drawn from the angular point. |