91929 between this and P,P,; then if r1, r, be the radii of the two small parallels pp., 192, the rest of notation as before, we shall have If the second arc q.q2 becomes P.P2, p' = 0 and cos. Every plane perpendicular to a radius at its extremity is a tangent to the sphere in that point. Let ZXY be a plane perpendicular to the radius OZ. Then ZXY touches the sphere in Z. Take any point P in the plane; join ZP; OP; Then (Prop. 6, Geom. of Planes) OP > OZ. Hence the point P is without the sphere; and, in like manner, it may be N Ꮓ X shown that every point in XYZ, except Z, is without the sphere. Therefore the plane XYZ is a tangent to the sphere. *These formulas are of frequent use in Astronomy, serving to express the relation between the distance moved on a parallel of declination and in right ascension of a star, and various other useful relations of a similar kind. PROP. IV. The angle formed by two arcs of great circles is equal to the angle contained by the tangents drawn to these arcs at their point of intersection, and is measured by the arc described from their point of intersection as a pole, and intercepted between the arcs containing the angle. Let ZPN, ZQN, arcs of great circles, intersect in Z. Draw ZT, ZT', tangents to the arcs at the point Z. With Z as pole, describe the arc PQ. Take O, the center of the sphere, and join OP, OQ. Then the spherical angle PZQ N Ꮓ T T is equal to the angle TZT', and is measured by the arc PQ. For the tangent ZT, drawn in the plane ZPN, is perpendicular to radius OZ; and the tangent ZT', drawn in the plane ZQN, is perpendicular to radius OZ; hence the angle TZT' is equal to the angle contained by these two planes (def. 6, Geom. of Planes), that is, to the spherical angle PZQ. Again; since the arcs ZP, ZQ are each of them equal to a quadrant; .. Each of the angles ZOP, ZOQ is a right angle, or OP and OQ are perpendicular to ZO. .. The angle QOP is the angle contained by the planes ZPN, ZQN. .. The arc PQ, which measures the angle POQ, measures the angle between the planes, that is, the spherical angle PZQ. Cor. 1. The angle under two great circles is measured by the distance between their poles. For the axes (def. in Prop. 2) of the great circles drawn through their poles being perpendicular to the planes of the circles, will be perpendicular to all lines of these planes, consequently, to the lines which measure the angles of the planes, and .. (see th. 65, Gen. Sch., 4°) the angles under these axes will be equal to the angle between the circles; but the angle under the axes is obviously measured by the arc which joins. their extremities, that is, by the distance between. their poles. Cor. 2. The angle under two great circles is measured by the arc of a common secondary intercepted between them. Cor. 3. Vertical spherical angles, such as QPW, RPS, are equal, for each of them is the angle formed by the planes QP R, WPS. Also, when two arcs cut each W other, the two adjacent angles QPW, QPS, when taken together, are always equal to two right angles. PROP. V. P Ri If from the angular points of a spherical triangle considered as poles, three arcs be described forming another triangle, then, reciprocally, the angular points of this last triangle will be the poles of the sides opposite to them in the first. Let ABC be a spherical triangle. From the points A, B, C, considered as poles, describe the arcs EF, DF, DE, forming the spherical triangle D EF. Then D will be the pole of H And, since C is the pole of DE, the distance from C to D is a quadrant. Thus, it appears that the point D is distant by a quadrant from the points B and C. .. (cor. 1, 2, Prop. 2) D is the pole of the arc BC. Similarly, it may be shown that E is the pole of AC, and F the pole of AB. Note. D having been shown to be the pole of the arc passing through the points B and C, it must be of the arc BC, because but one arc of a great circle can be made to pass through the two points B and C (cor. 5, Prop. 1). PROP. VI. The same things being given as in the last proposition, each angle in either of the triangles will be measured by the supplement of the side opposite to it in the other triangle. Produce BC to I and K, AB to G, and AC to H. Then, since A is the pole of EF, the angle A is measured by the arc GH at a quadrant's distance from A (Prop. 4). But, because F is the pole of AG, the arc FG is a quad rant. D I B E G H And, because E is the pole of AH, the arc EH is a quadrant. or ... EH + GF=180°, EF+GH180°; ... GH180° — EF. K In a similar manner, it may be proved that the angle B is measured by 180°-DF, and the angle C by 180°-DE. Again; since D is the pole of BC, the angle D is measured by IK. But, because B is the pole of DK, the arc BK is a quadrant. And, because C is the pole of DI, the arc CI is a quadrant. or .. IC + BK = 180°, .. IK = 180° — BC. But IK is the measure of the angle D (Prop. 4). In the same manner, it may be proved that the angle E is measured by 180°-AC, and the angle F by 180° - AB. These triangles ABC, DEF are, from their properties, usually called Polar triangles, or Supplemental triangles. PROP. VII. In any spherical triangle any one side is less than the sum of the other two. Let ABC be a spherical triangle, O the center of the sphere. Draw the radii OA, OB, OC. Then the three plane angles AOB, AOC, BOC form a trihedral angle at the point O, and these three angles are measured by the arcs AB, AC, BC. Α B But each of the plane angles which o form the trihedral angle is less than the sum of the two others (Prop. 1, Polyhedral Angles). Hence each of the arcs AB, AC, BC, which measures these angles, is less than the sum of the other two. PROP. VIII. The sum of the three sides of a spherical triangle is less than the circumference of a great circle. Let ABC be any spherical tri angle. Produce the sides AB, AC to Ieet in D. Then, since two great circles always bisect each other (Prop. 1, A cor. 3), the arcs ABD, ACD are semicircles. C B |