PROP. m. v.] 



MATHEMATICS SPHERICAL GEOMETRY. 



597 



Flg.J. 



sphere, we can draw a plane through them, and this 

 plane can be made to pass through any third point, ru., 

 through the centre of a sphere. The section of the 

 sphere made by this plane is a great circle, and the two 

 points clearly lie on it. 



These three points determine the plane, and . . we 

 cannot be sure of its passing through any other point, 

 whether on the surface of the sphere or not. Q. E. D. 



COR. It is plain that a small circle may be made to 

 pass through any three points. For a plane being drawn 

 through two points can be made to pass through a third ; 

 and if these three points are on the surface of a sphere, 

 the plane cuts the sphere in a circle on which these three 

 points lie. 



N. B. We shall henceforth assume that we can draw arcs 

 of great circles in any possible direction ; for instance, 

 through any two points. ' For this is merely equivalent 

 to drawing a plane through the centre of the sphere, 

 and those two points, which of course cuts the sphere 

 in the required great circle. 



PROPOSITION III. 



Two yreat circles bisect one another. 



For suppose A B C D, the 

 section made by the plane of 

 the paper, to be one circle, 

 and B P C, a section made by 

 any other plane BPCQ, to be 

 the other, then these planes 

 intersect in the straight line 

 B C, which passing through O, 

 the centre of the sphere B C 

 is a diameter of each of the 

 circles, and . . bisects each 

 of the circles, . . B P C = 

 CQB, and BAC =CDB, 

 or the two great circles bisect 

 each other. Q. E. D. 



PROPOSITION IV. 



The inclination of two great circles is the angle betieeen 

 the tnivients drawn to those circles at their point of 

 intersection. 



Let one great circle be that made by the plane of the 

 paper A Q B. Let A P B, the other circle, be that made 

 by any other plane, then the diameter A O B is the line 

 Fig. s. in which the planes inter- 



sect ; in plane of paper draw 

 A S perpendicular to A B ; 

 in the plane A PB draw A R 

 perpendicular to A B. Then 

 (Planes, ante, p. 690, Def. 4) 

 R A S is the angle between 

 the planes. But AR touches 

 the circle A PB at point A, 

 and A S touches the circle 

 AQB at point A (Euclid 

 III., 16), .'. the inclination 

 of two great circles is the 

 angle between their tangents 

 at the point of intersection. Q E.D. 

 N .15. The angle R A S, between the tangents R A, S A, 

 U generally supposed to be measured on the sphere, 

 and is called the angle P A Q. 



Der. VII. If from the centre of a sptare a line be 

 drawn perpendicular to thr plane of any circle, whether 

 great or small, and be produced both ways to meet the 

 surface of the sphere, the points in which that line meets 

 the surface of the sphere, are called the poles of the 

 circle. 



Thus, in Fig. 1, let C Q D be the plane of a great 

 circle, O being the centre of the sphere. Through O 

 draw O A perpendicular to the plane C Q D, and pro- 

 duce it to meet the surface of the sphere in F and A. 

 Then F and A are the poles of the great circle C Q D. 

 Again, if B P E be the plane of a small circle H P E, 



from O draw N perpendicular to that plane, produce 

 O N both ways to meet the surface of the sphere in A 

 and F. These are the poles of the small circle B P E. 

 It is usual to call A (the pole nearest to the small circle), 

 the pole of the circle. 



From the demonstration of Proposition I., it is plain 

 that N is the centre of the circle B P E. 



PROPOSITION V. 



If from the pole of a circle great circles be drawn to any 

 two points of that circle, the intercepted arcs are 

 equal. 



(1). In the case of a small circle, let A P B be the plane 

 of the small circle, and XUY 

 perpendicular to that plane, 

 and meeting it in N, then 

 X Y are tlie poles of the 

 circle, and N is its centre. 

 Let X A Y, the section made 

 by the plane of the paper, 

 be one great circle, X P Y 

 the section made by the plane 

 passing through any other 

 point P be the other great 

 circle. We have to prove 

 that the arc X P = arc XA. 

 For, since N is the centre of 

 the circle APR, we have 



A N = P N ; also, since O is the centre of the sphere 

 O A = O P, . . in the triangle A O N, P O N, we have 

 the sides A O, O N, = the sides P O, O N, each to each, 

 and the base A N = the base P N, .-. the angle A O N = 

 angle PON. Hut in equal circles, equal angles stand 

 on equal circumferences, .'. the arc A X = arc P X. 



(2). In the case of a great circle, let A P B be the 

 great circle, O its centre, and X O Y perpendicular to 



the plane APB. Then, 

 XY are the poles of the 

 circle. Let the section 

 made by the plane of the 

 paper X A Y be one of the 

 great circles, and let XPY 

 be the other ; we have to 

 show that arc A X = arc 

 PX join OA, OP. Then, 

 because X () is perpendi- 

 cular to the plane, X O A 

 and XOPare right angles, 

 and . . are equal to each 

 other. Hence as before, 

 arc A X = arc P X. 

 Q. E. D. 



COR. 1. Hence the pole of a circle is equally distant 

 from every point of that circle. Tlte distance bcinj 

 measured aloiuj a great circle. 



COR. 2. In the case of the pole of the great circle, it 

 is plain, since AOX is a right angle, that AX is the 

 fourth part (or quadrant) of the great circle A X B Y. 



COR. 3. Also any plane passing through the poles of 

 a great circle is clearly perpendicular to the plane of 

 that circle ; since the line joining the poles X O Y 

 (Fig. 5) is perpendicular to the plane APB. (See 

 Planes, p. 690, Def. 2). 



COR. 3 (Fig. 5). The inclination of the two great 

 circles X A Y, X P Y is clearly measured by the arc A P. 

 For P ( ) and A O are each perpendicular to X Y, the 

 line of intersection of the great circle, . . P O A is the 

 inclination of the great circles, and P O A is measured 

 by the arc A P. 



COR. 4. We have already seen that the angle AXP, 

 i.e., the angle between the tangents to the circles at the 

 point X, is the inclination between the planes, . . A P 

 measures the angle AXP. (See Prop. IV.) 



DEF. VIII. A spherical triangle is the portion of i 

 surface of a sphere contained by the arcs of three great 

 circles. 



Thus, let AXB, CZD, E YFbe three great circles 

 wiiich intersect in the points ZYX, then the space 



