69C 



MATHEMATICS. SPHERICAL GEOMETRY. 



[PROP, i., n 



CHAPTER VIII. 

 SPHERICAL GEOMETRY. 



l5TEODrcno!. This chapter on Spherical Geometry, 

 ii intended to be strictly introductory to the cognate 

 science of Spherical Trigonometry. To explain their 

 relation, the following, for our present purposes, will 

 suiliort. The Science of Geometry, as given in the first 

 ix books of Euclid's Geometry, contains, along with 

 others, a variety of propositions concerning the relations 

 between the sides and angles of plane triangles. And 

 these propositions may be directly applied to solve a 

 variety of problems by construction e. g., if we are 

 asked to construct an equilateral and equiangular pen- 

 n mi a given straight hn.-. <-.ui <1<> this, with ruli; 

 oompasses, by skilfully availing ourselves of certain 

 urtiu* of lines and angles which Euclid has proved. 



ut if the question were asked given that one side of 

 a triangle is so many feet long, and that the angles 

 adjacent to that side are respectively certain parts of a 

 right angle, how many feet long are the remaining sides ? 

 The question is one, not of construction, but of calcula- 

 tion, and we cannot solve it directly, but only by the 

 intervention of a science which shall give algebraical 

 expressions for the relations between the sides and angles 

 of triangles. Such a science has been invented, and is 

 called plane trigonometry. It clearly presupposes a know- 

 ledge of the relations which Euclid has established, and 

 assumes them as its basis. Now, suppose the triangles 

 to be described, not on a plane, but on the surface of a 

 sphere, spherical trigonometry is the science which gives 

 us the means of calculating from given data the sides 

 and angles of such triangles. This science, therefore, 

 stands to the spherical triangle in the same relation that 

 plane trigonometry stands to the plane triangle. And as 

 the latter science rests on that part of the science of 

 Geometry which treats of plane triangles as its basis, so 

 the former science must rest on another portion of the 

 science of Geometry, which shall treat of triangles de- 

 scribed on the surface of a sphere as its basis. 



The need of such a science as Spherical Trigonometry 

 will be apparent to any one who reflects on the circum- 

 stance that the surface of the globe is (very nearly) 

 spherical ; consequently, all the triangles calculated in 

 the course of a survey on a large scale are, when reduced 

 to the surface of the earth, spherical triangles. Hence, 

 surveying on a large scale (Geodesy) cannot be carried 

 on without the investigations of spherical trigonometry. 

 Again, in practical astronomy, the positions of all the 

 heavenly bodies are referred to the surface of the great 

 sphere that, namely, wliich has the centre of the earth, 

 supposed to be fixed, for its centre ; and thus the tri- 

 angles recognised in practical astronomy are spherical 

 triangles, and the requisite calculations cannot be carried 

 on except by means of the science of spherical trigouo- 



III' TV, 



Having thus explained that this so needful science 

 demands as its basis the investigation of certain pro- 

 perties of the spherical triangle, we will proceed to in- 

 vestigate those properties. As already stated, we shall 

 c n tine ourselves strictly to such propositions as we shall 

 hen-after need in treating of spherical trigonometry. 



We have already, in general terms, said that a spherical 

 triangle is one described on the surface of a sphere ; we 

 must, however, define this and other points more accu- 

 rately, which we shall do as wo proceed. 



It is to be observed, that we suppose that we can 

 draw any plums through any three given points ; or. 

 which is the same thing, through any straight line, and 

 through a point not on that same straight line ; also 

 that we can cut any given solid, by a plane, in any 

 direction whatever. 

 N-B. From Euclid VI., 33, it appears, that in a given 



circle, any angle at the centre is proportional to the 



arc on which it stands. The arc is therefore laid to 

 the angle. 



DEFINITION'S. 



I. 



A nlld is a space which has three dimensions namely, 

 length, breadth, and thickness. 



II. 



A sphere it a solid bounded by a surface, of which 

 every point is equally distant from a point within it 

 called the centre. 



in. 



The radius of a sphere is a straight line drawn from 

 the centre, to any point in the surface of the sphere. 



IV. 



A straight line drawn through the centre, and termi- 

 nated botti ways by the surface of the sphere, is called 

 the diameter. 



PROPOSITION I. 



Every section of a sphere made by a plane is a circle. 

 Let A B C D be the sphere ; draw O A any radius 



whatever ; let B P E N 

 be a plane cutting the 

 sphere's surface in the 

 line B P E. It is sup- 

 posed that the plane of 

 the paper passes through 

 the centre of the sphere 

 perpendicularly to this 

 cutting plane, which 

 also cuts the radius O A 

 in the point N ; and 

 suppose O A to be per- 

 pendicular to the plane ; 

 take P, any point in the 

 line B P E, join P N, P O, O K. Then because P N is 

 in the plane B P E, and O N is perpendicular to the 

 plane, .'. PNO is a right angle. For the same reason 

 O N E is a right angle, .'.in the triangles PNO, O .N E, 

 we have P N* + N O 2 = P O 2 and E N 2 + N O 2 = O E*. 

 Now, O P = O E, because each are radii of the sphere, 

 .'. P N 2 + N O 2 = E N 3 + N O 2 , .-. P X = E X. Simi- 

 larly of any other point in the line BPE,.'. BPEisa 

 circle, the centre of which is N. 



If the plane pass through the centre of the sphere, as 

 plane C Q D, take Q any point in the line in which the 

 plane cuts the surface of the sphere. Join O Q, O D. 

 Then O Q, O D are j-adii of tho sphere, .-. O Q = O D. 

 Similarly of any other point in the line C Q D, .'. the 

 section C Q D is a circle. Hence every section of a 

 sphere is a circle. Q. E. D. 



N. B. The figure, as thus drawn, will, it is hoped, be 

 quite comprehensible ; it may bo observed, however, 

 that the circle A C 1) is tho section of the sphere made 

 by the plane of the paper. The other sections of the 

 sphere are made by pianos which intersect thepl.-me 

 of the paper in straight lines e.g. , in B E, and in C 1 >, 

 and these sections are seen in perspective, as CQD, 



DKF. V. The section of a sphere made by a plane 

 which does not pass through the centre is callr.l a SHU/// 

 circle. Thus, in Fig. 1, B P K is a small circle. 



DKK. VI. The section of a sphere made by a plane 

 which passes tlirough the centre Ls called a great circle. 

 Thus, in Fig. 1, C Q D is a great circle. 



PROPOSITION II. THEOREM. 



A great circle may be drawn through any ticn points on the 

 surface of a splmre,but in general not through more tJmit 

 two points. 



For, taking any two points on the surface of the 



