PROP. i. xm.] 



MATHEMATICS. SPHERICAL GEOMETRY. 



599 



two right angles. 



Similarly 



L 2 : area hemisphere : : B : 2 right angles. 

 L 3 : area hemisphere : : C : 2 right angles. 

 .". L, + Lo + L s : area hemisphere : : A + B + C : two 

 right angles. Now, the three limes clearly make up the 

 hemisphere B A Ca6, together with the triangle abc. 

 Hence the three lunes are in all cases greater than a 

 hemisphere, and .'. the three angles of the triangle are 

 together greater than two right angles. 



Again, triangle abc is equal to triangle ABC (Remarks 

 on Def. VIII .),-' L, + L 3 -(- L 3 = area of hemisphere 

 + area of triangle ABC. Also, A + B -)- C = two 

 right angles -f- spherical excess ; .'. area hemisphere + 

 triangle ABC: area hemisphere : : two right angles -f- 

 spherical excess : two right angles, .'. area triangle ABC: 

 area hemisphere : : spherical excess 

 Q. E.D. 



DBF. XII. The tri- 

 angle formed by the 

 great circles which join 

 the poles at the sides of 

 a given triangle, is called 

 the polar or supple- 

 mental triangle. 



Thus, let A B C be a 

 given triangle ; then if 

 11, b, c, be respectively 

 the poles of the sides 

 B C, C A and A B, .160 

 is the polar triangle. 

 And clearly, if oA be 

 joined by the arc of a great circle, and this be produced 

 to meet B C in 1), then aD in a quadrant of a circle, and 

 A 1 is perpendicular to B C, and similarly of the other 

 poles. 



The relation between the given triangle and its polar 

 triangle is very important, as will be seen when we come 

 to employ its properties in Spherical Trigonometry ; the 

 properties on which its importance depends are proved 

 in the following propositions. 



PROPOSITION X. 



If tteo great circlet intersect, their point* of intersection 

 will be the poltt of the grtat circle which patset through 

 their polet. 



For (in Fig. 5). take the plane of the paper for the 

 plane of one circle, and Y P X for any other, so that X 

 and Y are the points of intersection of the two circles ; 

 through O draw a plane A P B perpendicular to X Y, 

 and . . perpendicular to both the planes X A B, and 

 X 1' Y, and . . the plane A P B will contain the lines 

 drawn perpendicular to those planes, and therefore will 

 contain the poles of the two given circles. Hence A P B 

 is the great circle joining the poles of the circles X A Y 

 and X P Y ; but Y O X i perpendicular to the plane 

 A P B, . . X and Y are the poles of the circle A P B ; t. e., 

 are the poles of the great circle which joins the poles of 

 the two given circles. 



PROPOSITION XL 



If ABC le o given triangle, and A'B'C', is iti polar 

 Fi 10 triangle, then is A B C the 



polar triangle of A'B'C'. 



For since C' is the pole of 

 A B, and B' is the pole of 

 AC, .'. (by last Prop.) the 

 point of intersection A of 

 A B and A C is the pole of 

 the great circle joining B'C', 

 i. e., A is the pole of B'C', 

 similarly B is the pole of 

 C C'A', and C the pole of A'B'. 

 Q. E. D. 



PROPOSITION XU. 



If A B C be a triangle, and A'B'C 7 its polar trianyle, then 

 the arc on the sphere, which measures the angle A, to- 

 gether with the side B'C', equals the semi-circumference 

 of a great circle. (See Fig. 10). 



For, produce A B, A C to meet B'C' in P and Q. 

 Then because A P and A Q are quadrants, P Q is the arc 

 that measures the angle A (Cor. 4, Prop. V) ; now B'C' 

 + P Q = B'P + C'Q + Q P=B'Q + C'P. But since B' 

 is the pole of A C, .'. B'Q is a quadrant. Similarly C'P 

 is a quadrant, and the two together are a semicircle, .'. 

 B'C', together with the arc on the great circle which 

 measures A, equals the semi-circumference of a great 

 circle. 



COR. 1. If for these arcs we substitute the angles 

 they measure, we may state the proposition as follows : 



A + E'CX = 2 right angles. 

 Similarly 



B + C'A' = 2 right angles, 

 C -f- A'B' = 2 right angles. 



COR. 2. And since ABC is the polar triangle of A'B'C > 

 we have 



A' + B C = 2 right angles. 

 B'-j- C A = 2 right angles. 

 C'-f- A B = 2 right angles. 



COR. 3. Hence, the sum of the angles of any triangle, 

 together with the sides of the polar triangle, = six right 

 angles. But the sides of the polar triangle must have 

 some magnitude, and must be less than four right angles 

 (Prop. VII.) Hence the three angles of a triangle must 

 be less than six, and greater than two right angles. 



It is plain, since the three angles of a spherical triangle 

 are greater than two right angles, and less than six 

 right angles, that a spherical triangle may have one, two, 

 or even three of its angles right angles. 



DBF. XIII. A right-angled spherical triangle is one 

 which has one or more right angles. 



DBF. XIV. A quadrantal triangle is one which has 

 at the least one side a right angle, t. ., the quadrant of 

 a great circle. 



PROPOSITION XIIL 



If A B C be a right-angled triangle, having a right angle 

 C, and A'B'C' be its polar triangle, then A'B'CX is a 

 quddrantal triangle, having the tide A'B' a quadrant. 



For by the last proposition (Cor. 1) 



C + A'B'= two right angles. 



Now, C is a right angle, .'. A'B' is a right angle, t. e., 

 is a quadrant, y. E. 1). 



COR. Hence, if all three angles, A'B'C, are right 

 angles, the sides of the polar triangle are all right angles. 

 For if two sides of a triangle are right angles, the third 

 side measures the opposite angle ; .'. if the third side be 

 also a right angle, all the angles are right angles. Hence 

 in the polar triangle the sides and angles are all right 

 angles. And since the angles of the polar triangle are 

 each right angles, the sides of A B C will be right angles 

 (last Prop., Cor. 2). Hence, if all the angles of any 

 triangle are right angles, the sides are right angles (i. ., 

 quadrants) also. 



With this chapter we conclude the Elements of Geo- 

 metry, both plane and spherical. In the succeeding one, 

 the subject of Practical Geometry will be treated on, as 

 well as Conic Sections. In the section on Astronomy, 

 the student will require a considerable knowledge of the 

 properties of the ellipse, parabola, &c. ; which are the 

 most important of the conic sections, properly so called ; 

 excluding, however, the circle, which has already been 

 discussed in the Elements of Euclid. The subject of 

 Trigonometry, plane and spherical, will follow Practical 

 Geometry. An extended series of the logarithms of 

 sines, tangents, etc., will be given at the end of this 

 section. ED. 



