ON NAPIERS RULES. 



329 



or the angle at C is a right angle. Q.E.D. 



c 



Fig. i. 



COR. It follows from this that the dihedral angle AC is a 

 right angle. 



Considering the figure OABG, we observe that it is in some 

 sort symmetrical with regard to the line OB. GB being at 

 right angles to the plane OA C, and OA at right angles to the 

 plane CAB ; and as the dihedral angle C is a right angle, so 

 also is the dihedral angle BA. 



Now the three lines OA, OB, OG evidently represent any 

 right-angled spherical triangle; and similarly the three lines 

 BO, BA, BG represent another, between which and the former 

 a certain relation exists. One angle, namely, the dihedral angle 

 BO, is common ; the side ABC is the complement of the dihe- 

 dral angle OA ; the hypothenuse OBG is the complement of 

 the side BOG', the side OB A is the complement of the hypo- 

 thenuse AOB\ and, lastly, the angle BG is the complement of 

 the side AOG. 



Hence this conclusion. If a v a 2 , a s , a 4 , a 5 represent the parts 

 of a right-angled spherical triangle taken in order, and begin- 

 ning at the hypothenuse, then | - a 3 , | - 4 , | - a B , | - a l9 



and a a are the parts also taken in order and beginning at the 

 hypothenuse of another right-angled spherical triangle. If 



