Interpretation of Formulce in Spherical Trigonometry. 19 



to give some instances of this kind, and shall commence with 

 the formula 



» cosfl —cos h cose 



cos A. — _ -. a 



sin sm c 

 which enables us to solve the problem in which the three sides 

 are given, to find the angles. Now, since, if Q be any arc or 

 angle, cos Q = cos (2 tt — Q), it is plain, that if A' denote 

 one value of the angle A, there will be another value 2 tt — A', 

 unless it can be shown that this value is excluded for some 

 reason not indicated by the foregoing formula. Let the 

 former of these be less than two right angles ; then, if the 

 latter, which we may denote by A'', be admissible, it will be 

 greater than two right angles *. In like manner the corre- 

 sponding formulae would give two expressions, B' and B" 

 (= Stt -B'), for B; and two, C'and C"(= 2 7r-C0, for C. 

 Now the first of these. A', B', C, regarded ^s being each less 

 than two right angles, are those which are universally, and, 

 in a practical point of view, correctly employed. On consi- 

 deration, however, we shall find that the latter values are just 

 as admissible as the former. When a great circle is described 

 through two points on the surface of a sphere which are not 

 the extremities of a diameter, the points may be regarded as 

 hen\g joined by either of the arcs into which they divide the 

 circle : it is usual, however, to consider the smaller arc as the 

 one which joins them. Now, if we take on a spherical surface 

 three points which are not on the same great circle, and join 

 them in the way last mentioned, by three arcs, a, b, c, we shall 

 obviously divide the entire surface into two parts, each bounded 

 by three arcs of three great circles, and therefore each of them 

 a spherical triangle. The smaller of these is that which is 

 usually alone considered, and its angles are A', B', C\ The 

 greater has for angles A", B'', C; and, having a, b, c as sides, 

 it answers the conditions of the problem just as well as the 

 smaller triangle. The angles of the greater triangle are evi- 

 dently each greater than two right angles. 



* It is scarcely necessary to state, that there is no impropriety in re- 

 garding an angle as greater than two right angles. If we suppose one 

 radius of a circle to be fixed, and another to revolve from a state of coin- 

 cidence with it, the motion of the latter will make continual additions to 

 the quantity of angular space described ; and the angle made by the lines, 

 commencing from nothing, may be regarded as increasing so as to obtain 

 any magnitude we please. If the line revolve in one direction, the angle 

 may be regarded as positive ; if in the other, negative. It is plain also, that 

 if any angle Q has been described, the relative positions of the lines will 

 be the same again after the description, once or oftener, of four right 

 angles in either direction; or, as it may be expressed, the relative positions 

 of the lines will be the same, when the angle which they make is Q, as 

 when it Q + 2 nx, n being a whole number. 



D2 



