110 



SCIENCE. 



[N. S. Vol. XVIll. No. 447. 



methods, or constructions, analogous, but 

 differing somewhat from each other. The 

 first may be referred to as the gnomonic 

 method, and the second as the sterographic. 



Let ABC, Fig. 3, be the spherical tri- 

 angle. A plane tangent to the sphere at 

 A contains the lines AD, AE, whose lengths 

 are tan c, tan h. Imagine the triangle 

 ODE revolved about BE into the plane 

 ADE. In the plane quadrilateral ADOEA 

 thus obtained, 0D = sec c, OE^sec h, 

 0B = 0C = 1; also arc BC=^a, which 

 measures the angle 0. 



Assume now that we have scales of tan- 

 gents and secants. The quadrilateral con- 

 structed by aid of such scales and a pro- 

 tractor gives the angle A when a, i, c are 

 the known parts, or a when A, b, c are 

 known. In practice the quadrilateral is 

 not actually constructed; but the work of 

 finding the required unknown part of the 

 triangle is arranged in accordance with the 

 diagram in Fig. 3, which shows A and 

 as coinciding. More particulars about this 

 arrangement may be gathered from the 



Given the two sides 6, c of a spherical 

 triangle and the included angle A, find the 

 side a by means of the above-described 

 apparatus. 



Lay off with the scale of tangents when 

 pivoted to the drawing board a distance 

 along the initial direction which reads c; 

 this fixes on the board the point D. Lay 

 off towards A, as found on the margin of 

 the board, a distance which reads h on the 

 tangent scale; this fixes on the board the 

 point E. On beam compasses set the dis- 

 tance DE, or mark it off upon a strip of 

 paper. Next remove the scale of tangents 

 and pivot to the board the scale of secants. 

 Lay off on the secant scale, and beyond 

 D, a distance which reads c; this fixes the 

 point Dj. With the beam compasses cen- 

 tered at D^, describe an arc. Lay off on 

 the secant scale, still pivoted to the board 



(and revolving about 0) a distance which 

 reads h and note where it intersects the 

 arc just drawn; this fixes the point E^^. 

 The angle read oft' on the margin of the 

 board is the side a. 



The same construction taken in a slightly 

 different order serves for finding A where 

 a, b, c are the known parts. 



We now take up the stereographic 

 method. The problems to be considered 

 are the same as those to which the gno- 

 monic method applies. The advantage of 

 the ster.eographic method lies in the fact 

 that tan ^ x does not approach infinity 

 until X approaches 180°. 



A plane tangent to the sphere at A, Fig. 

 4, contains the lines AD', AE' whose 

 lengths are 2 tan -J c, 2 tan ^ b. Imagine 

 the triangle O'D'E' revolved about D'E' 

 into the plane AD'E'. In the plane quad- 

 rilateral AD'O'E'A thus obtained, O'D' = 

 2 sec i c, O'E' = 2 sec i b, O'B = 2 cos ^ c, 

 0'C^2 cos ib and line BC = 2 sin i a. 



With suitable scales and a protractor 

 the quadrilateral AD'O'E'A could be con- 

 structed and the required part of the 

 spherical triangle could be thus deter- 

 mined; but the more practical arrange- 

 ment is that shown in the figure where 0' 

 is made to fall upon A. Moreover, it is 

 convenient to omit the factor 2 before 

 tangents and secants. 



Given h, c, A to find a by means of the 

 above-described apparatus. 



Lay off by means of the scale labeled 

 tan J X, pivoted to the drawing board, a 

 distance along the initial direction which 

 reads c; this fixes a point D'. Lay off 

 on this scale now directed towards A, as. 

 found on the margin of the board, a dis- 

 tance which reads h ; this fixes on the 

 board a point E'. On beam compasses 

 set the distance D'E', or mark it off upon 

 a strip of paper. Next remove the tan -J x 

 scale and pivot to the board the double 

 scale shown in Fig. 2. Lay off with the 



