JiLV 24, 1903.] 



SCIENCE. 



Ill 



scale labeled sec i x, and beyond D', a 

 distance which reads c ; this fixes a point 

 Z>,'. With the beam compasses centered 

 at D^', describe an arc. Lay oif on the 

 sec i X scale, still pivoted to the board 

 (and revolving about 0'), a distance which 

 reads b and note where it intersects the 

 arc just drawn; this fixes a point £/. 

 Along the directions ADi and AE.^' locate 

 the points B, C by laying oft" distances 

 which read c and h upon the scale labeled 

 2 cos i X. The reading of the distance 

 BC upon the scale labeled 2 sin i j is the 

 value of the side a. 



The same construction taken in a slightly 

 different order serves for finding A when 

 a, b, c are the known parts. 



In passing, it may be well to note that 

 plane trigonometry applied to the triangles 

 ADE and OED gives 



DE^ =AI)' + AE' — 2AD AE cos A 



= OD' + 6e^~ 20D OB cos a 



. •. tan' c + tan' b — 2 tan c tan b coa A = sec' c 



+ sec' 6 — 2 sec c sec 6 cos a. 



Noting that sec- = 1 + tan- and multiply- 

 ing through by cos h cos c, we have, after 

 transposing, 



cos a = cos b cos c -^ sin i gin c cos A . 



The same equation follows from the sec- 

 ond method by noting that 



D'E" = Aiy^ + At"-- 2Aiy AE' cos A ; 

 and 



IPE' : BC= sec i c : 008 J 6 



where BC = 2 sin | a, and later on that 



2 Bin' J a =1 — cos a, 2 cos' i a = 1 -(- coe a. 



POLAR TRIANGLES. 



Cases analogous to any of the above, 

 but having sides and angles interchanged 

 throughout, can be solved by the foregoing 

 methods provided we first subtract all 



known parts from 180°, interchanging 

 capital and small letters, and then after 

 having solved this polar triangle, subtract 

 the parts from 180°, and finally inter- 

 change the capital and snuiU letters. 



APPLICATIONS. 



The methods already described may be 

 used to advantage in some classes of plane- 

 table work. To obtain the azimuth of the 

 sun by one mechanical solution of the tri- 

 angle, it is necessary that the telescope of 

 the alidade be supplied with a vertical 

 circle. The direction of the sun could be 

 ascertained by means of a good watch, but 

 the spherical triangle would then have to 

 be solved for two of its parts. 



The azimuth and hour angle of the sun 

 or other heavenly body can be obtained 

 from an observed altitude with sufficient 

 accuracy for enabling one to lay down a 

 Sumner line at the assumed or dead-reckon- 

 ing latitude, and for ascertaining the varia- 

 tion of the magnetic needle at sea. 



Tables of sunrise and sunset can be com- 

 puted with great facility by means of the 

 stereographic method. In fact, the true 

 zenith distance (BC) of the rising or set- 

 ting body is a constant for all latitudes 

 and dates. The distance between pole and 

 zenith at any particular latitude is con- 

 stant for all dates. (That is, D', B and D,' 

 are fixed points for a given latitude, and 

 about B, the zenith, a circle can be de- 

 scribed with 2 sin ^ a as radius.) Since 

 the polar distance of sun or moon never 

 exceeds 120°, scales of moderate length will 

 suffice for all possible cases. 



Consider now the question of great 

 circle sailing. The distance and initial 

 course between two points specified by 

 their latitudes and longitudes require the 

 solution for two parts of a triangle whose 

 given parts are two sides and the included 

 angle. The longitude and latitude of the 



