136 Goniometric Frohlcms. 



convenient line is considered as the radius, and then the 

 relation subsisting between it and the rest of the lines of the 

 figure in terms of this radius is found out by means of plane 

 trigonometry; after which, if the radius be made equal to 

 unity, all the others become merely the trigonometric lines 

 of our common tables. This has been the method of pro- 

 ceeding in solving the foregoing problems; and hence the 

 reason why no linear dimensions are retained in the results. 

 Thus in solving Problem VUl, the side CD has been 

 made the radius, and all the other lines of the figure have 

 been found in terms of it ; but as the equation was through- 

 out equally affected by CD, we put it equal to unity, and 

 the whole is immediately referred to a sphere, having its 

 radius equal to 1. 



It will be found that the next problem to this, viz. 

 the IXlh, contains all the fundamental principles of spheric 

 trigonometry. 



If the earth be considered as a sphere, the three angles of 

 a triangle on its surface, being reduced to the same distance 

 from the centre, will represent those of a spheric triangle, 

 the planes of whose three sides pass through the centre of the 

 earth ; therefore when the proper redactions have been ap- 

 plied to the observed angles for this purpose by means of 

 the foregoing problems, the resolution of the triangle be- 

 comes extremely easy by spheric trigonometry. 



As radii drawn from the centre of a sphere to the circum- 

 ference, or beyond, continually diverge or recede from each 

 other, therefore when a chain of triangles are observed near 

 the earth's surface, it is clear they nmst be reduced so as to 

 have all their angular points at the same distance from the 

 earth's centre, when considered as a sphere ; but if the 

 earth's figure be taken as a spheroid of revolution, these 

 triangles are connnonly reduced to the level of the sea in 

 tile same latitude. 



The three angles of each triangle are then added together; 

 and since this triangle is in fact spherical, therefore the 

 hum of the angles must necessarily surpass 180". This ex- 

 cess, which is also affected by the error of observation, is 

 to be divided by 3^ and one-third part is to be taken from 



each 



