3S2 Mr Sang 07i Optimum Surveying. 



The ratio which the observed spherical excess bears to 720°, is 

 that which the entire area of the polygon bears to the surface of 

 the globe. But, unfortunately, that spherical excess is found 

 to be so small as to merge among the errors of observation, 

 so that nothing but the rudest approximation to the size of the 

 earth could be expected from this source. Unless, indeed, the 

 survey were very extensive, embracing many thousand square 

 miles, some other element than the horizontal angles must be 

 introduced, the relative inclinations of the horizons at various 

 stations form the best additional data ; and as these cannot be 

 had from direct observation, on account of the great curvature 

 of rays of light nearly horizontal, we must deduce them from 

 observations on stars near the zenith. 



Here, then, is the method which circumstances compel us to 

 adopt in measuring the dimensions of the earth. We first con- 

 nect, by a system of trigons, one station with another at the 

 distance of fifty miles or upwards, and then determine astrono- 

 mically the latitudes and longitudes of these stations. The 

 data thus obtained are sufficient for our purpose. 



The distance, determined astronomically, has to be compared 

 with that determined by triangulation, but here arises a diffi- 

 culty ; as the triangles are spherical, the sides cannot be com- 

 puted by the usual formula of plane trigonometry ; thus, ha- 

 ving measured a base and its two adjacent angles, we are not in 

 a condition to compute the other two sides, unless we know the 

 dimensions of the spheroid upon which we work ; but the di- 

 mensions of that sphere are the very objects of cur research. 



The method followed in the trigonometrical survey is this : 

 assuming that the degree of the meridian (or equator) is 

 60859-1 fathoms, the spherical excess for each trigon has been 

 computed from a previous approximate determination of its 

 area, and this spherical excess is employed to assist in the com- 

 putation of the unknown sides. Now, essentially, this opera- 

 tion is one of approximation merely, and the error caused by it 

 is a function of the error in the original supposition ; this ought 

 to have been rigorously scrutinized, and the computations re- 

 done according to the Rule of False of our common treatises 

 on arithmetic; until the computers were satisfied that the error 

 is too small to be noticed. 



