340 Mr Sang on Optimum Surveying, 



have been taken at one extremity or at both, while at some sta- 

 tions, the angles only have been read. Astronomical observa- 

 tions, too, have settled the latitudes and longitudes of many 

 stations. This whole field-work having been completed, we 

 proceed to draw from it, regarded as one integer operation, the 

 figure of the earth, and the true relative dimensions of the dis- 

 trict, and that with all the exactitude of which the matter is 

 susceptible : and then comparing the measured lengths of the 

 various bases with their geographical lengths, we discover the 

 true ratio which the standard yard bears to the axis of the 

 earth. The measurement of the base would not then be the 

 foundation of the survey ; and the error in it would not be aug- 

 mented in proportion to the extent of the district. Nor would 

 we need to compute from one base for so many miles, and from 

 the average of that and another for so many more. There 

 would be no selection of one set of trigons, nor any arbitrary 

 checking of the results ; our computations would be divested of 

 their hypothetical character, and unity both of design and of 

 execution would be given to the whole process. 



The general aspect of such a geodetical operation is this, 

 — The ratio of the earth's Polar to its Equatorial radius, the 

 geographical latitude and longitude of each station, would be 

 the unknown quantities ; and the question would be so to de- 

 termine each one of these, as that no change could be made on 

 it without augmenting the general measure of inaccuracy, the 

 sum of the squares of the errors of observation. 



To carry through such a computation, we would need to be 

 furnished with data for the degree of confidence to which each 

 observation is entitled, and must therefore call for a sheet ex- 

 hibiting, not the coincidences of the survey, but its discrepan- 

 cies ; not the nearness with which one base has been deter- 

 mined from another, but the probable amount of error in the 

 intermediate angles, as well as those observations which have 

 been rejected on account of their disagreement with the rest, 

 in order that we may throw the whole into the pan of an im- 

 partial balance, which will strike a fair average of them all. 



Supposing the origin of co-ordinates to be at the centre of 

 the earth : x being the intersection of the equator with the 

 prime meridian ; y that of the equator with a perpendicular 



