Gonlomet'r'ic Prnllems. £27 



Hence it is evident that the point B (fig. 1.) viewed at A 

 will appear elevated to D, and that A seen from B will be 

 elevated to E. Lei two observers, at the same moment, take 

 tl^e 2enilh distances of these two objects. Let the zenith 

 distance of B, observed at A. be, on accotint of the refrac- 

 tion, ZiVD ; and that of A from B, xBE : now, supposing 

 A "and B to be at eqiuil distances from C, then by drawing 

 the line AG through A parallel to EF, it is evident, sliico 

 the angle GAC = ACF, 



that BAG = .jo^' - i ACB, and , 

 DAC = 1 80 - ZAD ; 

 therefore taking their difference BAD = PO" + i AGB — 

 ZAD; whence ZAB = 9U° + 4- C, or the true zenith di- 

 stance of one object seen from the other is equal to go" plus 

 half the contained terrestrial arc. Therefore by comparing 

 the observed angles with this when the earth's radius and; 

 the distance of the objects arc known, the effect of the hori- 

 zontal refraction will be had. 



When one of the places is above the earth's surface, as 

 at E, then, by nearly the t^ame mode of reasoning, we find 

 ZEB + xBE = ISO" + C: but in this case AE and BK 

 are easily obtained, which will give the angles CEB and 

 ABE, and consequently ZEB and x;BE; whence by com- 

 parison, as before mentioned, the horizontal refraction is 

 obtained. 



The usual method of allowing for the horizontal refraction 

 is to take it as some fractional part of the terrestrial arc, or 

 angle formed at the earth's centre by the distance betwcei> 

 the object and the observer. Thus a distance of forty miles 

 subtends an angle at the earth's centre of about 17' 20"; 

 consequently, the zenith distance of an object distant forty 

 miles would be about 90"' &' 40", or it would be depressed 

 8' 40" below the horizon : now s\ippose he had found this 

 angle by his instrutrient to be go" 7' 26", the difference be- 

 tween this apparent and the true angle is l' 14", which is 

 about the -r':fth part of 17' 20", and from this instance we 

 should take the horizontal refraction at about y,-th of the 

 ponlained terrestrial arc. 



After a number of observations of the quantity of this 



horizontal 



