CHAP. XI.] HEIGHTS 259 



to the theodolite that we must look to give us a sufficiency of 

 elevations. 



Before dealing with the method of calculation of heights, Refrac- 

 we must refer to the effects of refraction. ^^°^' 



The apparent position of one object from another, as seen 

 through our atmosphere, appears higher, whether we look up 

 or down. The amount varies with the difference of densities 

 of the various strata of the air, which are constantly changing. 



All we can do is to take the mean refraction, and it has been 

 found by experiment that by taking one- twelfth of the distance, 

 regarded as minutes and seconds of arc, and applying this to 

 the observed angle of elevation, it Avill give us a fair mean 

 result for the true angle of elevation when this is small, as 

 in all practical cases it is. It follows from this unknown 

 amount of error in the coefficient of refraction that, when 

 possible, objects should not be observed for elevation or 

 depression at more than a few miles' distance. We cannot 

 always command the maintenance of tliis limit, any more 

 than we can many other theoretical points in practical hydro- 

 graphical work ; but when circumstances are favourable, they 

 must be regarded. 



Looking upwards, or from a denser into a rarer medium, 

 the effect of refraction is to increase the apparent elevation. 

 This correction is therefore to be subtracted from elevations. 

 As the effect, when looking downwards, is also to raise the 

 object, or, in other words, to decrease the angle of depression, 

 the correction for refraction must be added to angles of depres- 

 sion. 



The angle of elevation measured by a theodolite, or the Result of 

 sextant angle when corrected for height of eye above the sea, Yorm of 

 is the angle between the tangent to the earth's surface at the *^e Earth, 

 observer's position and the line drawn from him to the object. 

 If the surface of the earth was a plane, all that would be 

 necessary to obtain the height would be to work out in a right- 

 angled triangle, Perp. = Base x Tan angle of elevation, after the 

 latter had been first corrected for the effects of refraction ; 

 but as the earth is a sphere, the tangent to it, produced, will 

 cut the line representing the height we want, not at the point 

 where it leaves the earth, but somewhere above that, depend- 

 ing upon the distance. The perpendicular, therefore, as 



17—2 



