Trigonometrical Surveying, 111 



log rf = i (3.609990 -i- log h) = 1.804995 -f- i log A (1.) 



In natural numbers, this becomes rf=: 68".8256<v/A = l'.064y'A. 

 If the general eifect of refraction be taken at 0.08 of the arc, 



then d =: -j^ a/ h ; and consequently 



h= 1.031 a^ (2.) 



in which d is the depression in minutes of arc. 



This formula, for practical purposes, may be thus expressed : 



h = d^ + % nearly, (3.) 



or the height in English feet is equal to the square of the depres- 

 sion of the horizon of the sea in minutes, increased by one-thirtieth 

 part neai'ly. 



This is applicable to observations made with the dip-sector, the 

 theodolite, and the spirit-level provided with a divided scale on a 

 slip of mother-of-pearl, when within the limits of the instrument. 



To those possessing fine instruments reading to seconds, the fol- 

 lowing rule, derived from the same principles, will be useful : 



To the constant log. 6.456838, add twice the log. of the depres- 

 sion in seconds ; the sum will be the log. of the elevation in Eng- 

 lish feet. 



Problem II. 



To determine the height of any point from a known horizontal 

 base, and the angles of elevation at each extremity in the same 

 vertical plane. 



Let a be the base, and » and /3 the angles of elevation ^ the e%- 

 tremities; then, 



A = sin* sin/3 cosec (««dt/3)ai (4.) 



The upper sign must be used when et and /3 are both interior, the un- 

 der when et, is exterior and /3 interior. 



In this latter case, it may be remarked, that the height h is equi- 

 valent to the base a, divided by the diiference of the natural cotaB- 

 gents of et, and /3. This formula will also determine the perpendi- 

 cular breadth of lakes, rivers, &c. when the angles are horizontal. 



Problem III. 

 To determine the height of a point from a horizontal base, when its 

 extremities and the point are not in the same vertical plane. 

 Let A and B be the horizontal angles measured at the extremi- 

 ties of the base a, and « and /3 the corresponding angles of eleva- 

 tion ; then 



