332 Mr Anderson on the Method qf’fmdhig the Dip 
Let r denote the radius of the earth, corresponding to the 
mean parallel of 45°, in which case it is 20972190 feet; then, 
if h be the height of the observer, in feet, above the surface of 
the ocean, and t the tangent at the point A'. 
# = VDA'x AA' = V ^ ^ ^ 
But if the arch AE, which, from what we have shewn, must 
measure the dip, be represented by D, then, from the well known 
expression for an arch, in terms of its tangent, 
D ^ 5 4“ ^ — 5 
If t be expressed in terms of the radius, considered as unity, 
then h must be divided by 20972190, by which means it will 
become a very small fraction, and it will be quite unnecessary to 
retain more than two terms of the above series. We shall thus 
have t— and 
1+^}. 
This expression will give the dip in minutes, if it be multi- 
plied by 3437.75, the number of minutes equal to the radius, 
when the circumference is 360° ; but before it can be used for 
practical purposes, it must be corrected for atmospherical refrac- 
tion. At present it is sufficient to state, that, according to La 
Place and De Lambre, the effect of refraction, in the ordinary 
condition of the atmosphere, is to increase angles of elevation 
observed near the surface of the earth, j § o intercepted 
terrestrial arch between the object and the place of the observer. 
Hence the angle AA^E must be multiplied by igf to reduce it 
to its proper magnitude divested of refraction. We thus ob- 
tain 
D = S4S7.75 X 2 A + | 1 + | 
For elevations not exceeding 400 feet, it will be sufficiently 
coiTect to give this expression the form 
D =: 3183 V^XT^^r even D = 3183 V 2 h. 
Thus if A = 25 feet, then h = ^ — - = -OOGOOliggOS. 
20972iy0 
And D = 31 83 V 0000023841 = 3183 x 00154 = 4' -9. 
D=(2A + A^)^+| 
orD = ^2A + /t2 I 
