or Depression of the Horizm> 33S 
It appears by the approximated expression, D = 3183 V 2 
that the dip for different elevations above the level of the sea, is 
nearly proportional to the square root of the height ; and since 
we have found it 4'.9 for ^5 feet, we obtain for any other height 
in feet H, 
D V H = V H nearly. 
o 
Hence the dip in minutes is very nearly equal to the square root of 
the altitude in feet. This method of finding the dip, the simplest, 
we believe, that has yet been given, will seldom differ in result 
above four or five seconds from the most rigid calculation, and, 
considering the changeable nature of the height of the obseiver 
at sea, may be regarded as sufficiently correct for all practical 
purposes. We deduce from ti the following practical rule for the 
calculation of the dip : Take half the logarithm of the height of 
the eye of the observer in feet, and it will be the logarithm of 
the dip in minutes. 
Example. — What is the dip of the horizon for an observa- 
tion taken at the height of 45 feet above the ocean ? 
Log. 45 = 1.65321 
I of which is *82660 = Log. of 6*7. 
Hence the Dip is 6'.7, or 6'. 42". 
The relation deduced in the preceding investigation, is one 
of the few coincidences among the objects of science, which are 
readily impressed on the mind, by their singularity and simplicity. 
Those who are acquainted with the allowance for the depression 
of the earth'^s surface below the tangent at a particular point, will 
at once recognize a similarity in point of simplicity between the 
above expression for dip, and the usual correction for curvature, 
in conducting the operations of levelling. In that case, the tan- 
gent line, or the direction indicated by an accurate level, is ele- 
vated above the true level, by a quantity expressed in feet, 
which is obtained by taking two-thirds of the square of the dis- 
tance in miles. The relation between the height of the eye and 
the dip is still more curious, and not less simple ; but the sin- 
gularity of the coincidence, in the two cases, is the more remark- 
able, as they both depend on the particular magnitude of the 
earth, and the accidental length of the English foot. 
