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Re: Time sights for Jim T
From: Frank Reed CT
Date: 2004 Sep 26, 22:45 EDT
From: Frank Reed CT
Date: 2004 Sep 26, 22:45 EDT
Oops! I've discovered a mistake in my notes. There are a couple of factors of 1/2 in the wrong place in the Bowditch time sight description. The 19th century Bowditch time sight instructions should read as follows:
>>>
For your amusement, here is a method given in Bowditch in the 19th century for performing the same calculation:
First, calculate the Sun's polar distance: pdist = 90 - dec.
Next take half the sum of the altitude, the latitude, and the polar distance. Call that the "half_sum". From the half-sum subtract the altitude. Call that the "remainder". Add up the following four logarithms: logsec(Lat), logcsc(pdist), lcos(half_sum), logsin(remainder). Divide that sum by two. The result is equal to the logsin(LHA/2). So we look up the sum in the logsin column and take out the corresponding angle and multiply it by two (many tables included this in time units, but if not you convert to time by dividing by 15 degrees).
<<<
To repeat, this is (seemingly) complicated only because of the needs of logarithmic computation. The short formula for cos(LHA) in the earlier post is equivalent.
Frank R
[ ] Mystic, Connecticut
[X] Chicago, Illinois
>>>
For your amusement, here is a method given in Bowditch in the 19th century for performing the same calculation:
First, calculate the Sun's polar distance: pdist = 90 - dec.
Next take half the sum of the altitude, the latitude, and the polar distance. Call that the "half_sum". From the half-sum subtract the altitude. Call that the "remainder". Add up the following four logarithms: logsec(Lat), logcsc(pdist), lcos(half_sum), logsin(remainder). Divide that sum by two. The result is equal to the logsin(LHA/2). So we look up the sum in the logsin column and take out the corresponding angle and multiply it by two (many tables included this in time units, but if not you convert to time by dividing by 15 degrees).
<<<
To repeat, this is (seemingly) complicated only because of the needs of logarithmic computation. The short formula for cos(LHA) in the earlier post is equivalent.
Frank R
[ ] Mystic, Connecticut
[X] Chicago, Illinois
