A Community Devoted to the Preservation and Practice of Celestial Navigation and Other Methods of Traditional Wayfinding
From: Sean C
Date: 2016 May 18, 08:45 -0700
Excellent news about the GHA/dec. table! And timely, as well. The recent discussions of 19th century navigation methods piqued my interest in that topic once again. I had made several half-hearted attempts in the past at understanding the traditional method of using logs of trig functions to work time sights. But each time, I failed to grasp it. Like Francis, I just don't have the time or money to travel and attend Frank's workshop, unfortunately. (Even though it would be far easier for me than for Francis.)
I resolved to give it another shot and to put forth my best effort at wrapping my mind around how it was done...regardless of how averse I am to maths. ;) A little googling and I found the book Practical Navigation by Warren W. Sheppard & Charles Carroll Soule. On page 281, I found the following quote:
When the navigator takes an observation for longitude he adds together the secant of the latitude (by dead reckoning), cosecant of the polar distance, cosine of the half sum and sine of the half sum minus the altitude (see longitude). These four logs. are added together and divided by 2 (unless the table 45 of haversines is used), the answer being called "Sine of apparent time at the ship." With this log. sine enter table 44, and on the page where the sine is found will be read in the time column on the left-hand side of the page, the apparent at the ship at the instant the sight was taken. If the observation be taken in the morning, it will be found in the a.m. column. If in the afternoon, in the p.m. column.
...That was the first piece of the puzzle I needed: a clear, straightforward explanation of the steps involved. So, I cracked open my 2002 ed. of Bowditch and turned to table 3: "Common Logarithms of Triganometric Functions (offset +10)". Easy enough. But, how do I determine the time without an "a.m." and "p.m." column? And what in the world is a "half-sum"? The next piece of the puzzle was to be found in Frank's post on Dec. 24th, 2012. Now I knew what a "half-sum" was. Another post (which eludes me now) clued me in to the fact that the "tens" need to be dropped from the sum of the logs before dividing it by two. Almost there...
I noticed in Frank's post that he had attached a snippet of a table with those "a.m." and "p.m." columns. But how did he arrive at those values? Without the full tables, it would be a challenge to figure out. A little experimentation and I found that by multiplying the degrees and minutes, which resulted from treating the "half-total" as a sine, by two and then dividing by 15 gave me the apparent time. Now it was all coming together. Comparing what I had read, Frank's partial table and the tables in Bowditch allowed me to fully reconstruct Frank's tables. I decided to print them out and bind them in a book for future reference.
I had already planned on putting the old table 4 in with the log tables. Now, it will be useful for a much longer period. I aslo threw in the current "Q" corrections for Polaris and the GHA of Aries tables. It's a nice little system that I was able to fit on only about fifteen pages.