NavList:

There are many resources available when you want to decipher historical navigation calculations. Just a reminder to everyone that you can access the majority of editions of Bowditch from the 19th century as well as editions of Norie and other standard navigation manuals at various online archives including Google books and archive.org. There is an extensive though not complete list available directly from the main menus on the NavList web site: click on "Resources" and then "Bowditch and more" or follow this direct link. One catch is that you have to think about the terminology they used instead of the twentieth century ephemisms that remain with us now. Back then they didn't normally call these "time sights" though that's certainly an apt name. Look for the chapter in Bowditch (e.g. the 10th edition of 1837) entitled "To Find the Time at Sea, and Regulate a Watch, by the Sun's Altitude". You could skip the introductory paragraphs and jump right to "First Method" (on p.209; copy attached). The required tables, of course, are included in the volume. You can easily reproduce them in a modern spreadsheet. Of course this 19th century method employs a solution of the standard spherical triangle that was uniquely suited to logarithms.

If you're doing this today, and you don't care about logarithms, then you can do it more transparently using the "fundamental formula" of spherical trigonometry, or in other words, the law of cosines. The apparent time by the Sun is simply the hour angle of the Sun. Draw the usual "ZPS" triangle and solve for hour angle. The inputs are the same as the logarithmic version of the problem (of course, as must be the case): latitude, declination, and the corrected altitude yield hour angle. The business of calculating the half-sum and the remainder are not in any way fundamental to the problem. They're just mathematical setup for the logarithmic solution.

Frank Reed
ReedNavigation.com
Conanicut Island USA