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    Re: Latitude by law of cosines
    From: Frank Reed
    Date: 2018 Jan 18, 21:23 -0800

    Chris Caswell, you wrote:
    "I was exploring it for a thought experiment of being "dropped" somewhere in the world could I find out where I was and what I would need.  It seems I would need a sextant, watch set to correct GMT, Pub 229, Nautical Almanac, and a scientific calculator or slide rule, if I wished to not use any electronics.  I realize the slide rule, with only 3 decimal places, would be less accurate."

    Ok, that's worth consideration. Even if the thought experiment would never happen, there is a real scenario that might occur which could closely resemble the thought experiment. For example, suppose you're digging through someone's old notes and you come across a bit of scrap paper with a few sextant observations scribbled down including date and time, but no DR or other situational information or maybe nothing but some very general information like "sailing from Boston to the English Channel, we took these sights". Could you figure out an anonymous navigator's position from those notes?

    I want to assure you, first of all, that the equations you previously referenced in John Karl's book are not the answer... and they're not an answer. I am not being disingenuous when I tell you that there's no real use for that particular set of equations in a scenario like this (or, indeed, any practical scenario). But just so you know, there is an algorithm in Karl's book which some people cite as a tool for solving exactly this sort of middle-of-nowhere mystery. Its at the end of chapter 7 on "special sights", his formulas 7.5a-f. There are different printings of this book, and the pagination of yours doesn't match mine (as I have alread discovered) so I won't give a page number. I hope that description is enough to find the formulas in question, and you can explore those as you like. They are straight-forward.

    But really, that's the obscure approach to this question --a robotic application of a set of formulae. A potentially more illuminating approach that will teach you something when you apply it is the process of analog computation. To do this, you will require an analog computer for spherical trigonometry, also known as a rubber ball. Even an orange will do. Mark up your rubber ball with an equator and a north pole and a prime meridian. Get a strip of paper and stretch it from equator to pole along the surface of the ball, and then mark that distance up with five or ten degree intervals to cover the full 90 degrees. For each sight, place a dot on the globe at the latitude and longitude where the body is directly overhead --this is the declination and GHA at the instant of the sight. For each sight, take the altitude, subtract from 90 and measure that distance out from the plotted point. Measure out like this from the central dot multiple times until you have enough marks to connect to form a circle. This is your circle of position for each sight. You'll have two or more circles, and where the circles cross -- that's your latitude and longitude. This really works. It is fundamental celestial navigation, and with some basic attention to detail, you can get your position this way accurate to a few degrees. With that initial position estimate in mind, you can apply the usual intercept method, or any other methodology, to get an exact fix if required.

    Frank Reed

       
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