From: Frank Reed
Date: 2016 Jan 23, 20:40 -0800
Mark Coady, you wrote:
" I kept googling up the question various ways with everything coming back that you just draw a perpendicular to the line opposite my vertex. Yup...ok..makes perfect sense.....thats exactly what last math site said...BUT HOW ??"
Just a quick thought here... Mark, I have a hunch --and I could be wrong-- that you're making a mistake that a lot of people make when learning this material. You may be confusing a purely mathematical derivation with a practical "procedure". When these sources talk about "drawing a perpendicular", they're not telling you that this is something you have to do in order to use these equations. They're merely outlining the mathematical derivation as background to the toolkit. So when they say "draw a perpendicular line", no, you don't have to do this. You just use the equations as given, which were originally derived by that construction. On the other hand, if you're looking for the mathematical proof... well, that's another hobby entirely.
Another thought (for the group)... some of you out there see two different sets of equations for solving a spherical triangle and imagine that these are "different solutions". Well, yes and no. They can be computationally different, of course, one set more practical with logarithms, for example, but there's really only one solution to a spherical triangle, and it's been known for centuries. You can solve pretty much anything that is solvable with the spherical law of cosines and the spherical law of sines. And really, that's the end of the story, so long as modern computing tools are available. After that it's just "trig identities" which are employed to make the work more convenient with certain olde-timey computing tools.
I also would like to address an idea that Bob Goethe has raised in a number of posts. In a recent one, Bob wrote:
"That is part and parcel of the reason that spherical geometry solutions are hard to come up with...and people keep mulling over different solutions for centuries at a time."
Mostly, I would have to say that this is not true. Spherical triangle solutions are not at all difficult to come up with. This is "high school math". The only trick is finding ones that might be computationally efficient in situations where we purposely tie our own hands --for our amusement. Naturally, we do this a lot in the "game play" side of the navigation hobby. It can be fun, and there are "neat tricks" to discover or re-discover. For example, if you try to force common slide rules to do things that they were never used for historically, you might have to dig up an old form or an unusual mathematical identity to get things done using those limited tools. Note that this isn't "historical navigation", but maybe you could call it "historical navigation as it should have been". But the mathematical solution is the same as always. The equations underlying the Bygrave slide rule (which, so there's no confusion, was used historically for sight reduction for a couple of decades) are not a distinct solution of the problem of spherical trigonometry. It's the law of cosines turned around with some trig identities. They're a different way of writing it out. A different computational algorithm that the device demands. That's all.
Conanicut Island USA