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## A Community Devoted to the Preservation and Practice of Celestial Navigation and Other Methods of Traditional Wayfinding

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Re: Bygrave Equations with a Calculator
From: Frank Reed
Date: 2020 Dec 25, 07:52 -0800

"Are there ranges that should be avoided when doing Bygrave equations with an electronic calculator?"

No.

Of course, if you have a calculator, there's no reason to use the Bygrave equations except to confirm your paper calculations. But if you're checking paper calculations, then go ahead and run them up on your calculator without worries.

Maybe it's worth mentioning that the Bygrave equations are not a new solution to the problems of spherical trigonometry. They're a different algorithm, a series of steps with practical advantages in certain computing technologies (like the eponymous cylindrical slide rule), but in no other way "better" than, for example, the "law of cosines" since the things are mathematically identical. We're looking at trigonometric identities here. They're the same thing, and this can be proved by a finite set of exact mathematical steps and substitutions. Even if you can't (or don't want to) work through all of those trig identity steps, a good test of this is to try out different variations on the basic problem, like calculating Hc, in a spreadsheet side-by-side. You'll discover that they match perfectly, with something like 15 digits of precision for any and all inputs. Just try three or four random cases, and if two separate sets of equations yield the same results, then you can be nearly certain that you're looking at mathematically identical solutions that only look different on the outside. By contrast, if you have two algorithms that are related by some simplification or approximation (NOT a pure trigonometric identity), then you'll discover that you get matching results only to perhaps three or four significant digits, depending on the case.

Frank Reed

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