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    Re: Longitude by observation of the Sun, 19th Century
    From: Frank Reed
    Date: 2013 Jun 12, 09:56 -0700

    Jacques, you wrote:
    "In France, we have not haversine table"

    Well, that doesn't matter, and that's not what I was referring to in any case. The equation which you call "Borda's formula" is simply "the solution" for true time in nearly every English-language navigation manual from the late 18th right through the mid-20th centuries. It didn't need a name. It is sometimes called the "haversine formula" when we need to distinguish it from the common "cosine formula" for solving a spherical triangle. And I should add, for anyone else following along, when worked up on a modern computing device, it makes exactly ZERO difference. They are mathematically identical. All we're talking about here is the solution of the most basic spherical triangle problem. The haversine/Borda solution was preferred historically simply becaus it is easier to work up in logarithms with fewer cases. You can spot its "signature" as I noted in a post to Peter, by looking for three angles added up and then divided by two, and then four logarithms added up. There's more to it, but that's a good initial test. If he sees that in his time sights, then we can be confident that they are simply "common" time sights worked by the overwhelmingly most common method (whether Borda's name should be linked to it or not).

    Getting back to our lunars conversation, it was the application of this haversine/Borda computational "trick" that distinguished Bowditch's original method (known as the method of "Mendoza Rios" in Britain) from Lyons' method. They replaced the "cosine formula" solution of the triangles for the "corner cosines" with the "haversine formula". I call it a "trick" because, as above, there's no intrinsic mathematical superiority --one is derived from the other by simple trigonometric identities. It is merely that the haversine (what you call Borda) formula saves us from an "embarrassment of cases", as they used to say. It's a little longer in the paperwork, but you don't have think as hard! And that's always a good thing for the poor practical navigator tossing about in miserable weather on the open ocean.


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