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    Re: Longitude Without Time - 1977 Luce Article
    From: Frank Reed
    Date: 2018 Jul 29, 21:45 -0700

    Mark Coady, you wrote:
    "By Chichester's day, while Bowditch and others and dropped lunars from their later editions, it seems that lunars had been simplified to the point where they weren't such an intimidating exercise?  Various corner cosine methods, etc. had been worked out and made life a bit easier."

    A lot easier! In fact, all of that had been done by the late 18th century. The series methods (involving those factors of proportion which I have called "corner cosines") were available and popular right at the beginning of the lunars "era", such as it was. So what had changed by Chichester's day or John Letcher's later day, and why didn't they simply use one of those nice easy, accurate lunars methods?

    In Letcher's case, he did describe one of those nice, easy accurate methods for working real lunars! He advocated a method in his book which he appears to have assembled himself, presumably after some basic study of historical lunars methods. Letcher's method is a simple variant on the series or "corner cosines" approach, and if a copy of his book had been thrown into a time vortex and ended up in 1835, a Bowditch or Thomson would have recognized it immediately. Letcher then went on in his book to advocate this "time by lunar altitudes" method which so many people re-discovered/re-invented over the centuries. And he advocates it for the same reason that most of them do: on the face of it, it seems easier to teach to potential students who would presumably be familiar already with the nitty-gritty of calculations for altitudes. That's only true up to a point, only in certain contexts with particular "rules of the game", and it ignores the fact that lunar altitudes methods require special techniques (that will have to be taught separately!) if they're going to achieve anything like the accuracy of a "real" lunar. For example, it's important to know that lunar altitudes for GMT are only really viable when the "horns" of the Moon are relatively horizontal. 

    For other navigators who re-discovered or wrote about the concept of using lunar altitudes as a check on the Greenwich Time, it's also important to remember that real lunars had vanished into history much earlier than the almanac tables might suggest. It's true that you could find tables of lunar distances in nautical almanacs in the first decade of the 20th century, but for practical maritime use, they were fifty years in the past with only rare exceptions in the second half of the 19th century. Lunars were the stuff of legend by 1925. Sure, you might know an "old salt" who said he had once taken a lunar decades earlier and bragged about it, but that was the end of it.

    If you were a navigator trying to use the Moon for GMT in the 1930s, nearly all of the potential resources were in ancient books whose very navigational jargon was already becoming dificult to decipher. And how many navigators had access to a library of old books in the first place? As Lecky wrote in "Wrinkles", lunars were "as dead as Julius Caesar" in 1883 (and the methods might as well have been written up in Caesar's Latin, they had become so inscrutable by then). Furthermore, as lunars faded into history, the last methods recommended and the last published resources tended frequently toward the theoretical, heavily mathematical, and difficult end of the spectrum. Practical mariners were not expected to be prime users of lunars. The remaining user groups might conceivably include teams working in the high Arctic trying to achieve inordinately accurate longitudes, so fussy corrections for temperature and the oblateness of the globe were emphasized. Mathematical exactness was emphasized over practicality in many resources in the closing decades of the 19th century and into the early 20th (not always --the British Abridged Nautical Almanac included a nice practical emergency method that targeted practical navigators).

    Mark, you wrote:
    "When I did Frank's lunar class I learned that lunars were no longer a source of mathamatical pain and suffering, and that their discontinuation was not so much the result mathamatical gyrations as the result of the common availability of timepieces of high accuracy. I found the Bowditch-Thompson method or others cookbook enough not to be intimidating."

    Quite so. Bowditch-Thompson is genuinely easy, like most of the series (or "corner cosine") approaches to clearing lunars. And it's the same methodology that Francis Upchurch has grown to love, too. And as he has discovered, the required accuracy of the computations, when done using a series method (like Thompson's original, Bowditch-Thompson, or Lyons, or Turner, or a whole list of others) is really quite low, contrary to nearly all expectations of lunars, and it's quite possible to work them up on a basic handheld calculator, or with basic log tables, or on a common cheap slide rule (the Moon's "corner cosine" only tells us the fraction of the Moon's altitude correction that acts along the lunar arc and since that altitude correction is going to be on the order of one degree and given to the nearest tenth of a minute of arc --so one part in 600--, we only need the corner cosine accurate to a tenth of a percent, which is within the range of computational accuracy of common slide rules or paper calculations with common log tables --which was overwhelmingly the normal tool historically).

    It should be remembered though that using real lunars without access to a computing device did still have one roadblock through most of the 20th century --there were no tabulated lunar distances in the almanacs. Today this is no problem. It's just a great circle distance problem. Given the GHA and Dec of the Moon and the GHA and Dec of the other body at any hour of time, we can roll our own lunar distance tables just as easily as calculating the distance on a great circle from the Empire State Building in NYC to the Eiffel Tower in Paris. Without those tables, "GMT by lunar altitude" has one other mark in its favor.

    Frank Reed

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