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    Re: Lunars.
    From: Steven Wepster
    Date: 2001 Jul 13, 8:25 AM

     Dear Herbert,
    Herbert Prinz wrote:
    > I am excited to hear about your dissertation about LDs and Mayer and I am
    > looking forward to the time when it will be available to the public. I am not
    > aware of any exhaustive treatment of the subject of LDs since Cotter's
    > excellent book, "A History of  Nautical Astronomy", 1968. My best wishes to
    > you.
    Thank you very much.
    > I am actually surprised to hear that Mayer contributed to lunar theory. I
    > always assumed (without checking) that his merit was mainly to go to the
    > of  casting Euler's theory into the format of tables. Can you give
    us a lead to
    > relevant literature?
    It is commonly held that that is what Mayer did, and it will
    be the/a major point in my thesis that he did more than that. The
    rumour might have something to do with the prize of 300 pounds that
    the British parliament awarded to Euler at the time they awarded 3000
    pounds to Mayer's widow and 5000 punds to Harison.
     Mayer did learn many of the necessary mathematical techniques from
    Euler, especially from his "Opuscula (varii argumenti)" and from his
    papers on the perturbations in the motions of Jupiter and Saturnus
    [see Eric Forbes: The Euler-Mayer correspondence, 1971]. Mayer himself has
    always been expressive of how much he owed to Euler, who was the first
    to cast the problem of the motion of three bodies in the form of
    differential equations. However, the hardest part is to 'solve' these
    equations, whatever you mean by 'solve'. There I think Mayer has his
    own unique method, though I must express myself carefully because I
    haven't seen that much of 18th century lunar theories yet. Certainly
    what Mayer does in his account "Theoria Lunae juxta systema
    Newtonianum" (publ. 1767, but written 1754-5) has very little to do with
    Euler's first lunar theory "Theoria motus lunae" (1753). As far as I
    know the only published study that really looks into Mayer's "theoria
    Lunae" is Gautier's monumental "Essay Historique sur le Probleme de
    Trois Corps" (1817). Articles on Mayer's lunar theory by Eric Forbes
    and/or Curtis Wilson don't delve into the technicalities, which you
    have to do when you want to say something about the ideas behind it.
    > Did Mayer actually believe that his tables could be directly used at sea,
    > without any further transformation? (Obviously, the fact that Maskelyne could
    > use them did not prove their usefulness for just any odd navigator). My
    > impression was they were just meant to prove the principal
    possibility of using
    > LDs and to assist the Board in having suitable ephemeris or distance tables
    > produced. The impracticability at sea was the reason why the Board originally
    > denied him an award.
    That is an interesting question. Mayer applied for the longitude prize
    on the instigation of others, mainly Johann David Michaelis and
    Euler. He realized that having an accurate lunar theory was not
    enough, so he included an instrument of his own design (the repeating
    circle) and a method of calculation, written down in an article
    "Methodus longitudinum promota", of which I have up til now only
    second-hand information. Putting it all together it _was_ a method for
    finding longitude at sea; but first the repeating circle was rejected
    by Capt. John Campbell, and then Nevil Maskelyne devised his own
    method of calculation (well, he adopted an idea from LaCaille).
    The feasibility of the method was not only demonstrated by Maskelyne
    but also by Carsten Niebuhr, a scholar of Mayer, on a Danish
    expedition to Arabia Felix ( = Yemen ).
    Returning to the question: it might be possible that Mayer merely wanted
    to indicate that Lunars could be done, leaving the practical details to
    someone else. It doesn't sound very convincing to me; Mayer had a
    practical inclination and he was versed in the practical aspects of
    astronomy and geography. Yet the LD method doesn't seem to occupy him
    that much of his short life: lunar theory did, however.
    > Where the 14-plus 'equations' are concerned: In fact they were just for
    > longitude. Add 11 more for latitude and then some for parallaxe. But I never
    > suggested that iteration was a good method for manual evaluation. I did say
    > 'for modern use'. I still think it's the only way to go.
    True. I was referring to the 1753 version of the tables, which had
    only two eqns for latitude and three for parallax. Later he added more
    eqns for latitude (and only then was his theory accurate enough for
    Indeed you did say 'for modern use' and it was me who made the link to
    Mayer. When you make use of electronic calculation then iteration
    might be the most accurate way to go. But in almost all cases a linear
    interpolation over a time span of an hour will be accurate
    enough. Certainly in the light of measurement error. Linear
    interpolation over three hours was done using the Nautical Almanac and
    even then second order corrections were only necessary in rare
    cases. I wonder why you think that iteration is the only way to go.
     Best regards

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