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    Re: Time of meridian passage accuracy, Smart, and Cotter.
    From: Douglas Denny
    Date: 2009 Sep 28, 17:03 -0700

    A couple of points in reply: (though I am becoming satiated with navigational 
    detail now and must desist or I shall go mad with it).
    Quote - George: 
    "Do not take the values from Smart equation 32, if you
    want precise answers for Equation of Time. Instead, go to Meeus, where you
    will find all the details to get it exactly right."
    Well as always,  there is 'exact' ... and then there is "exact"! 
    ..... it is all relative depending on what you mean by 'exact'.
    You are too quick to deride Smart's formula.  Meeus in fact suggests using it 
    - and gives it as an alternative to his first formula.
    Smart's equation can incorporate the modern constants easily enough, and does 
    so for the most important parameter (Mean longitude) used in Smart's anyway.
    Also, Meuus uses it.  After giving one (slightly simpler than Smart's) formula for Eqn of Time, he says:-
    "the Eqn of Time can be obtained with somewhat less accuracy by the following 
    formula given by Smart....."
    ...and goes on to derive values for the EQ of Time by Smart's formula applied 
    to a previous example,  comparing it to the first method.
    In the first formula he ends up with: E = +3.427 351 degrees ( i.e 13 minutes 42.6 secs of time)
    and using Smart's formula he finds:  E = +3.427 752 degrees. (i.e. 13 minutes 42.7 secs of time).
    I don't mind the 0.1 second of time difference in Smart's formula for the 
    'somewhat less accuracy' involved: it is proabably less than my reflex action 
    to operate a stopwatch could manage.
    Ref Captain Charles H. Cotter. Ex.C. BSc.(London) MSc.(WAles) F.Inst Nav.
    Senior Lecturer,  Dept. of Maritime Studies. Cardiff.
    Quote George:
    " .......As a former student of Cotter, I would be interested in any comments
    he may have about the matter.
    I was not a formal student of Charles Cotter.
    It is a measure of the generosity and kindlyness of the man, that when as a 
    student at Cardiff in a different subject entirely than Maritime Studies, and 
    I wandered into the Maritime department on a Wednesday afternoon 
    (traditionaly when I should have been playing rugby instead on the afternoon 
    off from studies), with the idiotic idea I could perhaps ask someone and 
    learn what all this astronomical navigation was about in an afternoon! (as I 
    used to sail and had a vague interest in navigation) - and I was directed to 
    the door of one Charles H. Cotter who taught "Nautical Astronomy" (as it 
    should be more properly called if you are old fashioned as I am).
    After listening to my request, instead of dismissing me out of hand as some 
    rather stupid nitwit with no idea of what he was asking, Captain Cotter just 
    looked rather surprised for a moment and then said: "Come in and sit down. 
    Have you an hour of your time available?" to which I said "Yes", and he 
    proceeded to given me an hour long lecture on the basics of his subject. The 
    hour incidentally, seemed like five minutes it was so fascinating, and I was 
    quite disappointed and not a little surprised when he stopped, looked at his 
    watch, and said "Well that's an hour, which will do for now. Would you like 
    to come back next week to continue?"  And you know my answer - of course.
    The first lesson I remember well was Latitude by Meridian altitude of the Sun 
    (or star) using declination from the Nautical Almanac!
    Thereafter for a few weeks on Wednesday afternoons he taught me the essentials 
    of astronomical navigation up to the obtaining a sight by Marcq St Hilaire.
    He could have sent me away with a flea in my ear.  He did not.
    He could have just given me references to some books. He did not.
    He was clearly so keen on his subject, and such a generous man, (and a 
    brilliant teacher) - he willingly gave up his time to teaching an individual 
    such as myself for no other reason than merely because I had asked!  That is 
    true dedication! for which I am very grateful and always shall be. 
    A truly great man, as a person and a teacher, in my opinion.
    His books that I posess:-
    'A History of Nautical Astronomy'  1968. Mine is personally annotated on the 
    frontispiece:  "With every good wish and safe landfalls at all times"
    'The Complete Nautical Astronomer' 1969.  Annotated: To Douglas Denny .. with every good wish.
    'The Complete Coastal Navigator' 1964.  Annotated:  What is beyond the horizon? .. Let us discover.
    'The Elements of Navigation'  first published 1953. reprinted 1965.
    'The Astronomical and Mathematical Foundations of Geography'  1966.
    'The Physical Geography of The Oceans'  1965.
    'The Atlantic Ocean'  1974.
    I note your corrections for 'History of NA'.  Yes they are useful corrections, 
    and thank you for the reference to them.  Some are trifling quibbles, though 
    some are quite important like the correction to Borda's method for Lunar 
    I am quite sure had Captain Cotter had the chance for a revised edition, most 
    if not all would have been dealt with. It is notable that there were no 
    further editions of his books; I suspect and believe probably because he 
    could not get them published anymore. Navigation was changing rapidly and 
    'old' methods of using sextants was no longer of interest or probably being 
    taught to any degree of detail by then, so his books were not necessary 
    reading matter for students.
    He retired shortly after I left university; and died suddenly a few years 
    later; probably in the late 1970s though I cannot now remember.  I suspect 
    his books became obsolete for teaching aids, as the Maritime profession by 
    the late 1960s was becoming more inclined to modern radio fixing aids. 
    Transit satellite was then available and GPS just coming into existence, with 
    radio aids such as Loran C;  Omega and Decca Navigator in full swing.  Radar 
    was a primary fixing aid, and still is, for coastal work.   Getting out the 
    sextant was by then  'old hat'.
    Douglas Denny.
    Chichester. England.
    Original Post:
    Good. Douglas Denny and I appear to have converged to the point where our
    differences are resolved.
    Two by-products of that discussion have emerged, which it may be profitable
    to explore further-
    The first relates to Smart's predictions for Equation of Time in chapter 6,
    on "Time", in his Text-book on spherical astronomy.
    Smart gives a general formula, in his equation 29, in my 1971 edition. That
    equation has also been reproduced in Meeus, as eq. 28.3. Meeus quotes
    Smart's 1956 edition, presumably the copy that was on his shelf. I expect
    it's also in the same form in Douglas Denny's 1977 edition of Smart, as it
    applies always. Meeus gives a way to calculate precisely the "constants" in
    that equation, as they are not actually constant, but vary slowly with time.
    In my 1971 edition, Smart also provides , in equation 32, a simpler
    expression, in which he has substituted the values of those "constants", as
    they happened to be in 1931. Douglas tells us that in his 1977 edition,
    those values had been updated to epoch 1975. It matters, to use up-to-date
    values, if any precision is being attempted.
    Meeus shows that over the century from 1900, the July value of EoT changed
    by all of 13 seconds of time. That would be enough to throw Sun GHA
    positions out by over 3 arc-minutes. Not a lot, perhaps, but in order to
    predict Sun positions to within "Almanac accuracy" of 0.1 arc-minutes, you
    can put up with no more that half a second of error in EoT. If you tried to
    use the Smart equation 32 for that purpose, it would already be that far out
    within 3 years of the 1975 epoch. Douglas tells us he has updated the
    parameters accordingly, which is the right thing to do. Otherwise, those
    1975 predictions would be out of date by 34 years already, in 2009.
    So my warning stands. Do not take the values from Smart equation 32, if you
    want precise answers for Equation of Time. Instead, go to Meeus, where you
    will find all the details to get it exactly right.
    The second topic relates to Charles H Cotter, and his book, "History of
    Nautical Astronomy", (1968).
    In an earlier posting, responding to Douglas' reference to Cotter's
    "Complete Nautical Astronomer", (1969), I wrote-
    "Thanks for bringing that Cotter book to my attention. I hadn't come across
    it before. Cotter also treats these questions in his "History of Nautical
    Astronomy",(1968), but some aspects are confused, and several of his
    equations are wrong. Perhaps he had got things clearer by 1969."
    To which Douglas Denny replied-
    "I have not yet looked to compare Cotter's 'History of Nautical Astronomy'
    with his 'Complete Nautical Astronomy' and check your claim that he was
    inaccurate in some way.   I have all of his books, some with personal
    annotations to me as I knew him from my Cardiff student days.  It is due to
    Charles Cotter that I have an interest in navigation and nautical astronomy
    today.  He was a most charming and generous man and an excellent teacher.
    I would be very surprised indeed if he is in any way "Confused" or with
    "wrong equations" as he was a recognised expert in the theroy of nautical
    Douglas should consider the attached list of perceived errors which list
    members have discovered in "History of Nautical Astronomy", and which have
    been disseminated on this list and elsewhere. He may find it useful to keep
    as an errata list with that volume; that's what it was intended for. A few
    of those defects noted may be rather trivial, but many others are certainly
    not so. If he can shoot any down, we would be pleased to make amendments. If
    he discovers other errors, no doubt he will pass them on, and we can add
    them. As a former student of Cotter, I would be interested in any comments
    he may have about the matter.
    contact George Huxtable,
    Douglas Denny.
    Chichester. England.
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