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    Errors in Cotter's "History of Nautical Astronomy, once more
    From: Jan Kalivoda
    Date: 2003 Mar 26, 13:03 +0100

    Hello, all,
    some months ago several members of the group were prepared to search for the 
    old, but very valuable book "A History of Nautical Astronomy" from Charles H. 
    Cotter (London 1968). George Huxtable has published the list of its errors 
    and misprints in this group.
    In the last days I have ascertained that the interest in this book awoke again 
    in the group. So I send to you the list of other errors I have found or I 
    suppose to have found in this title. Perhaps it can be useful for some of 
    you. If not, neglect the rest.
    Maybe in some cases I am wrong, not Cotter. If so, you will make me wiser.
    Both lists, the mine and George's (I will insert this last to the end of this 
    message for your convenience), seem to be very long together. I don't want to 
    detract from Cotter's merits. His book was a goldmine of invaluable 
    information about the history of navigation for me. The author really had 
    read the old handbooks from 18th - 19th centuries (not older, I guess) and 
    compiled very precious text from them. But he made it a bit hastily and the 
    junctures are to be seen somewhere.
    So let's go. The second number marks the line of the page - excuse me, if I 
    had missed the numbering a bit.
    83,10 from above
    Alae sui -> Alae sive
    84,15 from below
    the longest side -> the lower shorter side ???
    101,1 from below
    the true zenith distance theta_1 -> the apparent zenith distance theta_1 at 
    the point O_1 (i.e. "O with the index 1")
    107,5 from below
    To the formula at this line a remark would be useful: "(mi - 1) = U". 
    Otherwise the deduction from the formulas at the page 105 is not clear.
    108,3 from above
    Here I am not certain. But the deduction of this formula from the formulas III 
    and IV on the page 105 seems to be wrong. Cotter proceeds, as "2 PZ" on page 
    105 would equal "lambda" (geographical latitude !) in the picture 2, page 
    104. But this not the case!
    Can anybody help with this derivation? Maybe it's my fault.
    110,11 from above
    the first edition (of Maskelyne's "Requisite Tables") -> the second edition
    113,6 from above
    One would add to this line: "And h much less than R" (otherwise R/(R+H) would not equal to (R-h)/R)
    121,11 from below
    1/4 (approx.) -> 4/1 (approx.)
    15th century -> 16th century
    151, 5-7 from above
    The words from "Moreover, ..." to the end of paragraph seem to be wrong to me. 
    When reducing the measured altitude of the Sun to the time of the first 
    observation by the run and azimuth of the Sun, the observer should not make 
    any other reductions acording to the run between observations?
    158,16 from above
    ZY is the great circle -> XY is the great circle
    163,12 from below
    after the meridian altitude -> before the meridian altitude ???
    217,16-18 from below
    This was true only if the navigator used the table of "logarithmic 
    differences", giving the value of the equation X from the page 216 by 
    inspection. Such tables were in use, but introduced an error of 3-6 
    arc-seconds into result. This was tolerable in the times when lunar positions 
    itself (hence the true lunar distances, too) were tabulated with the error of 
    15-30 arc-seconds in alamanacs, owing to deficiences of the used theories of 
    lunar motion.
    225 below, 231 passim
    Cotter explains the fundamentals of the lunar distances clearly and 
    sufficiently. But his historical sketch of their evolution is very 
    unsatisfactory. With exception of Borda's method and his direct successors, 
    he describes only the methods of ending 19th century and neglects the most 
    esteemed methods from the first half of 19th century when the importance of 
    lunars was the greatest. Elford, Bowditch, Turner, Thomson, Cambridge Tables 
    - all are missing. Maybe, I will write a modest supplement to this subject.
    In particular, the Dunthorne's method mentioned on these pages was not a 
    indirect method, but the first and the most succesful direct method, 
    preceding the Borda's in time and a simpler one. It was the most common 
    method of lunars in the continental Europe, apart from France.
    239,6 from below
    prop. log SMALL delta being given in the almanac -> prop.log. GREAT delta being given in the almanac  !!!
    241,8 from above
    the value 50' of error when neglecting the second differences is grossly 
    exaggerated - George Huxtable emphasized it rightly some time ago.
    248,5 from above
    vers (PX � PX) -> vers (PZ � PX)             (in the numerator on the right side of the equation)
    265,7 from below
    delta d ->  delta h
    265,3 from below
    tan l cos h ->  tan l csc h
    266,2 from above
    cos h -> csc h
    266,9-13 from below
    This text seems completely confused - either by Cotter or by the author of 
    original. One could try to correct it, but it would better to verify it in 
    the Nautical Magazine for 1848, which I haven't at my disposal.
    272,19 from below
    if the two hour angles -> if the difference of two hour angles
    308,3 from below
    Now the least important -> Not the least important
    The manual of Pedro Nunez (mentioned elsewhere in the book) "De arte et 
    ratione navigandi" from 1522 is missing - it was very important and very 
    early publication, the example for many other authors
    The manual of Martin Cortes appeared in 1551
    Nos 238 and 240 of bibliography seem uncomplete to me. The first (1767?) and 
    fourth (1811?) editions of Maskelyne's "Requisite Tables" are not mentioned.
    Jan Kalivoda
    Here's an updated list of some things I suspect are wrong in Charles H
    Cotter's otherwise-excellent book "A History of Nautical Astronomy". Jan
    Kalivoda has kindly helped by confirming some earlier diagnoses and adding
    others of his own. If any listmember has others to add, I would be pleased
    to receive them.
    I doubt if my list, below, is at all exhaustive: I have not deliberately
    searched out errors; these are just the ones I have stumbled across in my
    own reading.
    The paging corresponds to my Hollis & Carter 1968 edition, the last page of
    its index being 387.. I think there was also a US edition: the paging was
    probably identical. I doubt if there were any later editions.
    Here goes-
    page 49. The third paragraph starts- "The civil day at sea commenced at
    midnight", which is correct. In the next paragraph Cotter states "The civil
    day commenced when the Mean Sun culminated at noon." which is
    contradictory, and wrong
    page 118, foot of. Cotter states
    "Augmentation = Moon's semidiameter x sine apparent altitude". This is wrong.
    It would be roughly true to state instead-
    Augmentation (in minutes) = Moon's semidiam. (in degrees) x sine apparent
    but more accurate to say-
    Augmentation (in minutes) = Moon''s semidiam. (in minutes) x sine apparent
    altitude / 55.
    page 120, 2nd line, Cotter says- "...  body Y, which has the same APPARENT
    place as body X". but fig 5 shows body Y at the same TRUE place as body X.
    page 210-212, Borda's method. Here I think Cotter has got into a real mess
    with his trig. The equation that precedes equation (Y) is given as -
    (sin D/2)^2 = sin{(M+S)/2 + theta} sin {(M+S)/2 - theta}
    Here, he has got the last term the wrong way round and it should be-
    (sin D/2)^2 = sin{(M+S)/2 + theta} sin {theta- (M+S)/2}
    so in consequence, in equation (Y), the second sine term in the product of
    two sines is also reversed.
    Similarly in the last equation on page 210, for log sin D/2, the last term
    in the sum should end up as log sin (theta- (M+S)/2), not log sin ((M+S)/2
    - theta), as Cotter gives it. If you slavishly follow Cotter's steps, you
    will end up taking the log of a negative quantity, which is an
    I think Cotter has realised there's something wrong, without being sure
    what it is, because on page 211 he states the rules in words for clearing
    the distance, and in rule 5 he says- "Find the sum of and difference
    between theta and phi". Because he hasn't defined here which way round to
    take that difference, the navigator will presume that he should subtract in
    such a direction as to give a positive answer. So that bit of "fudging" has
    got Cotter out of his problem. In fact the subtraction should ALWAYS be
    theta - phi, and NEVER as stated at the foot of 210, phi - theta.
    On line 3 of page 212, that's what he has written down in the calculation,
    theta - phi, just as it should be.
    There's an additional error in Rule 5, page 211, in that the last sentence
    should not read-
    "The result is the sine of half the true lunar distance, that is D/2.",
    but instead-
    "The result is the LOG sine of half the true lunar distance, that is D/2."
    page 226, Dunthorne's method. Dunthorne's is in fact a rigorous method of
    clearing the lunar distance, but Cotter has included it in his list of
    approximate methods
    page 237. For a navigator, it may be useful to know that alpha Aquilae is
    more familiar as Altair, alpha Arietis as Hamal, and alpha Pegasi as
    page 250.  The four equations shown on this page all use the quantity s,
    but I cannot find any definition of s. I presume that s is half the
    perimeter of the PZX triangle, so-
    s = 1/2 (ZX + PZ + PX)
    In the third expression, for cos P/2, a quantity s with a subscript 2
    appears. That little 2 appears to be a misprint should be erased.
    page 264. Cotter says, about finding the moment of noon by equal Sun altitudes-
    "By taking the equal-altitude sights shortly before and after noon the
    necessity for applying a correction for the change in the Sun's declination
    in the interval is obviated, since any such change will be trifling."
    I disagree with Cotter's analysis here. It seems to me that the correction
    necessary for a changing declination does not reduce as the interval chosen
    gets closer to noon.
    page 265. Re Hall's rule. Cotter introduces delta-d as the correction in
    seconds of time..
    This ought to be delta-h
    and in the equation near the foot of the page, cos h should be cosec h.
    page 266. Similarly, in the equation at the top of this page cos h should
    be cosec h.
    page 354. Napier's rule. I suspect that the second expression, shown as-
    cos x = cos y cos z  is wrong, and should be-
    cos y = cos x cos z
    It's rather disappointing that so many errors can be found in Cotter, and
    indicates some degree of carelessness in the checking and proofreading. But
    we can all get things wrong, as I have good reason to know...
    The prevalence of these detected errors leaves me suspecting that there may
    be many more, lurking as-yet unseen.
    They detract somewhat, but not a lot, from the value of Cotter's book,
    which remains by far the best source I know to treat the development of
    astronavigation, and is a wonderful goldmine of references.
    George Huxtable.

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