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    Re: Contents of Maskelyne's Tables Requisite
    From: George Huxtable
    Date: 2003 Jun 7, 08:01 +0100

    This is a response to Ken Meldrew, who said (inter alia)-.
    >I'm interested in learning about the navigational techniques used by
    >land geographers in the late 18th and early 19th centuries;
    >particularly those used by the fur traders exploring the Canadian
    >West at that time (such as David Thompson, Peter Fidler, Philip
    >Turnor, etc.).
    and later-
    >Sorry for following up my own post but it seems I was mistaken in
    >thinking that Merrifield's approximate method was used by these
    >explorers as it wasn't developed until about 50 years later (Cotter, A
    >History of Nautical Astronomy, 1968). Jeff Godfred apparently uses
    >William Hall's approximate method in his Northwest Journal article on
    >David Thompson, although this method was only published in 1903.
    >It doesn't much matter if one is using a calculator, but in order to get
    >a feeling for the difficulty and time required to do the calculation, the
    >actual method needs to be used.
    >Cotter says that two approximate methods (the first due to Lyons
    >and the second due to Dunthorne) were published in the original
    >Nautical Almanac. A brief description of Dunthorne's method is
    >given in Cotter and I am also slogging through a paper by Mendoza
    >from the Philosophical Transaction of the Royal Society published in
    >1797 where 40 exact methods are given and about 10 pages,
    >generously sprinkled with algebra, are devoted to approximate
    >methods. If anyone knows of other sources for the approximate
    >methods used ca. 1800 (especially those discussed in Robertson's
    >text, Elements of Navigation, as Thompson was known to have
    >possessed a copy) I would be very grateful for suggestions.
    Reply from George Huxtable.
    Don't believe everything you read in Cotter. Jan Kalivoda and I, with other
    help, have put together a long list of known (or at least thought-to-be)
    errors in Cotter's History of Nautical Astronomy. The book contains a lot
    of careless errors, and some others which I take to be failures of
    understanding, though its many virtues remain. I will copy a
    recently-updated error list below.
    Dunthorne's method isn't an approximate method, as Cotter states, but a
    rigorous one. That's one of the Cotter errors.
    Are you aware of Arthur Pearson's website    www.ld-DEADLINK-com
    which contains many useful references?
    You refer to Jeff Gottfred's papers on Thompson.. This information is
    valuable and useful, but I have made a number of comments and queries, the
    first being this-
    ******Gottfred promises, at the very end of Art. 7, to show "how he uses
    this time to compute longitude" and again in the heading of Art.8 promises
    to do this using the data from Nov 3, but the vital last step of going from
    lunar distance and LAT to longitude seems to have got missed.*********
    This matter is relevant to your statement "where a longitude calculation is
    fully worked". Trouble is, although that's promised, it doesn't happen.
    Do you know of the interesting paper by the late Richard.S.Preston,"the
    accuracy of the Astronomical Observations of Lewis and Clark", downloadable
    from www.aps-pub.com/proceedings/jun00/Preston.pdf
    In that paper, Preston refers to an "astronomical notebook" manuscript
    compiled by the astronomer Robert.M.Patterson, which Lewis and Clark took
    with them for advice. To make that notebook more readily available I have
    transcribed it with a commentary to make it more intelligible to modern
    readers. It's on www.huxtable.u-net.com/lewis01.htm
    >the excellent articles in the archives, as well as a remarkable paper
    >by Jeff Gottfred in the Northwest Journal
    >(http://www.northwestjournal.ca/dtnav.html) where a longitude
    >calculation is fully worked, I have come to have an understanding of
    >the technique.
    >I would really like to gain an appreciation of the mechanical aspects
    >of the calculation, as it was done ca. 1800. I know that they used
    >Merrifield's approximate method for clearing the lunar distance,
    >using calculated altitudes of the moon and star (with a double
    >altitude of the sun to establish latitude).
    Not sure I agree with all the above
    >But I don't know what their
    >method of manual calculation was (although I have used log tables to
    >perform mechanical calculations...long ago). I would like to know
    >what tables were used by these navigators. Does anyone here know
    >what tables were included in Maskelyne's Tables Requisite as it
    >would have appeared in the late 1700's and what precision the
    >numbers were given to?
    I have in photocopy Maskelyne's Tables Requisite of 1767 and also his
    British Mariner's guide of 1763.. Neither contains the log trig tables that
    I think you are asking for. Those are required to be obtained separately.
    In the Tables requisite there are "proportional logs", a special trick for
    easing interpolation within the three hours between times for tabulated
    lunar distances, but those are not what you are asking about, I think.
    In the British Mariner's Guide Maskelyne says on pages 47 "In making these
    calculations seven places of figures must be used, besides the index, and
    proportion must be made for seconds, except Gardiner's logarithms be used,
    out of which the logarithms may be taken at sight, to the nearest ten
    seconds", In the following table he shows log sin 25deg 53' 24" as being
    9.6401282., and so on.
    (In my own opinion 6 figures are probably adequate in most nautical
    The quote above shows that 7-figure tables were available at that time. I
    have never seen Gardiner's logarithms, or handled any such ancient tables
    in a library. It's surprising to me that such accurate tables were
    available at that date. Presumably, they were intended for astronomers. Who
    else would require such accuracy at that early date?
    Here is a list of known or suspected errors in Cotter's "History of
    Nautical Astronomy", compiled by Jan Kalivoda and George Huxtable, with
    additions from Herbert Prinz. I plan to edit this list shortly into full
    page-order and repost it to Nav-L.
    This first part is from Jan.
    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
    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
    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".
    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
    Below are some additions to the error-list, which have recently come to light.
    page 213, log computation, bottom half of page. The logs column is
    mis-ordered. The line to the right, Ar comp cos, which is placed alongside
    the value for d, relates to s, and should be placed alongside that
    page 247, near bottom of page. Equation for hav P should have its
    square-root sign removed.
    page 250. Mid-page. Cotter says "It can be demonstrated that the first is
    suitable for cases in which P is near 90 degrees." It seems to me that the
    first formula is completely UNsuitable in those circumstances, giving an
    inaccurate and ambiguous result.
    page 275, paragraph quoting from Sumner. Herbert Prinz has pointed out that
    "defect" should become "difficulty", and "observation" should become
    end of list of errors in Cotter.
    George Huxtable
    contact George Huxtable by email at george@huxtable.u-net.com, by phone at
    01865 820222 (from outside UK, +44 1865 820222), or by mail at 1 Sandy
    Lane, Southmoor, Abingdon, Oxon OX13 5HX, UK.

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