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    Re: Simple celestial navigation in 1897
    From: George Huxtable
    Date: 2006 Mar 6, 00:15 -0000

    First, I would like to add my appreciation (to Frank's) for the fine detective work that D.Walden
    has done on 2 March in transcribing and understanding those somewhat cryptic numbers in the Charles
    W Morgan
    workbook of 1897.
    
    He wrote  "Corrections, additions, comments are invited.", and I have the odd quibble and comment,
    if he doesn't mind.
    
    He states- "First, there seems to be, in fact three separate sight reductions on the page.  (The
    lines and boxes don't seem to accurately separate the three calculations, so they may have been
    added later.)"
    
    I agree completely. The next, similat, sight reduction seems to have been made a day
    later. Then there appears to be a gap of 3 days, in which no such calculations were made (cloudy
    weather, perhaps?) followed by  the final sight-reduction on that page. That is judging by the
    change in declination from day to day.
    
    Now for the upper calculation, the one that D.Walden has analysed. It's been labelled "Feb 19th
    1897". I take it to be an afternoon observation, somewhere South of Japan.
    
    The Greenwich time/date of 6h 34m 31s am / Feb 19th, is perfectly plausible, by the reckoning of
    those days, in which the almanac used astronomical time, in which a day extended from one noon to
    the next. What we would now reckon, in civil time, to be the morning of Feb 20, the almanac would
    then have reckoned as the morning of Feb 19th, which had followed on, after the previous midnight,
    from the evening of Feb 19th. Only at noon on the 20th, by modern reckoning, would the almanac-day
    of 20th Feb commence. When I try combine that with the date line, and the shift in time due to
    longitude differences, my head usually starts to hurt somewhat.
    
    I think that the workbook has these timings correct. What jars, though, is Walden's quotation from
    the "USNO on-line navigation site, as follows-
    
      ___________________________________________________________________________
    
      U.S. Naval Observatory
      Astronomical Applications Department
    
      Celestial Navigation Data
    
    
    
    
                  Celestial Navigation Data for 1897 Feb 19 at  6:34:31 UT
    
                       For Assumed Position:  Latitude    N  25 18.0
                                              Longitude   E 129 33.0
    
                              Almanac Data                |    Altitude Corrections
       Object       GHA        Dec         Hc       Zn    |   Refr   SD    PA    Sum
                    o   '      o   '      o   '       o   |     '     '     '     '
       SUN        275 07.7   S11 09.2   +33 14.0   235.6  |   -1.5  16.2   0.1  14.8
    
      __________________________________________________________________________
    
    By comparing those numbers with my pocket-calculator almanac, it seems to me that they apply to a
    time/date, stated in modern civil terms, of  6:34:31 UT on Feb 20 1897, not Feb 19 as stated. Has D.
    Walden made the apropriate adjustment for us, without saying so?  If so, then the use of a 24-hour
    clock giving UT time would be inappropriate. It's a bit confusing.
    
    He continues-
    
    =============================
    
      Taking only the first and uppermost calculation, I first transcribe the handwritten numbers below
    for easier reference:
      ___________________________________________________________________________
    
      6 32"50            33"03                            11"04"09
         1"41               12           04379                7" 8
      _______            _____                            ________
      6 34"31            33"15           00825            11"11"17
                         25"18           24677                2"34
                        101"08           86116            ________
                        ______           _____            11"08"43
      129"33            159"41           15997            90
                        ______           _____            ________
      Feb 19th 1897      79"50                           101"08"43
                         33"15
                        ______
                         46"35           2"58"46
                                           13"58
                                       _________
                                        15"12"44
                                         6"34"31
                                       _________
                                         8"38"13
    
    ==================================
    
    He adds- "The USNO's Sun's declination South, of 11 09.2 is close to the 1897 value of 11"08"43
    (almanac 11"04"09 + 7"8, correction for 8 hours after tabulated time (=the daily difference /3) -
    2"34 correction for 1hr 26min less than 8 hours after tabulated time)."
    
    Although this process gets to the right answer, I don't think the explanation is quite right. 11deg
    04' 09" must correspond to the Sun's tabulated declination at the following Greenwich noon,  5h 25m
    29s after the observation. The daily change in declination is a reduction of about 21' 12".   Layton
    seems to go about the process of interpolating for the change in latitude in a curious way. First he
    adds 7' 08", which Walden takes to be a correction for 8 hours, a third of a day, which would take
    him back to 8 hours before noon, or 4 am. Then he has to adjust further for the interval from 4 am
    to the time of the sight, at 6h 34m 31s, so he has next to allow for the change of dec in 2h 34m
    31s, and subtract it, because dec is decreasing. Layton subtracts 2' 34", which I take to be 3
    hours' worth (not the 1 hour 26m that Walden refers to, which appears to be an error, and not 2h
    34m, as it ought to be). My guess is that Layton is correcting for changing declination only to the
    nearest hour, which is (arguably) accurate enough, to the nearest half-minute. But his procedure for
    interpolating seems remarkably ham-handed.
    
    ==============================
    
    Layton had to use logs to work out how far from noon his time-sight was. However, for those with a
    calculator with trig functions, there's no longer any need to indulge in logs at all. The formula
    quoted by Walden can be conveniently rearranged for calculator into-
    
    t(in degrees) = arc cos ( 1 - 2 cos s sin (s-h) / (col L sin p))   (though I haven't checked that
    out)
    
    To get the time in terms of hours and fractions of an hour, divide t by 15.
    
    =====================
    
    There are some unusual features about the way Layton uses logs for his calculation.
    
    
      ___________________________________________________________________________
    
      After some guessing and iterating, using the fact that the Morgan is likely in the North Pacific
    at this time, I take 6"34"31 to be GMT (chronometer + correction), 25"18 to be Latitude North,
    129"33 to be the final result for Longitude East.  Using this date (obvious), time, Latitude, and
    Longitude, I go the USNO online navigation site, and get the following:
    
      ___________________________________________________________________________
    
      U.S. Naval Observatory
      Astronomical Applications Department
    
      Celestial Navigation Data
    
    
    
    
                  Celestial Navigation Data for 1897 Feb 19 at  6:34:31 UT
    
                       For Assumed Position:  Latitude    N  25 18.0
                                              Longitude   E 129 33.0
    
                              Almanac Data                |    Altitude Corrections
       Object       GHA        Dec         Hc       Zn    |   Refr   SD    PA    Sum
                    o   '      o   '      o   '       o   |     '     '     '     '
       SUN        275 07.7   S11 09.2   +33 14.0   235.6  |   -1.5  16.2   0.1  14.8
    
      __________________________________________________________________________
    
      Things are now looking very good.  The USNO's Sun's declination South, of 11 09.2 is close to the
    1897 value of 11"08"43 (almanac 11"04"09 + 7"8, correction for 8 hours after tabulated time (=the
    daily difference /3) - 2"34 correction for 1hr 26min less than 8 hours after tabulated time).
    
      The USNO's calculated altitude, +33 14.0, is close to the 1897 observation of 33"15 (33"03 from
    the sextant + "12, the 'universal' refraction+dip+semi-diameter correction described by Frank Reed)
    ((Close to the USNO value of 14.8' without dip correction))
    
      Now, for the actual meridian angle calculation using the 'time sight' method.  The equation used
    here is:
    
      hav t = sec L csc p cos s sin(s-h)
    
      where:
      hav=haversine (also useful is hav=(1-cos)/2
      sec=secant (sec=1/cos)
      csc=cosecant (csc=1/sin)
      cos=cosine
      sin=sine
    
      h=altitude
      t=meridian angle
      L=Latitude
      d=declination
      p=90-d  if L and d same name
      p=90+d  if L and d contrary name
      s=1/2 (h+L+p)
    
      We recognize 101"08"43 on the far right bottom as p=90+d.
      159"41 is 2 times s or (H+L+p) listed just above as: 33"15, 25:18, and 101"08 transferred from the
    far right bottom.
      79"50 is 159"41 divided by 2 or s.
      Below 79"50, 33"15 is repeated from above to facilitate calculation of the needed (s-h).
      46"35 is s-h.
    
      Now for some table (Bowditch Table 44 and 45 seem possible) look-ups (but only 4 entries for
    natural to log, and 1 for log to natural!)  The use of the formula above has the significant
    advantage of including NO addition of subtraction, so one only switches to logs once to multiply,
    then back for the answer and you're done.
    
      So, starting at the top of the third column, 04379 is 100,000 times the log base 10 of sec L. In
    detail by calculator for the first one:
    
      L=25"18=25+18/60=25.30 deg  or   25.3*pi/180=.4415683 radians
      cos L= 0.90408255
      sec L=1/cos L=1.1060937
      log10 1.1060934=.0437919
      100,000*.0437919=4379  to 4 digits
    
      00825 is 100,000 times log10 csc p.
    
      24677 is 100,000 times [(log10 cos s)+1], the +1, the standard method to avoid negative logs.
      86116 is 100,000 times [(log10 sin(s-h)+1].
      15997 is the sum of the logs, which gives the log of the product.
    
      Entering the log to natural haversine table with 15997 yields 2"58"46 in hour, minute, second
    notation. Useful equation for calculators; if x=hav t, t=acos(1-2x).
    
      13"58 must be the almanac value for the equation of time, needed for mean to apparent sun.  USNO
    above, gives GHA of Sun as 275-7.7 at 6:34:31 UT 2/19/1897.  Convert 275-7.7 degrees to time,
    subtract 12+GMT gives equation of time=14-0.
      12hr=180deg is added as required by "the rule" (it's not written down; neither the rule nor the
    12).
      The GMT 6"34"31 transferred from above is subtracted, giving the final answer for longitude,
    8"38"13 in hours, minutes, seconds.  Times 15 for degrees gives:
      8+38/60+13/3600=8.636944
      8.636944*15=129.5541deg=129deg-33min  QED.
    
      Ref: Bowditch 1962, Cugle 1936 (a great book, 'underappreciated').
    
    
    
    
    
    
    
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