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    Re: David Thompson's Navigational Technique
    From: Ken Muldrew
    Date: 2004 May 31, 10:54 -0600

    On 29 May 2004 at 22:41, George Huxtable wrote:
    
    > Congratulations to Ken Muldrew for a masterly account of David
    > Thompson's navigation. It qualifies him for full membership of the
    > Band of Lunartics.
    
    The moon has been pulling me toward this mad pursuit. It is quite
    inexplicable. At least I have found company with those similarly
    afflicted.  ;-)
    
    > I have a few comments and questions about this Thompson paper.
    >
    > 1. Ken said-
    
    > 2. In Ken's transcript, a capital-pi symbol sometimes appears, which
    > presumably indicates some fraction of a mile, as in-
    >
    > "Our Co to the crossing Place of the Clear Water River may be about
    > SEbE 2 ? M."  Is this a symbol for "1/2"? In other places, "1/2" is
    > spelled out, and comes over correctly.
    >
    > Ken, please clarify this. Just for interest, would these be magnetic
    > courses? and if so, is the variation measured and stated, anywhere?
    
    The odd symbol appears as 1/2 on my computer and I suppose it was
    inserted by an auto-replace function in Word, since I typed out 1/2
    in each instance. Sorry for not catching that.
    
    The courses are by the sun except for the one case noted in the text.
    He does measure the variation of the compass, but not on this
    journey. There are instances noted in Belyea's transcription
    "Columbia Journals", but the details will have been omitted in that
    book. I'll keep my eye out for a description in Fidler's journals
    which I'm going through right now. I'll be sure to post one when if I
    come upon it.
    
    > 3. Thompson's lunar distance sequences are very closely spaced in time
    > and very smoothly varying in lunar distance. Clearly, he was a highly
    > skilled observer; much more so than Lewis & Clark.
    
    He had lots of practice. Also, when one works by candlelight in -30?C
    temperatures, good technique is demanded (on this trip The
    temperature is quite mild, it being October, but Thompson continues
    his astronomical observations throughout the winter when he is
    travelling).
    
    > 4. Ken wrote-
    > >Right ascension and declination for both the sun and moon are reduced
    > >from the Greenwich time that results from adding the longitude by
    > >account (converted to h:m:s) to the local time (as well as the
    > >equation of time if the nautical almanac used mean time in 1800,
    > >although perhaps they still used sun time then).
    >
    > They still used Sun time. The argument of the Nautical Almanac
    > remained as Apparent Time until 1834, when it switched to Mean Time.
    >
    > 5. >The true altitudes
    > >of both the sun and the moon are then calculated (the following
    > >method comes from Patterson's notebook that Lewis & Clark carried):
    >
    > My transcription of that manuscript notebook (with commentary) is
    > available at- 
    
    That's where I found this method. I have greatly appreciated your
    transcription of those notes.
    
    > 6. Ken wrote-
    > >Thompson then subtracts the cleared distance (71?13'30") from the
    > >true distance that he obtained from the almanac (71?13'54") to get a
    > >difference of 24" (D by account being greater than the cleared,
    > >measured distance). 24" in distance corresponds to 12" in time which
    > >converts to 3 minutes of longitude. He then subtracts 3' from his
    > >longitude by account (subtract because his D by account was greater)
    > >to get a corrected longitude of 114?45'.
    >
    > I'm a bit worried by this procedure, as Ken describes it, in two
    > respects.
    >
    > To start with, the "nominal" speed of the Moon across the sky
    > background is something like 30 arc-minutes per hour (but note the
    > qualification below), which allows it to go right around the sky in a
    > month. But at that rate, wouldn't an angle of 24" in lunar-distance
    > convert to 48" (not 12") in time, which would convert to 12 minutes
    > (not 3 minutes) of longitude? Am I misunderstanding something here? Or
    > was Thompson? Or is there a transcription error?
    
    The error is mine. I had worked through his courses to begin with and
    was expecting a 3' longitude correction. I suppose I just carelessly
    went from 24" to 180" in my head. My sincerest apologies. The
    correction should be 12' of longitude, not 3'.
    
    This leads to the question of why Thompson's courses add up to 3' of
    longitude but his lunar places his error at 12'. Looking over my work
    there are some other assumptions that I made. I used a sun
    semidiameter of 16'. The Online Nautical Almanac gives 16.1' for that
    date, so that reduces Thompson's cleared value by about 6". Also, I
    used an average time of lunar speed of 30"/min (just because it
    seemed to work, although it did seem uncharacteristic of Thompson's
    careful nature). Thompson's data here show a speed of the apparent
    moon of 23"/min. It seems more likely that he would have used this
    value rather than 30". The fact that this notebook has been written
    well after the fact, and that the purpose of the notes is to provide
    accurate knowledge about the country (not an accurate account of
    Thompson's specific activities) means that changes could have been
    placed into the re-copied text to avoid errors in the future. For
    example, at the end of the notebook Thompson records 10 lunars and 4
    latitudes that were taken at Rocky Mountain House in Feb., March,
    April, and December of 1801. The average latitude from these
    measurements is 51?21'30" and the average longitude is 114?48'20". In
    all his courses of 1801 he uses the same starting latitude and
    longitude for Rocky Mountain House (51?21'30" N, 114?52' W). So a
    year after the journey that I transcribed (a year in which he has
    made relevant measurements), he is using the same values that appear
    in his 1801 notebook. I think it likely that he has probably altered
    his courses to reflect his best knowledge of where he went based on
    these later measurements.
    
    So his courses probably can't be used to definitively describe his
    technique after all, but the main conclusion about comparing a
    cleared distance with an almanac distance to correct his position
    still stands. The D value that he records in his notes can be found
    by using the right ascensions and declinations that he records,
    rather than clearing the distance that he measures.
    
    > But, leaving that matter aside, there's a weakness in this procedure,
    > as I see it. It appears to assume that the lunar distance changes at a
    > reasonably constant rate (of 30 arc-minutes per hour). That is only
    > very approximately true. The effects of our old friend "parallactic
    > retardation" can, under some circumstances when the Moon is high in
    > the sky, cause the speed of the Moon with respect to the Sun or stars
    > to drop to somewhere near half that value. If measuring to a star
    > which is badly out of line with the Moon's direction of travel, it
    > could reduce even further.
    
    This was an assumption on my part based on some sloppy work. The
    evidence that I used to make the assumption (that the courses could
    be used to find an assumed longitude that could be corrected to the
    measured longitude) is no longer valid since it was based on using a
    factor of 1/2 instead of 2.
    
    > 7. Thompson uses a curious procedure for establishing Sun altitude for
    > his time-sight of the Sun, that I haven't seen before. He makes a
    > series of timed measurements of Sun lower-limb altitudes, and then
    > another series, of upper-limb altitudes, and processes each set
    > separately to obtain two values of clock error, which is then
    > averaged. It works, but I wonder why he does it that way?
    
    When he takes lunars at night he uses 1 star for each lunar (i.e. if
    he only takes one lunar then he will only take 1 altitude, but if he
    takes 2 lunars, then he also takes 2 altitudes and averages the
    difference). With the sun, he always does the LL and UL separately
    even though he's only taking a single lunar. I don't know why.
    
    Ken Muldrew.
    
    
    

       
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