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    David Thompson's Navigational Technique
    From: Ken Muldrew
    Date: 2004 May 28, 11:35 -0600

    As promised to Bruce Stark earlier in the week, here is an account of
    how David Thompson used celestial navigation to figure out where he
    was (Thompson in particular, although the same methods were used by
    Peter Fidler and Philip Turnor).
    
    This article is intended as a supplement to the lengthy and
    informative discussion of David Thompson's navigational technique
    written by Jeff Gottfred and published in the Northwest Journal
    (Gottfred, Jeff. How David Thompson Navigated. Northwest Journal Vol.
    9. ISSN 1206-4203. Available at:
    http://www.northwestjournal.ca/dtnav.html). Although Gottfred covers
    almost the entire subject in this excellent article, there are a few
    minor points that can be clarified for completeness. Many will be
    uninterested in such minutiae, but for others, the way these
    navigators of old approached the subjects of time and computation are
    so foreign to our modern way of thinking, that only a careful
    recreation of their methods can capture their mindset (and their
    accomplishment). Some sample data from a few days at the beginning of
    Thompson's Journey to the Kootanaes of 1800 will be used to
    demonstrate the most common elements of Thompson's technique.
    Thompson is on foot here so his courses are easy to transcribe and
    compute (when he travels by canoe he can fill an entire page with the
    courses of a day's travel). Also, I happen to have photographs taken
    from the microfilmed copy of this journal, so there is no need to
    rely on someone else's transcription. Images of the journal entries
    used here can be found temporarily at
    http://www.ucalgary.ca/~kmuldrew/dt.html. There is also a copy of his
    great map there, but it is hastily patched together from separate
    images, so it isn't a great copy, but it's good enough to see how he
    reduced the immense number of observations that he took on his
    journeys into a practical document. He used Cook's survey of the
    Pacific coast for this map, but otherwise it was almost all from his
    own data.
    
    How David Thompson came to learn navigation is a story that is well
    known, but very briefly, David Thompson was taught practical
    astronomy by Philip Turnor, astronomer for the Hudson's Bay Company.
    Turnor had been a student at Christ's Hospital Mathematical School,
    of which William Wales was the master. Wales recommended Turnor for
    the position which he began in 1778. During his tenure with the HBC
    Turnor instructed David Thompson, Peter Fidler, and Malcholm Ross in
    practical astronomy. Ross drowned before he could accomplish much,
    but Thompson and Fidler both had outstanding careers as explorers.
    Turnor returned to London in 1792 or 3 and worked for Maskelyne as a
    computer for the Nautical Almanac. He died unexpectedly and Maskelyne
    was forced to pay off his debts in order to retrieve the books and
    materials that he needed to perform his calculations. Thompson,
    educated by Turnor in 1789, defected to the Northwest Company in 1797
    and remained with them until his retirement. After surveying much of
    what is now Western Canada, he constructed his great map of the North
    West. He later surveyed the boundary between the United States and
    Canada as far as the Lake of the Woods.
    
    The Journey to the Kootanaes of 1800 began at Rocky Mountain House,
    the Northwest Company fort along the North Saskatchewan River where
    the town of the same name sits today. Thompson had over-wintered
    there and had about 10-15 lunars and several meridian altitudes of
    the sun at that location, so he was very confident of its position at
    51?21'30" N and 114?52' W. Below are his journal entries for the
    period of October 5th (the start of the journey) to October 12th
    (where he takes his first lunar). The transcription is my own (from
    photographs I took of the microfilmed copy of Thompson's journal
    borrowed from the Archives of Ontario), but I consulted Belyea's
    edition (Thompson, David. Columbia Journals. Barbara Belyea, ed.
    McGill-Queen's : Montreal, 1994.) for help with some of the illegible
    words (though many minor differences remain). The table of courses
    appears at the end of the Journey to the Kootenaes trip. I have only
    transcribed the portion from October 5 to 12. No changes to spelling
    or punctuation have been made (I would normally change "fs" to "ss"
    for legibility but here it is left as written). It's also wise to
    keep in mind that Thompson's journals are "fair copies". These are
    not the original notes written in the field, but rather the condensed
    version that is copied out later. Thompson, like Turnor, keeps a very
    spare, professional record of the events that occur. Very little
    emotion is allowed onto the page, sometimes he even neglects to
    mention the birth of one of his children. Fidler is more emotive, and
    much more graphical, committing geographical information from natives
    into roughly sketched maps in his journals. Fidler is also an order
    of magnitude easier to read.
    
    [in the following, * stands for a circle with a dot (the sun), )
    stands for the moon, and letters that follow  ^ are superscripted]
    
    Thompson's Journal - Journey to the Kootanaes. Rocky Mountain.
    
    October 1800
    
    October 5th. Sunday. A fine cloudy Day. At 8 Am the Men crossed the
    River, La Gafs?, Beauchamp, Morrin, Pierre Daniel, Boulais & myself,
    with the He Dog, a Cree, and the Old Bear, a Pekenow Indian, our
    Guide. We had an afsortment of Goods, amounting to about 300 Skins,
    each of us a light Horse, belonging to himself, and 3 Horses of the
    Company's to carry the Baggage. We met several Blood Indians going in
    to trade. Our Co to the crossing Place of the Clear Water River may
    be about SEbE 2 ? M. After crossing that Stream we went on about SE 1
    ? M to the parting of the Roads where finding we had forgot to take a
    Kettle with us, I sent La Gafs? back again to the House for one. mean
    Time we went on to the Bridge, which is a few Sticks laid acrofs a
    Brook. Our Co during this Time thro' mostly thick Woods of Pine and
    Aspins may have been SEbS 1 ? M to a small Brook with very little
    water and which we crofsed. It goes into the Clear Water River then
    SbW 10 1/2M to the Brook Bridge, here we put up to wait La Gafs?, who
    came in the Evening with 2 Kettles - fine weather.
    
    October 6th Monday In the Morn Cloudy, with a small shower of Rain -
    afterwards fine - At 6 Am set off. lost ? Hour in crofsing the
    Bridge, which we found very bad - we went on thro' a willow Plain
    about SE 4M, then we entered the woods, then Co SE 4M South 3M very
    bad swampy Ground thick Woods of Pines. Co SbE 2M small Plains, saw a
    Herd of Cows - end of Co stopped an Hour at 10 Am to refresh our
    Horses and take Breakfast at 11 Am we set off and went SE ? M to a
    bold Brook. Co along it mostly SEbE 1M when we crofsed it. Then Co SE
    6M to a Plain in which we went abt S 3M, at end of a Rill of Water.
    crofsed it. Co SbE 4M when we came to 5 Tents of Pekenow Indians,
    with whom we staid to smoke about ? H. we then went on SbE 1 1/4M and
    crofsed a Rivulet, which a small Distance below us falls into the Red
    Deers River. Co SSW 2M to the Red Deers River, which we also crofsed,
    we then went on up along the River, mostly on the Gravel Banks, which
    formerly in high Water, were part of the Bed of the River. SWbS 2M SW
    2M in these Cos several crofsed & recrofsed the small Channels of the
    River, as they came in our Way and at end of Co recrofsed the River
    altogether, and went on thro' a tolerable fine Plain SW 2M to a bold
    Brook, which falls into the last mentioned River, here we had a grand
    view of the Rocky Mountains forming a concave segment of a Circle,
    and lying from one Point to another about SbE & NbW all it's snowy
    cliffs to the Southward were bright with the Beams of the Sun, while
    the most northern were darkened by a Tempest, & those Cliffs in the
    Concave were alternately brightened by the Sun & Obscured by the
    Storm which spent its Force only on the Summits. All the above Cos by
    the Sun. we then crossed a Plain, abounding with small Willows. Co
    SbSW 6M by the Compafs, to the Foot of a high woody Hill extending
    along the Mountain, where we found 5 Tents of Pekenow Indians - Into
    one of them belonging to our Guide we went & put up at 5 Pm.
    It is surprising what a quantity of Ground in some Places it was not
    less than 500 Yds broad, by different Channels, with gravel Banks
    between them, while at present it is contracted into a Stream of from
    40 to 50 Yds and its Depth upon a Medium about 2 ft at abt 3 1/2M pr
    Hour, with here and there a few small insignificant Channels
    occupying rarely more than 200 Yds and in general much lefs. Let us
    ask The Cause of this. is it that the heavy Rains and melting of the
    Snows have carried away such Quantities of the Particles of the
    Mountain as greatly to have diminished its' height, and therefore
    does not attract the Clouds & Vapours so strongly as formerly; or
    that the Earth and Ocean in these Climes do not yield the Vapours so
    freely as of Old; or if they do, are they driven by some unknown
    Cause to break and difsolve before they reach the Mountains. whatever
    Opinion we may form, the Fact is certain, that at present and for
    several Years past the Mountains do not send forth above two Thirds
    of the Water they did formerly for we see upon the Banks of all the
    Rivers large Trees that have been carried down by the Stream, and
    left either a great way from their present Boundaries, or a great
    Height upon the Banks far above the greatest known Level of the
    present Times - These Trees are not only to be found singly, but in
    vast Numbers, piled so intricately together that it is next to
    impofsible to disentangle them.
    
    October 7th Tuesday In the Night an exceeding heavy Fall of Rain,
    which in the Morning changed to Snow, and continues all Day. in the
    Even the weather moderated. The Snow is now about 1 foot deep.
    
    October 8th Wednesday A Cloudy Day, with at times small light Snow.
    Went a hunting with a Pekenow Indian. Killed a Jumping Deer, very
    fat, & my Companion killed another, which we brought with us to the
    Tents, where we arrived in the Evening - In this Excursion we crofsed
    the Red Deers River which here, is mostly confined to one Channel of
    about 40 yards & very strong current, with Banks of Rock. found the
    Country very bad, full of Large Swamps and high Knowls covered with
    thick Woods, that were in many Places burnt. Animals of all Kinds
    were numerous: but the Weather was too Calm for Hunting.
    
    October 9.th Thursday. A very fine Day. We wait a Pekenow Indian who
    is to come with us by his Promise as our Guide. In the Afternoon he
    came, but I soon found by his Conversation, that his Company like the
    rest of his Nation now present was intended only for the Spot, for
    the sake of Smoking and what else they can get - They are so jealous
    of the Kootanaes coming in to Trade, that they do all they can to
    persuade me to return, afsuring me that it is impofsible for me to
    find them, and that in endeavouring to search them out, our Horses
    will fall by Fatigue and Hunger, and perhaps also ourselves. At Noon
    Obsd Merid Altde of *LL 63?-30'1/2 error 22'-30" Lat^de 51?-47'-21" N
    Dec^n 6?-23'-59" S
    
    October 10.th Friday A cloudy stormy Day, with high Drift & Snow
    'till 10 Am when it cleared & became tolerable fine. Went a Hunting
    with our Guide & a young Man killed a Bull of which we brought 2
    Horse Loads to the Tents - Every where thick Woods of Pines with
    Spots of Aspin, and much, very much deep swampy Ground - The Indians
    difsuade us all they can from going any further, but our Guide tells
    me, They purposely misrepresent the Country for their own private
    Views.
    
    October 11th Saturday A very fine Day, but the Snow thawed very
    little. At 10 Am we set off & went about SbW 3M SSW 1M SbW 2M end of
    Co pafsed a small Brook, which falls close by us into the Red Deers
    River, which Last may be about SSE 1M from us. put up at end of Co -
    but I went a hunting with La Gafs? and our Guide on the Heights of
    the River - where I killed a Bull, with Horns of a remarkable Length,
    measuring 35 inches along the Curve - we brought most of the Meat to
    the Tents where we arrived in the Evening. -Cloudy-.
    
    October 12th Sunday.
    Latde by Acct. 51?-42' N
           #
    *AR  - 13-10'-28"
     Dec -  7-29 S
    )AR  -131-44-36
     Dec - 23-4 ? N
     SD  - 15..10
     HP  - 55..39
    *TA  - 17-52-39
     AA  - 17-55-26
    )TA  - 58-20-12
     AA  - 57-51-11
    D  --- 71-13-54
    +2'+19" -2'-1" +2"
    Longde 114?..45' W
    
    October 11.th
    Distance of * & ) NL
     #
    20-47'-32" -- 71?..5'..15"
       48..12  --      5  ~~
       48..56  --      4..30
       49..36  --      4..15
       50..18  --      4..15
       50..56  --      4  ~~
       51..32  --      3..45
       52.. 4  --      3..30
    ------------------------
    20..49..53 -- 71..4..19
        -2..53      -22..15
    ------------------------
    20..47..~~    70..42..4
    
    Double Altitudes
     #         *UL
    20..57'..8" -- 38?..27'..15"
        57..52  --      37..~~
        58..32  --      46..45
    ------------------------
    20..57..51  -- 38..37..~~
        -2..57        -22..15
    ------------------------
    20..54..54  -- 38..14..45
    
    #         *LL
    21..~~'..~~" -- 38?..3'..45"
        ~~ ..35  --     11..45
         1..12  --     20..45
    ------------------------
    21.. 0..36  -- 38- 12.. 5
        -2..49        -22..15
    ------------------------
    20..57..47  -- 37..49..50
    
    Courses
    Co by * Dist M  N S     E       W       Latitude        longitude
                                            52?21'30"       114?52'         Rocky Mountain House
    SEbE    2.5     1.39    2.07            52?20'27"       114?48'14"      Crofsed the Clear Water
    River
    SbE     1.5     1.06    1.06            52?19'38"       114?46'18"      Parting of the Roads
    SEbS    1.5     1.24    0.84            52?18'42"       114?44'46"      Woods to a Brook. Crofsed
    it.
    SbW     10.5    10.3            2.05    52?10'50"       114?48'7"       The Bridge and Brook.
    SE      4       2.83    2.83            52?8'41"        114?42'58"      A Plain, full of willows, &c.
    SE      4       2.83    2.83            52?6'32"        114?37'50"      Thick woody Pine & Swamps.
    S       3       3                       52?4'15"        114?37'50"        " -- 
                        very Swampy.
    SbE     2       1.96    0.39            52?2'46"        114?37'7"       Small willow Plains
    SE      0.5     0.36    0.36            52?0'29"        114?36'28"      A bold Brook which we crofsed
    SEbE    1       0.56    0.83            52?0'3"         114?32'57"      At end of this Co
    SE      6       4.24    4.24            51?58'50"       114?27'18"      Thick woods.
    S       3       3                       51?56'33"       114?27'18"      Plain 
    - narrow. End of Co Rill with
    water
    SbE     4       3.92    0.78            51?53'32"       114?25'53"      Thick woods to 5 tents of
    Pekenow Indians
    SbE     1.25    1.22    0.25            51?52'35"       114?25'26"      Crofsed a strong Rivulet
    SSW     2       1.85            0.77    51?57'8"        114?26'41"      Horse Plain - end of Co entrance
    the Red Deers River
    SWbS    2       1.66            1.11    51?49'51"       114?28'29"      Upon the Gravel Banks of ? ? "
    SW      2       1.41            1.41    51?48'46"       114?30'46"        " --         end of Co
    crofsed the Red Deers River
    SW      2       1.41            1.41    51?47'41"       114?33'3"       a fine small Plain. End of Co
    crofsed a Rivulet
    SbSW    6       0.42            5.99    51?47'21"       114?42'48"      a Spring with willows - to the
    ? - at the bridge ? ?
                                                                    Obsd for Lat^de
    SbW     3       2.94            0.59    51?44'40"       114?43'45"      ? & Plain with a small Brook,
    near the Red Deers River
    SSW     1       0.92            0.38    51?43'50"       114?44'22"        "     "     "
    SbW     2       1.96            0.39    51?42'3"        114?45'         
    crofsed a small Brook, which falls
    in ? ? River at SSE 1M
                                                                    Obsd for Long^de
    
    The latitudes and longitudes given for the above courses have already
    been corrected for the latitude measurement on the 9th and the
    longitude measurement on the 12th. If we update the latitude and
    longitude strictly from the courses (using a latitude of 52? to
    calculate longitude-it would be far too much work to look up the
    cosine of each latitude with an end result that might differ by about
    2 or 3 seconds of longitude, well below the expected error), we get
    the following table:
    
    Co by *         Dist M  N       S       E       W       Latitude        longitude
                                                    52?21'30"       114?52'
    SEbE    2.5             1.39    2.07            52?20'17"       114?48'38"
    SbE     1.5             1.06    1.06            52?19'22"       114?46'54"
    SEbS    1.5             1.24    0.84            52?18'17"       114?45'32"
    SbW     10.5            10.3            2.05    52?9'20"        114?48'51"
    SE      4               2.83    2.83            52?6'52"        114?44'15"
    SE      4               2.83    2.83            52?4'24"        114?39'39"
    S       3               3                       52?1'48"        114?39'39"
    SbE     2               1.96    0.39            52?0'6" 114?39'
    SE      0.5             0.36    0.36            51?59'47"       114?38'24"
    SEbE    1               0.56    0.83            51?59'18"       114?37'3"
    SE      6               4.24    4.24            51?55'36"       114?30'9"
    S       3               3                       51?53'  114?30'8"
    SbE     4               3.92    0.78            51?49'35"       114?28'51"
    SbE     1.25            1.22    0.25            51?48'32"       114?28'26"
    SSW     2               1.85            0.77    51?46'55"       114?29'41"
    SWbS    2               1.66            1.11    51?45'28"       114?31'29"
    SW      2               1.41            1.41    51?44'15"       114?33'46"
    SW      2               1.41            1.41    51?43'1"        114?36'3"
    SbSW    6               0.42            5.99    51?42'39"       114?45'46"
    SbW     3               2.94            0.59    51?40'6"        114?46'43"
    SSW     1               0.92            0.38    51?39'18"       114?47'20"
    SbW     2               1.96            0.39    51?37'36"       114?47'58"
    
    Presumably this is what Thompson has in his field notes. When he
    comes to calculate his longitude from his lunar on the 12th he has
    already updated his latitude from the measurement on the 9th, so by
    account he figures his position as 51?42' N 114?48' W on the 12th.
    
    To get a lunar distance Thompson takes eight sights between the near
    limbs of the moon and the sun and records the measured distance and
    the time by his watch. He then measures the altitudes of the sun's
    upper and lower limbs (he uses the term "double altitudes" because he
    is using a mercury artificial horizon, he is not finding his latitude
    by sighting two time-separated altitudes (the classic double altitude
    technique)). From the upper limb altitude he finds his watch is 2'
    57" fast and from the lower limb altitude he finds his watch is 2'
    49" fast. He averages these to get a watch error of  2' 53" fast.
    Then he averages the time and distance measurements from his lunar
    and subtracts the watch error from the average time and the index
    error from the average distance. The time for his lunar, 20h 47min is
    now used to get information from the nautical almanac.
    
    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). 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):
    
    1. Find the hour angle of the body for the estimated Greenwich time
    and take the log secant. Add that to the log tangent of the
    declination and, removing 10 from the index, this is the tangent of
    an angle A.
    2. When the latitude and the declination are of different names, or
    the hour angle is greater than 90, add the latitude to the angle A,
    otherwise subtract, to get an angle B.
    3. The sum of the log cosine of B, the log cosecant of A and the log
    sine of the declination, rejecting 20 in the index, is the sine of
    the true altitude.
    
    When I do this for the values given by Thompson I get an altitude of
    17?52'54" for the sun and an altitude of 58?20'8" for the moon. I
    can't account for the slight differences between Thompson's values
    and my own (I used Raper's Nautical Tables, Thompson would have used
    either Moore's or Maskelyne's). The apparent altitudes are calculated
    by reversing the typical operations of accounting for refraction and
    parallax.
    
    He then reduces the true distance between the sun and the moon to the
    assumed Greenwich time. I don't have an 1800 almanac so I can't check
    that directly, but if I calculate the true distance given the right
    ascensions and declinations given above, I get a true distance of
    71?13'55". This seems an odd thing to do but I have checked several
    of Thompson's lunars and the D value that he writes down is always
    the true distance from the almanac for the assumed time. The typical
    procedure (as far as I understand it) would be to clear the observed
    lunar distance and then use the almanac to find the Greenwich time
    that corresponds to that distance. Thompson, however, always uses his
    assumed time (the local time adjusted by his longitude by account to
    find Greenwich time) to get a D value that corresponds to his assumed
    position.
    
    He clears his observed distance using Witchell's method. Moore (New
    Practical Navigator, 1796) describes the method thusly:
    
    {begin quote}
    First add the sun or star's and moon's apparent altitudes together,
    and take half the sum; then subtract the less from the greater, and
    take half the difference; then add together:
      The cotan of half the sum,
      The tan of half the difference, and
      The cotan of half the apparent distance,
      Their sum, rejecting 20 in the index, will be the log tan of an
    angle A.
    
    Second, when the sun or star's altitude is greater than the moon's,
    take the difference between A and half the apparent distance, but if
    less, take their sum, then add together:
      The cotan of this sum or difference,
      The cotan of the sun or star's apparent altitude, and
      The proportional log of the correction of the sun or star's
    altitude;
      Their sum, rejecting 20 in the index, will be the proportional log
    of the 1st correction.
    
    Third, if the sum of A and half the apparent distance was taken in
    the last article, take now their difference; but if their difference,
    take now their sum. Then add together:
      The cotan of their sum or difference,
      The cotan of the moon's apparent altitude, and
      The proportional log of the correction of the moon's apparent
    altitude.
    Their sum, rejecting 20 in the index, will be the proportional log of
    the 2nd correction.
    
    Fourth, when A is less than half the apparent distance, the 1st
    correction must be added to, and the 2nd correction subtracted from
    the apparent distance; but when A is greater, their sum must be added
    to the apparent distance, when the sun or star's altitude is less
    than the moon's; but when the moon's altitude is less, their sum must
    be subtracted to give the corrected distance.
    
    Fifth, in table X [Moore's table numbering], look for this last
    corrected distance in the top column, and the correction of the
    moon's altitude in the left-hand side column; take out the number of
    seconds that stand under the former and opposite to the latter. Look
    again in the same table for the corrected distance in the top column,
    and the principal effect of the moon's parallax in the left hand side
    column, and take out the number of seconds that stand under the
    former and opposite the latter. The difference between these 2
    numbers must be added to the corrected distance if less than 90?, but
    subtracted from it if more than 90?; the sum or difference will be
    the true distance.
    {end quote}
    
    If we use Thompson's data to do this we find a first correction of
    2'19" and a second correction of  -2'-1". The table gives us a third
    correction of 2". I have done this for several of Thompson's lunars
    and the corrections always agree with those he writes under his
    almanac data, whereas all the other approximate methods that I have
    tried give different corrections. Thompson's cleared distance is his
    observed distance minus the index error plus the semidiameters of the
    moon and sun with the three corrections added. The cleared distance
    is 71?13'30".
    
    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'. He never bothers with a
    corrected Greenwich time because he deals only in local time.
    Greenwich time is merely for taking values out of the almanac.
    
    Having corrected his longitude he can now go back over the endpoints
    of all his courses and correct them proportionally so that when he
    goes to map out the landmarks on his journey, he will have accurate
    data.
    
    The practice of using distance to correct longitude confused me for
    some time since Thompson never writes down his d or "D" values (where
    "D" here refers to the cleared, observed distance, not the D that
    Thompson actually records). I could take his average sight and
    correct it for semidiameter and then apply the 3 corrections from
    clearing the distance, but that value never agreed with his recorded
    D value (although at times it was only out by a couple of seconds,
    other times it was out by many minutes). It's clear now though that
    Thompson is correcting his longitude based on a difference between
    assumed and measured distance, not based on a difference in time. Our
    current perspective on time is so different from that used by
    navigators of old that it can obscure our understanding a
    straightforward procedure.
    
    There are probably a lot of errors, omissions, or unintentional
    obfuscations in the above. I would welcome any corrections or
    comments where I haven't been clear.
    
    Ken Muldrew.
    
    
    

       
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