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    Re: Magnetic Variation - Lewis and Clark
    From: George Huxtable
    Date: 2004 Feb 19, 00:17 +0000

    Kieran Kelly described how Augustus Gregory, in Australia, calculated his
    magnetic variation. Then he asked how Lewis and Clark did that job in the
    United States.
    Exactly two hundred years ago, Lewis and Clark were in Winter camp, at Wood
    River (Camp Dubois), just across the Mississippi from where the Missouri
    joins, preparing to set off on the expedition-proper. To get there, they
    had travelled down the Ohio, then from the location of present-day Cairo,
    up the Miss. to that camp, a few miles above St Louis.
    From Cairo, they started taking astronomical sights, and I am following in
    their oar-strokes on paper, trying to disentangle their astronomy, and
    correct their many errors, as I go, 200 years behind. I hope to "arrive" at
    Camp Dubois well before the anniversary of the day they set off, May 14
    1804. However I am presently stuck, trying to make some sense of their
    first lunar-distance observations, which were taken near Kaskaskia, about
    50 miles below St. Louis.
    On that part of the journey so far, I have met with three attempts to
    measure variation. One was a few miles above Cairo, the others near
    Observation on 21 Dec 1803.
    On the next day after leaving Cairo, the expedition set off before dawn,
    and at 9:02am on their chronometer, on a sandbank, took a compass-bearing
    of the Sun as South 46deg 30' East (or in modern notation 133.5deg
    magnetic) and took an artificial-horizon doubled-altitude of the Sun,
    stated to be 45deg 42' 15". As with all their artificial horizon
    observations, the index correcton was in error, because they halved the
    reflected-altitude, then subtracted the sextant'sindex correction of 8'
    45", rather than doing it the other way round. Also, we can now deduce that
    although they stated it to be a lower-linb observation, it must have been
    upper-limb; a common confusion with L&C.
    And that's where they left it. They were, presumably, unable to work out
    the azimuth from that Sun observation, so they could not determine the
    variation. All their legs of each day's travels were noted down in terms of
    compass direction only. Unless they discovered, further along the voyage,
    how to obtain azimuth information, I presume that throughout the voyage
    they were unable to turn their compass bearings into true ones. Perhaps
    this will become clearer, higher up the Missouri. It's a serious matter,
    because there must have been major changes in the magnetic variation over
    their immense voyage to the Pacific.
    As with so many of their observations, L&C were content to observe and
    record, and leave it to others to work out the necessary calculations after
    their return (in the event, however that didn't happen). To be fair, that
    was what they had been advised to do, as far as any lunar-distance
    observations were concerned. To be fair again, they had been provided with
    a document written by the astronomer Patterson (available, transcribed, at
    ), which explained how to make certain
    astronomical computations, but didn't include azimuths.
    However, they have provided enough data for a modern computation of the
    variation near Cairo on 21 Nov 1803. We can make a good guess about their
    chronomter error from their recent equal-altitude observations, and from
    Sun dec. and equation-of-time for that day we can deduce that at the time
    of observation the Sun azimuth was 140.7deg, so the variation was 7.2deg
    West. The Sun altitude isn't needed if the azimuth is calculated by a
    modern method such as by-
    tan az = sin LHA / (cos LHA*sin lat - cos lat tan dec).
    If tan az is negative add 180deg to az. If LHA < 180deg, add another 180deg
    to az.
    This presumes that LHA is always measured Westerly, (i.e. GHA - Westerly
    longitude), and that North is positive for lat and dec. Then the azimuth
    comes out as 0 to 360deg, increasing clockwise from North.
    The formula above was quite unsuitable for the logarithmic methods that
    navigators of the 19th century were forced to use.
    Kieran Kelly refers to a method for finding az from its haversine. That's
    all very well, except at azimuths near 0deg and 180deg, when the haversine
    changes little with angle. An azimuth calculation is likely to be required
    in such circumstances, such as near meridian transit, or azimuth of
    Polaris, so some care is then needed, or a different formula.
    The measured Sun altitude wasn't necessary for finding azimuth, but it
    hasn't been wasted. It can be used, with the chronometer times, and the Sun
    equation-of-time and dec. from the almanac, to calculate a Sun altitude for
    an observer if he were based on Cairo. The observed altitude provides an
    intercept, and we can then calculate a position-line, offset from Cairo.
    Satisfactorily, this passes within a couple of miles of the L&C position
    for that observation, deduced from a recent map by Harlan and Denny of the
    Mississippi's banks at that date.
    2 Dec 1803.
    Vol 2 of the Moulton edition of the Lewis & Clark journals (from which all
    this information was extracted) records-
    "...3 miles W. of Kasskassais made the following observations-
    By circumpherenter- Azamuth of pole Star 7deg 47' 00" at 8h 11m 45s p.m. pr
    How the travellers could claim to measure a copmpass azimuth to that
    precision is beyond me. Discounting the 00" part of it" how did they even
    measure the 47'?
    There's a description of the "circumpherenter", which appears to be a
    6-inch diameter compass (presumably equipped with sighting vanes) and
    adjusted using a spirit level. It would need such a vaned sighting
    instrument to observe the azimuth of Polaris, which would have an altitude
    of about 38deg.
    For the time of observation, we can readily compute the azimuth of Polaris
    to be 0deg 56', which gives a magnetic variation of 6.9deg West, not far
    different from the 7.2deg of 21 Dec. Again, L&C make no attempt to deduce a
    figure for the variation, though they could easily have made at least a
    rough guess by presuming Polaris to be due North. We have to remember,
    though, that Polaris in those days was much farther from the true Pole than
    it is now, 1deg 40' rather than about 0deg 40' as at present.
    3 Dec 1803. still at the same Kaskaskais camp.
    Moulton records-
    "Sun's magnetic Azamuth by Circumpherenter- at 9h9m59sA.M.S.43deg45'East".
    Decoded, this should read "at 9h 9m 59s am [the Sun's azimuth was] South
    43deg 45'East", which works out as an observed azimuth (measured clockwise
    from North) of 136deg 15'.
    From an equal-altitude observation bracketing noon on that day, we can
    deduce that the Local Mean Time at the moment of the Sun compass
    observation was 9h 24m 45.1s am. From that, the Sun azimuth was 141.9deg.
    So the magnetic variation from this observation is 5.6deg W., or less by
    1.3 deg than the measurement of the previous day of 6.9deg W. Perhaps this
    indicates the limit of accuracy of reading that compass, even on land.
    As usual, however, L&C did not deduce a variation from their observation.
    Augustus Gregory, in Australia, and Lewis and Clark, in the USA, were in
    quite different leagues, as navigators.
    Of course, there was in interval of 50 years between them, in which period
    celestial navigation became much more systematic. Gregory was a real
    professional; L&C, in contrast, were bumbling around in the dark.
    The most important factor, in my view, is that L&C were given no time to
    learn, and to practice, the art of celestial navigation, but thrown in at
    the deep end with a crash course, and nobody with experience to learn from.
    What a difference it would have made, if the Corps of Discovery had
    accepted a marine-navigator into the outfit!
    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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