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    Re: Preston's paper on Lewis & Clark's Navigation (updated)
    From: George Huxtable
    Date: 2003 Jun 7, 23:57 +0100

    This is an update to my recent posting on the Preston paper about Lewis and
    Clark, which is downloadable from-
    I said there-
    "The table also compares L&C's calculated noon Sun altitudes (without
    telling us how they were calculated) with, in another column, those
    calculated by "Ellicott's method of 1803" (without telling us how they were
    Navigators had been calculating latitudes from noon Sun altitudes for
    hundreds of years, and there's no argument that I am aware of about how
    this should be done. It's a straightforward business. Yet if you look down
    the table, there are huge discrepancies in the results, between latitudes
    calculated by these two methods, whatever they may be. They should agree
    exactly, but they differ by (in minutes, starting at the top of the table)
    4, 30, 2, 0, -32, -30, 34, 7, 1, 40, 0, -7, -3, 10, 8, 8, 14, 14, 1, 32,
    35, 12. What is going on here? Neither L&C's set of latitudes, nor
    Ellicott's, agrees at all well with the Bergantino list of positions of
    observation site. Is there something here that I am completely
    I had presumed that the latitudes calculated by L&C and the latitudes
    calculated by Ellicott were both based on identical noon-altitude
    measurements for the Sun.
    After a more careful rereading of Preston's paper I realise that this
    wasn't so. At the foot of page 183, Preston describes L&Cs calculated
    latitude, in column 3 of the table, as "the latitude calculated from the
    measured altitude of the Sun at high noon...", exactly as I had presumed.
    But at the top of page 184, he refers to the 4th column as "the average
    values of latitude ... calculated by Ellicott's 1803 method, using only the
    ... equal-altitudes measurements."
    So, on each of these days, Sun altitudes were presumably measured, both at
    noon and at two (at least) other moments when the Sun was at the same
    altitude, going up and going down. L&C deduced the lat from the noon value,
    Ellicott from the two equal altitudes. The two analyses were based on
    different data, calculated in different ways, so one would not expect them
    to agree exactly (as I had thought they should).
    Still, they are both based on observation of the same Sun on the same day,
    and if they don't produce an identical value for latitude, they ought to be
    very close: MUCH closer than appears to be the case.
    Why might there be a difference (presuming that the trig formulae have been
    properly applied)?
    Here's one possibility. I have pointed out that the reflected altitudes
    give rise to octant angles that at noon are outside the 90? range of a
    normal instrument, so it has to be used in backsight mode and then the
    index error can't be checked.
    If the equal-angle method was used, however, the two equal altitude
    measurements could have been made at times in the morning and afternoon,
    such that the Sun altitude was less than 45?. In that case, the octant
    could be used in its normal mode, which would allow its index error to be
    checked and allowed for.
    If that was the reason for the discrepancy, then the equal-altitude method
    would be expected to give the more-accurate result, and errors in the L&C
    latitude might be expected to relate to the uncorrectable index errors. To
    examine whether that argument might be feasible, it would be interesting to
    learn, from the raw data, whether the equal altitudes were indeed
    deliberately taken at times when the Sun was lower than 45?.
    However, the big differences that occur between latitudes in column 2 (the
    presumed site coordinates) and column 4 (the Ellicott lats from equal
    altitudes) still remain unaccounted for.
    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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