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On 17 Feb 04, under the threadname "Magnetic Variation - Lewis and Clark",
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.

On 19 Feb 04, I posted a a reply, and then corrected it with a revised
posting that same day.

Hans Heynau is not a listmember, but he keeps a weather-eye open for web
postings about Lewis and Clark, and he has kindly contacted me to draw my
attention to some further errors in those postings.

Hans is a joint-author of the paper "Revisiting Fort Mandan's latitude",
about Lewis & Clark's astronavigation, published in the November 2001 issue
of "We Proceeded On" (journal of the Lewis and Clark trail heritage
foundation), and has been particularly helpful to me about Lewis and Clark
matters.

The errors are as follows.

I. I gave the date of the observation for variation, soon after leaving
(modern) Cairo, as 21 Dec 1803, but it should have been 21 Nov 1803 (all
the calculations were made for the correct date).

2. For that date, I had deduced, from a compass-bearing of the Sun of 133.5
deg magnetic, and a calculated Sun azimuth of 140.7 deg, that the magnetic
variation was 7.2deg West. That was wrong. From those observations, the
variation must have been 7.2deg East! I'm sorry to have made such an
elementary mistake, and rather disappointed that none of the Nav-L
navigators picked it up.

3. Similarly, for the observation on 3 Dec 1803, in which they reported a
magnetic bearing of the Sun of 136.3deg, the calculated Sun azimuth being
143.7deg, I deduced a variation of 7.4 deg. West, which should really have
been 7.4 East.

4. For the Polaris observation of 2 Dec 1803, the computed azimuth of
Polaris was 0deg 56'. The observed magnetic bearing has been transcribed as
7deg 47' 00" seconds in the edited journals, but Heynau suggests that the
manuscript entry may well have been 7deg 47' W, and without seeing that
manuscript, I agree that this is plausible. In that case, the magnetic
variation deduced from this measurement should not have been my figure of
6.9deg West, but instead 8.7deg East.

In the light of these errors, perhaps the best thing to do is to ask
listmembers to disregard those postings of 17 Feb, and replace them by a
corrected account, as follows-

===============================

LEWIS & CLARK: MAGNETIC VARIATION of late 1803, corrected.

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
Kaskaskia.

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's index 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
chronometer 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
East. 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, though convenient and accurate at all azimuths, was
quite unsuitable for the logarithmic methods that navigators of the 19th
century were forced to use.

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 about 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
chronometer."

There's a description of the "circumpherenter" on page 413 of Moulton vol
2; it 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.

How the travellers could claim to measure a compass azimuth to that
precision is beyond me. Discounting the 00" part of it" how did they even
measure the 47'? Also, that entry gives no clue as to which sector of the
compass it's to be found in.

Hans Heynau has suggested that this figure may be the result of a
mistranscription from Lewis's original manuscript entry, and without seeing
that manuscript (in his "Eastern Journal"), I tend to agree. The editor
would be accustomed to transcribing sextant altitudes, given in minutes and
seconds, and coming across this entry for azimuth, which would be in
degrees and minutes, and no seconds, but with a compass sector given with a
cardinal point (in this case presumably "W"), might have been tempted to
trancribe a badly -written "W" as 00". However, wihout a sight of the
Eastern Journal manuscript, this can be no more than intelligent
supposition.

For the time of observation, we can readily compute the azimuth of Polaris
to be 0deg 56' (clockwise from North, by modern convention), which results
in a magnetic variation of 8.7deg East. 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 Kaskaskia 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 Apparent Time at the moment of the Sun compass
observation was 9h 34m 50s am. From that, the Sun azimuth was 143.7deg.
So the magnetic variation from this observation is 7.4deg E.

As usual, however, L&C did not deduce a variation from their observation.

===============

To review their magnetic variations, then, the following values were obtained-

Near Cairo, 7.2deg East.

At Kaskasia, 8.7deg East, and 7.4deg East.

We can take the magnetic variation in that part of the journey, then, to be
8 degrees East, roughly speaking. We can apply that variation to all their
magnetic courses, recorded over the Mississippi part of the journey.

=============

Hans Heynau adds-

An East variation would be consistent with historical projections, such as
those indicated in the map model at the web site
http://clockwk.com/geomag.html
Alternate sites for the same model are
http://geomag.usgs.gov/flash/declination_fla.html
or http://geomag.usgs.gov/flash/declination_gif.html

=======================================================


George.




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