From Meridian Observation of the Sun
At noon on October 18, Lewis used his sextant and artificial horizon to obtain the meridian altitude of the sun's upper limb. This observation produced a double altitude of 68°57'30" from which the Lewis calculated a latitude of 46°15'13.9".1
Most of the 3'18"-difference just noted comes from a recurring mistake the captains made in their calculations when using the sextant and artificial horizon. Their procedure was to divide the observed altitude by two, then subtract the sextant's full index error. They either should have (a) subtracted the full index error from the observed altitude before dividing the altitude by two or (b) subtracted half the index error after dividing the altitude by two. This mistake, by itself, results in a latitude that is 4'22Ω" (5 statute miles) too far north.
But, as seen above, the captains' latitude for the observation of October 18 is only 3'18" too far north – not 4'22Ω". Somewhere in the process of calculating the latitude they also must have made what is called a "compensating error." Unfortunately, as they did not save their calculations it is not possible to find this "error." Most likely they either 1) made a simple mistake in adding, subtracting or dividing or 2) incorrectly determined refraction, parallax, sun's semidiameter or the sun's declination.
The 2 arc seconds difference between the recalculated latitude and that derived from map and aerial photo interpretation is equivalent to about 200 feet. Considering that the smallest angle that Lewis's sextant was capable of displaying was 7Ω", one might conclude that this day's Meridian Altitude observation either was first-rate and the sextant's index error continued to be +8'45" as it had been since the fall of 1803 or there were some unusual compensating errors in this observation.
From Double Altitudes of the Sun
At about 8 a.m. and 10 a.m. on October 18, Lewis took observations of the sun's altitude. Those two observations, together, are generally called Double Altitudes of the sun2 and commonly are used to determine latitude. These observation pairs, however, can be used to determine the chronometer's error on Local Time provided the latitude is known.
Because the captains took a Meridian Altitude observation of the sun less than two hours after the second observation of Double Altitudes, it is clear that they took this set of Double Altitude observations to find the error of the chronometer and its rate of loss since noon on October 17; see Lewis: 1805, July 20.3
When the sun's declination is changing rapidly (a month or so on either side of the Equinoxes) and the time between observations is more than about 3 hours, the declination of the sun should be determined for each observation. At other seasons, a simple average generally is adequate to obtain a latitude to within plus or minus a few arc minutes. Nevertheless, a latitude derived from a Double Altitude observation, even when made with great care, tends to be less reliable than a latitude derived from a Meridian observation.
1. The latitude of the mouth of Snake River is shown at about 46°15 N on the Lewis and Clark map of 1806, Clark's map of 1810, and the Lewis and Clark map of 1814. David Thompson, on July 8, 1811, obtained 46°12'35" for a point "close above" this junction. In his narrative for 1811 August 5, he gives the latitude of the junction, itself, as 46°12'15" N (longitude 119°31'33" W).
2. Not to be confused with the double altitude of the sun which results from the use of the artificial horizon.
3. "Having lost my post-meridian observations for equal altitudes in consequence of a cloud which obscured the sun for several minutes about that time, I had recourse to two altitudes of the sun with sextant."
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