Latitude--The Octant Observations
Lewis took three meridian altitude observations of the sun with his octant and one afternoon observation of the sun with his sextant, and from each of these observations he calculated a latitude. He then computed the mean (or average) of these observations at 44°35'28.1" North.
Meridian Observation of the Sun
If, as suggested, Fortunate Camp was at 44°59'36"N, how did Lewis, after averaging four observations, come up with a latitude 24 minutes too far south?
Although Lewis was without peer as an observer, his mathematical skills were less well developed. Not only that, in all his latitude calculations for 1805, when he used the octant in the back observation, he employed the wrong index error.7 On July 22, 1804, when he described his scientific instruments in detail, he stated that the index error of the octant in the back observation was +2°11'40.3"; that is, it read high by that amount. Actually, the index error was +4°23'20.6", but Lewis, when using the artificial horizon,8 always divided the observed angle by two before subtracting the instrument's index error, so 2°11'40.3", as used, was correct. The problem, however, was that in 1805 Lewis used the wrong index error in his calculations. That problem began with his first observation, April 12, 1805, at the mouth of the Little Missouri River. There Lewis recorded the index error of the octant in the back observation to be +2°40'00" and he used this erroneous index error until September when he could again use the sextant to take the meridian observations. This simple mistake, by itself, makes all of Lewis's latitude calculations, when using the octant, too far south by about 28'20". To Lewis's credit, he discovered this mistake while at Fort Clatsop during the winter of 1805–06 and recalculated some of the more important latitudes. Unfortunately, he either didn't tell Clark about this error or Clark forgot about it, because Clark used the incorrect latitudes for all of his post-expeditionary maps.
Lewis's meridian observation of the sun on the August 21contains an obvious error. Although he recorded an angle of 69°15' on the nineteenth, and 70°00' on the twentieth, he recorded an angle of 72° on the twenty-first. Because the sun's declination9 was decreasing by about 20' per day, Lewis's measured double angle should have increased by about 40' per day; instead, it increased by 45' between the 19th and the 20th and 2° 00' between the twentieth and twenty-first. So, how did he derive a latitude of 44°30'21.7" N from it? Likely, he actually read an angle of 70°32 or 70°42'' from which he made his calculations, but later miscopied it as 72° to his journal.
The latitudes recalculated from Lewis's three meridian observations of the sun taken at Fortunate Camp are: 44°57' (August 19), 44°59' (August 20) and 45°39' (August 21). If his observation on the twenty-first actually was 70°32' then the latitude becomes 44°55.5', and if the angle was 70°42' the latitude is 45°00.5'. Comparing these results with the latitude of 44°59'36" suggested for Fortunate Camp at the "forks of the Jefferson," Lewis's octant observations are right on the money.
Latitude from the Equal Altitudes Observations
Lewis's meridian altitude observations, for the most part, were exceptionally good despite his erroneous calculations. But what are we to make of the latitude from the "Hor of the P.M. observation" of the sun's center taken on August 20? The only angle that Lewis gives for the equal altitudes observation for that date is 68°30'. This angle is the "double altitude" of the sun's center above the horizon at 3:57:16 P.M. The altitude of the sun above the horizon calculated from that observation is derived as follows:
By averaging the time of the AM and PM Equal Altitudes observation and correcting for the sun's changing declination, the chronometer's error on Local Time can be determined.11 From this, then, the true Local Time of the observation can be found, and by using the astronomical tables for August 1805, the sun's altitude for this observation can be calculated. The calculation shows that the sun's center should have been at 34°04'15" above the horizon, not 34°09'37". This is a substantial difference and must indicate an error in the observation. Is this why Lewis calculated a latitude of 44°33'50.5" N (27' latitude or 30 miles too far south)? No, there must be some other problem, because making a calculation such as Lewis might have made with his own data yields a latitude considerably farther to the north.
The following steps might be the way Lewis prepared his observational data to calculate the latitude from an Equal Altitudes observation.
1) From the Equal Altitudes observation on the twentieth, determine the Chronometer's error on Local Apparent Time; that is the time shown by the sun on a sun dial.
Using the times Lewis recorded in his journal for this observation, he should have found that the chronometer was 22 minutes 32.8 seconds too fast compared to Local Apparent Time.
2) Correct the time of his afternoon Equal Altitudes observation to true Local Apparent Time.
Note: if we can assume that Lewis actually made his calculation on the twentieth, then he probably did not apply the chronometer's daily rate of loss because he would not have known it until after he calculated the chronometer's error from his Equal Altitudes observation on the twenty-first.
Correcting the chronometer's time by using the error it showed at noon on the twentieth should have yielded a time of 3:34:43 PM for the observation.
3) Find the difference from the time of Local Noon (12:00:00) and multiply it by 15: +3:34:43 x 15 = 53°40'45" = Local Hour Angle
4) Correct the observed sextant altitude of the sun's center for double altitude, index error, refraction and parallax.
This is how Lewis would have done it: Divide the double altitude 68°30' by 2 = 34°15'; from 34°15' subtract the sextant's index error of 8'45" = 34°06'15".
At this point Lewis would have made his first substantial error. Both he and Clark subtracted the sextant's full index error after dividing the observed angle by two when using the artificial horizon. They either should have a) subtracted the full error before halving the observed angle or b) subtracted half the index error after halving the observed angle. This erroneous procedure makes all their observed altitudes too low by 4'22.5" and latitudes too high by the same amount. Here Lewis should have derived an altitude of 34°10'37.5", but this calculation will be made as Lewis would have done.
From 34°06'15" subtract the refraction of 1'08" found in his book of tables = 34°05'07", finally add the parallax of 7" from that book of tables = 34°05'14".
5) From the Nautical Almanac determine the declination of the sun at the time of the observation: +12°23'31".
It is possible that Lewis used the sun's declination at noon Greenwich time instead of attempting to adjust it for his estimated longitude and observation time. The results of that possibility also will be shown.
At first one might suspect that Lewis's error derives from the incorrect methodology (see Step 4, above) that he and Clark used when they made calculations from an observation employing both the sextant and the artificial horizon.
But Lewis's calculation puts the latitude too far south, not north! Lewis doesn't show his calculations or explain how he derived his result. He may have made an honest try at the calculation, but it's possible that he was unable to work the mathematics and came up with a number that seemed to fit the results of his meridian altitude observations – and accepted that as a valid figure.
7. The index error (sometimes called instrument error) is the angular difference between what the instrument measures and what it actually should measure if it were in perfect alignment. The error usually occurs because the mirrors get slightly out of adjustment when the instrument is handled or mishandled. Most navigators merely determine the index error on a regular basis and factor that error into their calculations rather than taking the time and trouble to adjust the mirrors for every use.
8. On land rarely is there a true horizon from which to measure the altitude of a celestial object. Lewis, therefore, had recourse to one of three artificial horizons they carried with them. One of them was simply a tray filled with water and protected by a wind screen. The other two used mirrors as reflecting surfaces, but the mirrors had to be leveled with extreme care.
9. For an observer on earth the declination of a celestial body is the angular distance of that body north or south of the plane of the equator as projected out into space to that celestial body.
10. The earth's atmosphere bends a ray of light from a distant object unless that object is directly overhead; this bending of a light ray is called refraction. Refraction makes a body appear higher above the horizon than it actually is, and its effect must be removed from the observed altitude of a celestial body to derive a correct latitude. Finding the amount that must be subtracted is done either by consulting special tables, or by using an appropriate formula. The expedition carried Nevil Maskelyn's Tables Requisite to be Used with the Nautical Ephemeris for Finding the Latitude and Longitude at Sea (London, 1781), and Patrick Kelly's Nautical Almanac and Astronomical Ephemeris... (London 1781-1804).
Parallax, for a heavenly body, is the angular difference between the altitude of that body as an observer would see it if there were no refraction, and the body's altitude "seen" from the center of the earth at the same time. Parallax always makes a celestial body appear lower in the sky than it actually is. Lewis could find the value to be added for parallax from the Tables Requisite referenced above. In the example from Lewis's Equal Altitudes observation on the 20th, parallax (+7") was combined with refraction (-1' 08").
11. Lewis's chronometer was of the type known as an Arnold's watch; it was more like a large pocket watch than the typical chronometer pictured for use aboard ship. During the summer of 1805 Lewis's chronometer usually lost about 30 seconds per day. On the 20th the chronometer was 19 minutes 29.4 seconds fast, and this value can be verified by the altitude of the sun's center during the afternoon equal altitudes observation. On the 21st, however, the equal altitudes observation appears to show that the chronometer was only 18 minutes 33.6 seconds fast, a loss of 55.8 seconds per day. This suggests that either the observation for the 21st either contains an error, or the chronometer was beginning to run erratically again.