Shortly before 8 a.m. on October 17, the captains, with sextant and chronometer, began their observations of the Lunar Distance from the sun for longitude. They also took a second set of observations of the Lunar Distance from the sun the next morning. As usual, they did not calculate the longitude from their observations. The longitudes given below were calculated using the data that Clark's recorded.
Sequence of Calculations
1) A plot (graph) was made from the data pairs (angular distances and chronometer times) for each observation to evaluate how consistent the data were.
2) The data pairs that produced the longest, most-nearly straight line were selected and averaged.
3) The chronometer time for each observation average then was adjusted to Local Apparent Time based on calculations made from the Equal Altitudes observation and other observations for which the captains obtained the chronometer time of the sun's altitude.
4) The Greenwich Apparent Time of each observation average then was obtained assuming a longitude of 119°45' for the mouth of Snake River as is shown on Lewis and Clark's 1806 and 1814 maps. This makes a time difference from Greenwich of 7 hours 59 minutes.
5) The sun-moon data from the Nautical Almanac for October 1805 then were adjusted to what they would have been at the assumed Greenwich Apparent Time of the observation.
6) Calculations then were made to determine the sun and moon's true and apparent altitude.
7) The average angular separation of the sun and moon for each observation was corrected for the sextant's index error, the sun and moon's semidiameters and the moon's augmentation.
8) The calculation for true angular sun-moon distance was then made using Chevalier Jean Borda's method. The distance thus obtained was compared to the distances and times given in the Nautical Almanac for every three hours of Greenwich Time and the true Greenwich Apparent Time for the true separation was obtained.
9) The true Local Apparent Time of the observation was subtracted from the true Greenwich Apparent Time and the result, multiplied by 15 gives the longitude.
As can be seen below, the longitudes calculated by Steps 1-9 above are too far west by an average of 52 arc minutes.
As a check, longitude calculations were made using the true longitude of the 1805 mouth of Snake River of 119°02'29". These calculations moved the longitudes a mere 5 arc minutes farther to the east.
The captains' Lunar Distance observations with the sun usually produce longitudes within 25 arc minutes of the true longitude. In addition, their Lunar Distance observations made at mouth of Kansas, Fort Mandan, mouth of Marias, Three Forks and Clearwater Canoe Camp have produced longitude averages that were too far east.1 Why do both of the observations made at the mouth of Snake River result in a longitude that is too far west by about 52 arc minutes? The sun and moon on both days were aligned so as to make the observation an easy one to take, so that should not have been the problem. Furthermore is not likely that the captains misread the angles on the sextant's graduated arc by nearly the same amount on both days, nor is it likely that they misread the vernier in the same manner on both days. A possibility, however, is that the sextant's index error changed and no longer read high by 8'45". An arc minute or so in the angular distance read during the observation would make a sizeable difference in the calculated longitude. Using the average time and angular distance from the two Lunar Distance observations taken at the mouth of Snake River (see Step 2, above), what sextant index error would have given: a) the correct longitude and b) a longitude too far east by 30 arc minutes?:
a) about 6'55" too high and,
b) about 5'50" too high.
From the mouth of the Ohio River all the way to the Pacific, the captains always gave the sextant's index error as 8'45". Did they merely assume that this error remained unchanged or did they know it actually remained the same based on test observations?2 Lewis gives the first notice of a change in the sextant's index error on February 4, 1806, twenty-six months after leaving the mouth of the Ohio River: "By the mean of several observations found the error of the Sextant to be Subtractive – 5'45"." That is, the sextant read high by 5'45" and this amount must be subtracted from all angles measured. This index error, 5'45", looks suspiciously like a mis-copy for the sextant's standing error of 8'45", but that index error, in the original text, clearly is 5'45". Four months later, on June 9, 1806, Lewis gave yet another index error for his sextant: 6'15" too high. The large too-far-west error at the mouth of Snake River, however, may not be related to a change in the sextant's index error. The latitude re-calculated from the sextant observation Lewis took at the mouth of Snake River comes as close to the actual latitude as any the captains made. Inasmuch as it differs from the actual latitude by only 2 arc seconds, not 2 or 3 arc minutes, it suggests the index error, indeed, was still 8'45". Furthermore, it will be shown in the article on celestial observations taken at Station Camp (in present-day Washington nearly opposite Astoria, Oregon) that the longitudes calculated from the captains' Lunar Distance observations taken there by sextant are too far east again. Additionally, the latitude from their sextant observation at that location differs from the actual latitude by only 1 arc minute. The journal entries do not specify whether it was Lewis or Clark who took the Lunar Distance observations. If it was Clark, was he out of practice, bringing the images of the sun and moon's near limbs together incorrectly (too much overlap, making the angular distance less)? It appears that these observations are anomalous, but the reason for it is not clear at this time.
(Variation of the Needle)
The captains took two observations of the sun's altitude with the sextant and artificial horizon while simultaneously observing the sun's magnetic bearing with the circumferentor. Meanwhile another member of the expedition recorded the times shown by the chronometer for those observations. The captains did not calculate the magnetic declination of the compass from these observations. To determine the magnetic declination listed below, the two observations were averaged and a single calculation was made to find the sun's true azimuth at the averaged time.
1. One of the five observations at Fort Mandan was too far west by 27 arc minutes and one of the two observations at Fortunate Camp was too far west by 13 arc minutes.
2. A common and simple test for index error is to sight on a bright star through the sextant's horizon glass then find that same star in index mirror and bring the two images to match. The angle read on the graduated arm of the sextant is the index error. At sea or on the coast the sea horizon can be used, but the angle read on the graduated arc needs to be corrected for the height of the observer's eye above the level of the water.
3. U.S. Coast and Geodetic Survey, 1905, Lines of equal magnetic declination and of equal annual change in the United States for 1905, scale 1:7,000,000.
4. National Oceanic and Atmospheric Administration, Declination calculator. http://www.ngdc.noaa.gov/seg/geomag/jsp/USHistoric.jsp.
Funded in part by a grant from the National Park Service's Challenge Cost Share Program