The captains completed only one Equal Altitudes observation at the mouth of Snake River. Calculations made from this observation reveal that the chronometer, at solar noon, would have showed a time of 11:54:40 a.m (that is, it would have been 5m20s slow on Local Apparent Time but 9m15s fast on Local Mean Time). If Lewis's chronometer either kept perfect time or gained or lost time a consistent rate, this one observation would have provided all the data necessary to determine the true Local Apparent Times for the observations for Lunar Distance and Magnetic Declination taken at the mouth of Snake River. Lewis's chronometer, however, manifested a time-variable nature despite being housed in a special box and suspended on gimbals within it. Dust, extreme heat and cold, improper winding, severe jolts while being transported all took their toll on a device that was meant to travel in a navigator's cabin at sea. During 1805, the chronometer's daily rate of loss, as calculated from two or more Equal Altitude observations made at the same location, ranged was from 15 to 65 seconds per day. These rates, however, reflect the both the accuracy of the Equal Altitudes observation and the actual daily loss. The two cannot be distinguished easily.
Although the daily (or hourly) rate-of-loss of Lewis's chronometer cannot be determined from a single AM-PM Equal Altitudes observation, there are other means to approximate its rate-of-loss. The most common method is to calculate the Local Apparent Time of two observations taken several hours to several days apart at the same point of observation.1 In order to make this calculation it is necessary to know (or be able to calculate) the following:
1) the latitude correct to within several arc minutes,
2) the sextant's index error2 (to obtain the true altitude of the sun's center for the observation),
3) the refraction correction based on the altitude determined in item 3 above,
4) a good approximation of the observer's longitude.
If all the above values are reasonably well known, the calculated Local Apparent Time of the observation usually will be within five seconds of the actual time. The time so derived is then subtracted from the time the chronometer showed at the observation. The result is the chronometer's error on Local Apparent Time for that observation. Next it is necessary to find the chronometer's error for a second observation. Then, from the time elapsed between the two observations, the chronometer's rate of loss can be calculated and the Local Apparent Time of a time-dependant observation can be determined.3
In the 17m17.7s since the 1st AM Equal Altitudes the chronometer, thus, has lost another 1.3 seconds. Therefore it was 5m12.3s slow (5m11s + 0m1.3s) on Local Apparent Time at the average time of the Lunar Distance observation, and the "true" Local Apparent Time should have been 7:59:15.7 + 0:05:12.3 = 8:04:28.
The example above merely demonstrates the procedure. In actuality, the sun's altitude during the 1st AM Equal Altitudes observation on October 17, was still too low and still was being strongly influenced by refraction. This means that the sun's altitude, as calculated from this observation, might be subject to a significant error and might not provide a reliable Local Apparent Time. This observation probably would not have been used only if more reliable observations were available. Fortunately, several more reliable observations were available, and the calculations for longitude and magnetic declination in this article were made from an evaluation of the Local Apparent Times they provided. Nevertheless, after calculating the Local Apparent Time for all the observations for which the captains gave the sun's declination and double altitude and applying the results to the Lunar Distance observation of October 17, the projected true Local Apparent Time was 8:04:29.0—a difference of 1 second, and too small to make an appreciable difference in the longitude derived from that observation.
Time-of-observation calculations were made for the two AM and one PM Equal Altitudes observations and the averaged times for Magnetic Declination. These times, together with the calculated time for noon on October 17 from the Equal Altitudes observation, suggest that the chronometer's rate of loss for October 17 was 33 seconds per day on Local Mean Time. The time-of-observation calculations made from the two double altitudes observations, however, show a different rate of loss: only 10.6 seconds in 24 hours. This difference in rate may not be real inasmuch as only 2 hours had elapsed between the two observations whereas 7 hours and 42 minutes had elapsed between the first and last observations on the 17th. Nevertheless:
A difference of one second of time represents a difference of 15" (arc seconds) of longitude. If the chronometer's average rate of loss of 30 seconds per day was used for all observations, the error in longitude would be plus or minus 7.5' (arc minutes) solely from the time element of the observation. Considering all other possible errors in making observations and calculations for longitude, most navigators of the day would have been pleased with a longitude error of "only" 15 arc minutes (about 12 statute miles at 45° latitude).
1. See, for example, Robert Patterson; 1803, Astronomy Notebook for Meriwether Lewis, Problem III.
2. Since the mouth of the Ohio Rivers in November 1803, Lewis has stated that his sextant error is 8'45" (it reads too high by that amount). The latitude derived from the sextant observation of the sun's Meridian Altitude on the 18th, seems to confirm this error. But, if so, why do the longitudes obtained here fall nearly a degree west of the actual longitude?
3. For best results, the observation (Lunar distance, Magnetic Declination) for which the correct time is needed should lie between the two observations for which the Local Apparent Time has been calculated. If this is not possible there should be about 8 hours difference (or more) between the two calculated times in order to extrapolate the error rate to the observation for which the correct time is needed.
Funded in part by a grant from the National Park Service's Challenge Cost Share Program