Determing the Time of the Observation
Lewis had a number of opportunities for taking Lunar Distance observations for longitude other than the one he took on May 25, but duties of command, weather and lack of an 1806 Nautical Almanac hampered his attempts. Because Lewis did not take an Equal Altitudes observation on the 26th (or even the 27th), the time of the observation on May 25 is a problem.
There are two ways to try to resolve this problem. The first is to take the chronometer errors at local noon on May 25, June 5, and June 6 and plot them to derive the chronometer's rate of loss. If there was a good correlation one might assume that the rate-of-loss was constant, and from this one could make an approximation of the mean or mid-time of observation. The problem is that, although the line between the three points so plotted is close to a straight line, it doesn't pass through the time for which the chronometer was set on May 17 (assuming that it was set at local noon as it should have been). Additionally, a gap of eleven days between observations and a loss rate of 1 hour 17 minutes per days is a tenuous thread upon which to hang so important a calculation as the longitude.
The second method of deriving the true mean or mid-time of the observation is to calculate the time of the forenoon and afternoon Equal Altitudes observations on May 25 and compare them with the times shown by the chronometer. Next, these times are graphed or put into a linear regression together with the time the chronometer would have shown at noon based on its error at noon. This method also presents problems arising from uncertainties about the sextant's index error, and from the accuracy of the sun's observed altitude, but is the "safer" of the two methods to use.
The large observed angular distance between the sun and the moon of about 102½° is the yet another sign of trouble with this Lunar Distance observation. This is because observations with separations much over 90° are more difficult to take than those, say, between 40° and 80°. Still, with a steady hand and good eye they can provide an acceptable longitude.
The long and exacting preparatory calculations would have been considerably shortened and simplified if the captains had done two things. The first of these would have been to take another observation for Equal Altitudes on May 26 or May 27 to determine the chronometer's rate-of-loss more nearly. The second would have been to measure the altitude of the sun and moon at the beginning and end of the time-distance measurements. This, alone, would have made Steps 3 and 4, below, unnecessary.1
The first step is to take the recorded chronometer times for the observation and plot them against the observed angular separation of the sun and moon's nearest limbs. The plot reveals there is a large time gap between the first and second data sets. The first data set should be omitted. The plot also reveals that there is an error with the fourth data set: the time is later than for the next data set. The time for the fourth data set, in reality, appears to be 1:38:34, not 1:39:34, but it is best to omit this set, too, because data sets 2, 3, 5 and 6 make nearly a straight line. The average chronometer time for the four "good" data sets is 1:38:15 PM and the average angular sun-moon separation for those same four data sets is 102°29'15".
Second, find the true Local Apparent Time of the observation. In the discussion on latitude above it has been seen that by projecting the chronometer error backwards from June 6 through June 5 to May 25 that the chronometer's rate of loss for May 25 falls close to the resultant line, but for a lunar distance observation, close isn't good enough.
A calculation using an "average" latitude of 46°13'30" (derived from the meridian altitude observation of June 9 and the mean of the latitude from the two observations for Magnetic Declination—see Latitude, above) produces a "true" Local Apparent Time of 7:58:17 for the AM Equal Altitudes and a "true" Local Apparent Time of 4:01:57 for the PM Equal Altitudes. From these times and the chronometer error at noon, the "true" Local Apparent Time of the mean of the Lunar observation turns out to be 4:22:45 PM.
Now it is necessary to determine the hour angle of the sun. This is merely the hour-minutes-seconds of the observation times 15; the hour angle thus is 67°41'15". From this hour angle, the latitude and the sun's estimated declination, the calculated the true altitude the sun's center turns out to be 31°37'37". But, to make the calculation by the methods of the time, it is necessary to determine what the "observed' altitude might have been—to "uncorrect" the observation as it were; that is, add back in the effects of parallax and refraction. This operation produces an altitude of 31°39'00" for the sun.
Fourth, using the 1806 Nautical Almanac, one obtains the Right Ascension of the sun for Greenwich noon on May 25 and May 26 and from the estimated longitude of the camp, determines the sun's Right Ascension at the time of the observation. This procedure is followed by obtaining the moon's Right Ascension for noon and midnight of May 25 and determining its Right Ascension for the time of observation. Next the sun's Right Ascension is subtracted from that of the moon, giving the moon's hour angle. Continuing on, one obtains the moon's declination for noon and midnight from the Nautical Almanac and calculates the moon's declination at the time of observation. The hour angle, declination and latitude produce a true altitude of 33°42'02" for the moon's center. The moon's true altitude also must be "uncorrected" to what the captains might have observed. This is a more complicated operation because the moon's parallax is so great and usually changes at a rate arc seconds per hour. But finally the calculation is made and the "observed" altitude comes out to be 32°54'04".
There are still more calculations to be made before the true distance can be computed. The sextant's index error of 6'15" needs to be subtracted from the observed distance (102°29'15"—6'15" = 102°23'00"). To this the sun's and moon's semidiameter (calculated from data in the Nautical Almanac) are added, and also the moon's augmentation (derived also from the Tables Requisite). Now we are ready to begin the calculations to determine the true distance of the sun from the moon and then the longitude.
The data produced above are entered onto a form using a procedure essentially identical to that outlined by Robert Patterson, though modified to use real numbers instead of the logarithms that would have been used at the time.
1. At sea, with multiple sextants available and other personnel trained in making observations also available, these altitudes were regularly taken. On land, however, with a single sextant and using an artificial horizon for altitude, the process is considerably more difficult to manage. Besides, Robert Patterson recommended that, on land, the "observed" altitudes be calculated.
Funded in part by the Idaho Governor's Lewis and Clark Trail Committee.