While at Fort Mandan, Lewis made observations of the Lunar distance from the sun for longitude on four different dates. He recorded making only one attempt to calculate a longitude from them; this was on 23 February 1805. On that date Lewis completed the Lunar distance observation less than ten minutes before he began his Equal Altitudes observation to check his chronometer. From this Equal Altitudes observation he calculated that his chronometer was 2 hr 14 min 25.7 sec slow on Local Apparent Time at noon (using modern data I calculate 2:14:29.2 slow). He, however, had taken no Equal Altitude observations since 6 February to check the chronometer's daily rate of loss, but must have felt he could make a valid correction. Let's look at his efforts:
What longitude might Lewis have calculated from this observation? That longitude would depend upon 1) the method he used for "clearing the lunar," 2) how he interpolated the logarithms he needed to use and, of course, 3) the accuracy of his mathematics.
Robert Patterson provided Lewis with a method of clearing the lunar. Ellicott also must have given Lewis a method to use, and there were other methods including that by Nathaniel Bowditch. The method that was considered the most rigorous was that by Chevalier Jean Borda in 1787. Methods for clearing the lunar continued to be developed every few years up to about 1920 when the lunars became obsolete except for a steadfast handful of navigators who still practice (and improve upon) the method today.
Taking a Lunar Distance required practice, steady hands and good eyesight. Using the sextant, an observer sighted the sun, moon or star through the sextant's horizon glass. After finding the other body (sun or moon) in the index mirror, the two objects were optically brought together until they just touched. The observer called "Time," and the time shown by the chronometer was noted while the observer read the angle measured by the sextant. This operation was repeated until (usually) six to eight measurements had been obtained, often over a period of twenty minutes to one-half hour.
For the observation of 23 February 1805 the sun and moon were at 17.5° and 14.3° above the horizon, respectively. This meant that the sextant had to be held nearly horizontal to make the observation. The observer usually tried to keep the "working side" of the sextant upwards, but for some moon-star observations it was necessary to turn the "working side" downwards. Sextant positions or attitudes had much to do with the accuracy of the observation because of fatigue. Reading the angle measured on the index arm also could be difficult, especially at night using torchlight.
If accurate and reliable observations for Lunar Distance were difficult to take, making the calculations that converted the observed data into longitude was equally challenging--and much more time consuming.
Preliminary Calculations (as needed for
Lewis and Clark's Lunar Distances)
Step 1: The times and distances from the observation should be plotted to evaluate them and reject data pairs that look spurious,
Step 2: Using the valid data pairs, averages for the time and for the distance are calculated,
Step 3: The true Local Mean Time must be determined from the chronometer time – usually from data obtained by Equal Altitudes observations before and after the Lunar Distance observation,
Step 4: Convert Local Mean Time of the observation to Local Apparent Time using the Equation of Time ratioed to your estimated or dead-reckoned longitude,
Step 5: From the Local Apparent Time, calculate the sun's hour angle (15 x the number of hours before or since Local Apparent Noon),
Step 6: Using data from the Nautical Almanac find, for the moon and the sun or star its 1) Right Ascension, 2) Declination, 3) semidiameter (sun and moon only) and 4) moon's horizontal parallax. All these parameters must be adjusted (ratioed) to the time of the observation using the estimated longitude,
Step 7: Using the proper formula together with a book of trigonometric logarithms, calculate the altitude of the sun's center from its hour angle and declination and the known latitude,
Step 8: Using Tables Requisite find the refraction and parallax for the true altitude determined and calculate what the apparent altitude the sun would have been for this observation.
Step 9: From the sun's right ascension at the time of the observation, the time of the observation and the moon's right ascension, calculate the moon's hour angle,
Step 10: Using the proper formula together with a book of trigonometric logarithms, calculate the altitude of the moon's center from its hour angle and declination and the known latitude
Step 11: Using Tables Requisite find the refraction and parallax for the true altitude determined and calculate what the apparent altitude the moon would have been for this observation (if the Lunar distance observation was between the moon and a star, steps 8, 9 and 10 must be repeated),
Step 12: From Tables Requisite and the altitude for the moon obtained in step 10, find the moon's augmentation,
Step 13: From the average lunar distance, subtract the sextant's index error,
Step 14: If the observation was between the sun and moon, to the result of Step 13 add the sun's semidiameter and the moon's semidiameter plus augmentation. If the observation was between the moon's near limb and a star, add just the moon's semidiameter and augmentation. If, however, the observation was between the moon's far limb and a star, subtract the moon's semidiameter plus augmentation.
Clearing the Lunar
T he information needed to continue with the calculations depended upon what method one used to clear the lunar (that is, to find the true distance). All methods, however, required the apparent lunar distance (result of Step 14), the sun or star's apparent altitude (result of Step 8 or Step 11) and the moon's apparent altitude (result of Step 11). All involved using the logarithms of trigonometric functions to find sine, cosine, tangent, secant, cosine, cotangent, most of which required making interpolations for each angle or distance used. The answer derived from Clearing the Lunar was the true distance between the moon and the other object per that observation.
Finding the Longitude
W e're not done yet! After the true lunar distance has been calculated it is necessary to determine the Greenwich Apparent Time when that distance would have occurred. The Nautical Almanac gave the calculated distance between the moon and either the sun or one or more navigational stars for every three hours of the day. Using the sun-moon observation of 23 February 1805, for example, the cleared Lunar Distance was 66°32'16". The Nautical Almanac shows that the true sun-moon distance was 66°43'36" at III hours (3 pm or 1500 hours) and 65°22'04" at VI hours (6 pm or 1800 hours). By proportion we can calculate that the Greenwich Apparent Time would have been 3hr 25m 01.4s pm (15:25:01.4 hours) when the true sun-moon distance was 66°32'16". Then, as the true Local Apparent Time of the observation was 8:38:07.0 hours, the time difference would be 6 hours 46 minutes 54.4 seconds. Multiplying this by 15 gives a longitude of 101°43' 36" west, which should be rounded to 101°44' west or more reasonably, 101°owing to in the uncertainties introduced by methodology, the numerous calculations and the observation itself. Even as late as 1900, the Nautical Almanac implied that the maximum accuracy obtainable from a lunar distance observation was plus or minus 5 minutes of longitude. Until the mid-1800s, an error of 15 minutes longitude from a lunar distance observation was considered tolerable. It was also for this reason that it was recommended to take as many observations as possible at a given location and use the average.
1. This should be 66°23'45" as sextant can be read only to the nearest 1/8 minute (7Ω seconds).
2. This should be 66°21'22.5" as sextant can be read only to the nearest 1/8 minute (7Ω seconds).
3. This angular distance includes the sextant's index error of 8'45", which must be subtracted. Additionally, the sun's semidiameter and moon's semidiameter plus its augmentation need to be added to it.
4. To obtain this time, Lewis would be using a longitude of 99°24'55.5" west though probably it was 99°25', a rounded average of the two values he obtained from his observation of the eclipse.
5. At 15h15m57.7s Greenwich Apparent time the sun's declination should have been -9°49'15.9". The sun's declination was 9°52'10" south at Greenwich Apparent Noon the 23rd and 9°30'06" south at Greenwich noon the 24th.
6. Hour angle (or HA) = sun's hour angle. The sun's forenoon Hour Angle = [(12h - Apparent AM Time of observation) x 15]. Lewis, however, should have calculated 50°25'30" [(12 - 8h38m18s) x 15 = 3h21m42.0s x 15 = 50°25'30.0"].
7. The fault for the difference between the sun's altitude that Lewis derived from Ellicott's formula and Patterson's lies in Robert Patterson's Astronomy Notebook that Lewis carried with him. In Patterson's Problem 4th ("to compute the altitude of the sun, moon or any known star"), Example 1st, Form IV (A), step E, the instructions are to take the cosec (cosecant) of the sum of the Hour Angle and the Declination. Step E, however, should read cos (cosine). Using the cosine here produces an altitude of 17°24'19" for the sun's center; essentially the same as that obtained by Ellicott's formula. After obtaining these differences for the sun's altitude, Lewis stopped, uncertain which value to use.
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