# Longitude and Declination

Page 5 of 6

Observation of the Lunar Distances from Antares
to Find the Longitude

© 2005 by Robert N. Bergantino

## Longitude

The longitude that Lewis used to make his calculations for latitude and check his chronometer was an estimate. Jefferson's instructions required that Lewis take observations to obtain the data needed to calculate the longitude. Lewis's mainstay observation for longitude was called a Lunar Distance observation, or more commonly, a lunar. This observation required using the sextant to measure the angular distance between the moon and the sun or between the moon and a star. The time shown by the chronometer was noted and recorded for each angular-distance measurement. The calculations for longitude from the observation began by evaluating the distance-time data sets and discarding those that appeared spurious. Next the remaining distances were averaged as were the corresponding chronometer times. The average time then had to be correctedfor chronometer error on true Local Apparent Time.

On the morning of June 29, 1804, the captains took six sets of observations of the angular distance between the moon and the sun with the sextant. Each consisted of eight measurements of the angular separation between the sun and moon's nearest limbs and the time of each angular measurement. The moon, approaching its third quarter, was slowly closing the distance between itself and the sun to the east of it.

In order to evaluate the raw data, Lewis should have made a plot or graph of the times recorded versus the angular separation observed. From this plot the observation sets with the most consistent trends could be identified, using only those in the calculations for longitude. A plot of the data shows that Observation set No. 1 had the most consistent data, and Observation set No. 5 probably the second best.

Of all the celestial observations, the calculations for longitude were among the most complex, lengthy, tedious and subject to error. Lewis did not calculate the results of their observations during the expedition; we are going to make those calculations now. Using the data from Lunar Distance Observation set No.1 and the method for clearing1 the Lunar Distance by Robert Patterson (Astronomy Notebook, 1803), the longitude by this observation is 94°26' W. Using Jean Borda's more rigorous method (1787) the longitude is 96°24'W.2

## Comparisons and Conclusions

The Expedition's river survey together with the oldest detailed maps available and modern maps and aerial photos suggest that the Expedition's camp at the mouth of Kansas River was at 94°36½' West Longitude. The longitude derived from Observation set No. 1, averaging the results of Patterson's and Borda's methods is 94°25' West. The captain's longitude, thus, is 11½ arc minutes too far east of the longitude suggested above (94°36½' West). At 53.8 miles per degree of longitude at 39°N, these 11½' represent a distance of 10.3 miles. The Great Circle distance between Camp Dubois and the mouth of Kansas River is 242 miles. The 10-mile difference between the actual longitude and the longitude obtained from the captain's celestial observation makes an error of 4.3%. By the captains' river survey (estimated miles), the straight-line distance between Camp Dubois and the mouth of Kansas River is 271 miles—29 miles more than the actual 242 and an error of 12.1%. The longitude calculated from the captains' Lunar Distance observations, though imprecise by modern standards, is much more nearly correct than the distance obtained from their estimated river miles. Jefferson clearly understood that this would be the case and that the longitudes derived from the captains' celestial observations should be used to correct their compass-and-estimate river survey.

The 1804 Missouri-Kansas junction point as I have identified it
was at 39°07' N, 94°36½' W
The junction point per the Expedition's river survey corrected to
True North is at 39°25'35"N, 95°07'54" W
The Lewis and Clark map of 1814 shows the junction point to
be at 37°11'45"" N, 94°15' W (See map detail below.)
The Expedition's celestial observations put the junction point
at 39°05'33" N, 94°25" W3

## Inconsistencies

Detail From Clark's 1814 Map

To labels, point to the map.

Library of Congress, Geography and Map Division, Washington, D.C.

Cartographer Nicholas King prepared Clark's final map for publication in Nicholas Biddle's paraphrase of the captains' journals, which was published in 1814. For some unknown reason King placed the mouth of the Kansas River 1¼° west of its location on Clark's 1805 map. It appeared at 93°15' on the 1806 map, and about 94°20' on his 1810 revision.

Detail From Clark's 1805 Map

To labels, point to the map.

Library of Congress, Geography and Map Division, Washington, D.C.

## Magnetic Declination (Variation of the Compass Needle)

The needle of a magnetic compass points to True North only along a few ever-changing lines on the earth. A survey in the continental United States, based on Magnetic North, currently can depart from true north by as much as 15° West to as much as 20° East. This range of values was about the same for 1805. To correct a survey based on Magnetic North, the magnetic declination (or variation of the needle) must be known. The direction of True North can be found by several methods. In far northern latitudes during the winter the bearing of the sun can be taken at Local Apparent Noon (David Thompson usually did this). Another method is to take the bearing of the sun, its altitude and the chronometer time of the observation. Still another method is to observe the bearing of Polaris and the chronometer time of the observation. On June 27, 1804, the captains recorded the chronometer time for three observations of the bearing and altitude of the sun. These observations were intended to provide the data to calculate the magnetic declination at their camp at the mouth of Kansas River.

That same evening the captains also recorded the chronometer time for three observations of the bearing of Polaris, this also for magnetic declination. Lewis could calculate the true bearing of the sun from 1) the corrected time of the observation, 2) the latitude of the place of observation, 3) the declination of the sun for the estimated longitude and 4) the altitude of the sun's center. The calculated values then could be compared to the magnetic bearings that were observed at those times. The differences then would be averaged to reduce random errors, and this average would be the magnetic declination for 1804 at the mouth of Kansas River by their sun observations. Lewis could calculate the true bearing of Polaris from a) the corrected average time of the Polaris observation, b) the declination of Polaris, c) the right ascension4 of Polaris, and d) the latitude of the place of observation. The difference between the calculated true bearing and the magnetic bearing that was observed is the magnetic declination by that observation.

1. At the time of Lewis and Clark's expedition several methods were available to "clear" the lunar. Clearing the lunar was the mathematical process by which the true distance between the moon and the sun or star was determined. For Lewis and Clark's method of taking a lunar, this process involved calculating the altitude of the moon and sun (or star) and also calculating what their apparent altitudes would have been from the refraction and parallax associated with their altitude.

Once these altitudes had been calculated the observer or mathematician followed a series of equations by which the moon and the second body were placed in their true position with respect to the center of the earth. Once this true distance was obtained (that is, once the lunar was cleared) the observer or mathematician would calculate what Greenwich Apparent Time would have been when this distance (or separation) occurred. Then, knowing the Greenwich Time of the separation and the local time of the observation, the longitude could be calculated.

2. Jean-Charles de Borda (1733-1799) was a French naval officer, mathematician, astronomer, and military engineer renowned in his time for his development of navigation instruments, and his studies of the size and shape of the Earth. He briefly participated in the American Revolutionary War. He published his rigorous method for clearing the lunar distance in 1787; competent authorities considered it to be the best solution to the problem.

3. Lunar Distance Observation set No. 5 of June 29, 1804, produces a longitude of 94°17' W. The average longitude from Lunar Distance Observation set No. 1 and No. 5 is 94°20½', but the data from Observation set No. 1 appear more coherent than those from No. 5, so weight ought to be given to the longitude derived from it. Observation sets 2, 3, 4 and 6 all contain too many spurious data pairs to warrant calculations.

4. The Right Ascension of a celestial body is the angle expressed in hours-minutes-seconds between that body and a point in the sky where the sun will be or was at the moment it crossed the celestial equator during the spring equinox. This point is called both the first point of Aries and the vernal equinox; it has a Right Ascension of 0h 00m 00.0s. the Right Ascension of the sun, moon, planets and brighter stars are calculated years in advance and published in a book called a Nautical Almanac, a copy of which was in the expedition's portable library.

The difference in Right Ascension between two bodies is the difference in time that they cross the upper meridian ( line due south from a person in the northern hemisphere). This difference, multiplied by 15, gives the angular separation of the two bodies referenced to the celestial equator. The celestial equator is essentially the plane of the earth's equator projected to the celestial sphere, which, in turn, is the imaginary globe upon whose surface all celestial bodies are imagined to be affixed.

5. "N81°E": This bearing was either read or recorded incorrectly in the journals as S81°E; it should have been N81°E, which is used in the average here.

"Polaris": It was more difficult to take the bearing of Polaris with the circumferentor than it was to take the bearing of the sun. The value N7°55'W is the average of three observations, 8°W, 8°W and 7°45'W. If the circumferentor could be read only to the nearest½°, how did the captains obtain a bearing of N7°45'W for the third value of their observation with Polaris?

Funded in part by a grant from the National Park Service, Challenge-Cost Share Program