### Lewis And Clark Expedition

Lewis received three weeks of studies in celestial observations under Andrew Ellicott eminent astronomer-surveyor. He also received tutoring in botany, fossils and more lessons from Robert Patterson in the determination of latitude and longitude. Lewis and Clark carried the necessary instruments for measuring the altitude of celestial bodies (sextant and quadrant), a chronometer for determining longitude and compasses to determine course. They also carried the best available maps of the region, an Astronomical Ephemeris and Nautical Almanac, Practical Introduction to Spherics and Nautical Astronomy and tables for finding latitude and longitude. It is to be noted that Lewis and Clark used an artificial horizon in his celestial observations as the clear horizon was not always available and the terrain elevation above sea level was not generally known. At sea the sextant measured angle to a celestial body which was then corrected for height above sea level (dip angle). Clark maintained a daily record of courses and distances traveled and frequently mapped the regions encountered by taking bearings and estimating distances to references.

The team traveled on a 55 foot masted keel boat and canoes when on the waterways. When they reached their Pacific Coast destination in Oregon territory which they named Fort Clatsop (for the neighboring Indian tribe), Clark estimated that they had traveled 4,162 miles from the mouth of the Missouri to the Pacific. This estimate has been cited to be in error by ~ 1% (40 miles) of the actual distance traveled. It is claimed that Lewis' celestial observations at Fort Clatsop, when reduced to latitude and longitude, would locate the site within 4 miles of its actual position. How were these navigational accomplishments achieved?

a. Clark used dead reckoning and Lewis used meridian transits of the Sun for latitude and lunar distances to establish Greenwich time and longitude.

b. Lewis used eclipse tables to establish longitude and Polaris to establish latitude; Clark used inductive reckoning.

c. Lewis used Viking tables for longitude and meridian transits to establish latitude; Clark used dead reckoning.

d. All of the above.

The answer is a.

Clark, versed in surveying and map making, maintained a daily log of courses and distances traveled and transferred the information to his map. Courses were determined from his compass. He could determine the magnetic variation by comparing compass magnetic north to true north. True north could be obtained by taking a bearing of Polaris (which traced a circle approximately 1° in radius around the celestial pole). Knowing magnetic variation he could plot his dead reckoning position relative to true north. He could determine the speed of the boat by timing a log chip dropped in the river along the side of the boat. If the log chip traversed the boat's length in 71/3 seconds for example, he would know that he was traveling about 5 miles per hour. However, it is doubtful that Clark could achieve a dead reckoning error of 1% without compensatory errors. He relied upon a compass with an inherent error of at least a degree; his estimate of speed and distances was about 5% to 10% if not more.

Both Lewis and Clark obtained the data for determining latitude and longitude by making equal altitude measurements (before and after noon) of the Sun using the sextant or quadrant and chronometer and were capable of reducing the data. The actual reduction of the data (which was recorded on tabular forms) to establish longitude by lunar distances en route was accomplished at West Point by mathematicians after the expedition was completed. Lewis and Clark were instructed to measure the altitude of the Sun at least two hours before noon, set the instrument down, and wait until the altitude would return to the same altitude verified by observation. The two times were then averaged to establish the time the Sun was on their meridian. This was the local apparent noon. Subtracting the time of noon at Greenwich (obtained from the Nautical Almanac) from the time recorded for the local noon (in Greenwich time) would yield the difference in time of the two locations. Multiplying the time difference by 15°/hr would yield longitude. If the altitude of the Sun were taken and plotted periodically between the initial and final observations, one could determine latitude which would be calculated at the midpoint between the initial and final observations of the Sun when the Sun reached its highest ascension and was on the observer's meridian. This whole procedure was known as determining local apparent noon (Figure 1 and 2). One of the watches could be reset to local time on the basis of this procedure (allowing the chronometer to maintain Greenwich time). The chronometer was regulated prior to the expedition which meant that its error rate was known and could be acknowledged in the computations.

Longitude was to be obtained by performing measurements of lunar distances by Lewis and Clark. Lunar distances was a technique for determining Greenwich time and longitude by measuring the horizontal angle of the Moon to the Sun or one of the selected stars and measuring their altitudes using the sextant. A tedious calculation using a spherical triangle was employed to clear the distance of refraction and parallax effects for each measured altitude and other errors. This information was compared to tabular data in a table to obtain Greenwich time and longitude. This technique was conceived in the 15th century and underwent perfection over the centuries. It was a very difficult procedure for most navigators.

Figure 1. Establishing local apparent noon

Steps in obtaining longitude:

Conversion FactorsArc time 15°= 1 hour 1° = 4 minutes 15' = 1 minute 1' = 4 seconds |
Arc Equivalents° degree of arc ' minute of arc " second of arc 1°= 60' 1' = 60" |

17hr31min Greenwich time initial observation

39hr02min

Establish midpoint of observations for meridian transit :

39hr 02min/2= 19hr 31min GT(Sun is on your meridian)

Noon at Greenwich (0° longitude) from the tables was at 1204 1/2. Subtract the Greenwich noon time from Greenwich time of the local noon yields 7hr 261/2 min (use conversion factors above) convert the difference in time components to degrees and arc minutes and add them:

15°/hr X(7 hrs.) = 105°

0.25°/min. X (26 1/2min.) = 6° 38'

Longitude = 111° 38' W

Figure 2. Time diagram for determining longitude at Beavercreek fork

How latitude is obtained by meridian transit of the Sun is shown in Figure 3.

Steps in obtaining latitude:

Lewis determines that at meridian transit of the Sun, its elevation was 60° 02.4' (after corrections for refraction and semi-diameter)

Latitude = (90°- h) + d

Given declination "d" of Sun is 15° 15.4' N

h = 60° 02.4'

Therefore Lewis finds his latitude as 45° 13'N and longitude as 111° 38' W

In reality Lewis and Clark fully calculated longitude by lunar distances only once early in their journey up the Missouri River. It was left for the mathematicians at West Point to reduce the extensive data recorded on the expedition. So vexing was the task to reduce the data for longitude by lunar distances that F.R. Hassler, a West Point mathematics instructor, never succeeded in completing the calculations casting doubt as to whether Fort Catslop was located by lunar distances. If it were located with the four-mile accuracy claimed, it may have been achieved by meridian transit calculations. A lunar positions table (extracted from the British Almanac issued in 1766 for the year 1767) is shown in Figure 4. A diagram of the lunar distances spherical triangle showing the angles to be measured and the angle to be calculated is shown in Figure 5.

Figure 4. Table of Lunar Positions

Figure 5. Lunar Distances Spherical Triangle

Captain Vancouver (earlier a midshipman under Captain Cook), an experienced navigator, utilized lunar distances in establishing the longitude of Nootka a port on the west coast of Vancouver Island in 1792. He and his sailing master Lt. Whidbey made 13 sets of observations with an average difference from the mean value of 8.7 minutes of longitude (5.7 nmi at his latitude) and a standard deviation of the sets of 10.18 minutes of longitude. Each set was an average of from two to eight sets of lunar distances. The average number of observations in one set was 7.5. Since the scatter of a number of operations is reduced by the square root of the number of observations averaged, the standard deviation should be multiplied by the square root of 7.5 or 2.7. Therefore for a single lunar distance observation, the expected random error should have been about 30 minutes of longitude (19.5 nmi for latitude 49.5°). At sea one could expect not to obtain better than one degree accuracy from a single lunar distances observation.

Bibliography

Duncan, Dayton and Ken Burns. Lewis & Clark An Illustrated History. New York: Alfred A. Knopf 1999.

Emmott, Norman W. "Captain Vancouver and the Lunar Distances." Litton Avionics Newsletter, Vol. One No. Three ( March 1970): 27-39.

Jones, Landon Y. The Essential Lewis and Clark. New York: The ECCO Press An Imprint of HarperCollins Publishers, 2000.

Links

http://www.lib.virginia.edu/exhibits/lewis_clark/ch4.html and http://www.lib.virginia.edu/exhibits/lewis_clark/ch.5html