President Thomas
Jefferson whose intellect, scientific prowess and foresight led him
to realize in 1803 that our continent needed to be explored beyond its
border at the Mississippi River. With the acquisition of the Louisiana
Purchase, this need became more compelling. Jefferson believed that
there was a Northwest Passage, a waterway of adjoining rivers that could
connect the Missouri River to the Pacific. Congress approved $2500 for
this expedition. Jefferson selected his secretary, Captain Meriwether
Lewis, to organize the "Corps of Discovery" to establish this route
and bring back specimens from nature, maps and navigation data. Lewis
then chose his close friend Lieutenant William Clark to be his coleader
of the "Corps of Discovery. "
Lewis received
three weeks of studies in celestial observations under Andrew Ellicott
eminent astronomersurveyor. 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
Assume Lewis
and Clark conducted this observation of the Sun and established their
position:
Steps in
obtaining longitude:
Conversion
Factors
Arc 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"

Add initial
to final time of observation:
21hr31min Greenwich
time final observation
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.
Figure
3. Latitude by meridian transit of the Sun.
August 11,1805
just south of Dillon, Montana at the fork of the Beaverhead River.
Steps in
obtaining latitude:
Lewis determines
that at meridian transit of the Sun, its elevation was 60° 02.4' (after
corrections for refraction and semidiameter)
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 fourmile
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
The meticulous
preparation for the expedition, the use of the finest available maps
of the region, the creation of maps and charts en route and the recorded
data made it possible for Lewis and Clark to accomplish their goal and
preserve the Northwest region beyond the Louisiana Purchase for later
claim by the United States.
Afterword
on Lunar Distances
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.