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[ 168 ]
PROCEEDINGS OF THE AMERICAN PHILOSOPHICAL SOCIETY
VOL. 144, NO. 2, JUNE 2000
The Accuracy of the Astronomical
Observations of Lewis and Clark*
RICHARD S. PRESTON
Physics Department, Northern Illinois University
How Did Lewis and Clark Know Where They Were?
*Five of the six principal figures connected with these measurements were members of the
American Philosophical Society: Andrew Ellicott, Ferdinand Hassler, Thomas Jefferson,
Meriwether Lewis, and Robert M. Patterson.
T
HE FAMOUS expedition of Meriwether Lewis and William
Clark up the Missouri River, over the Continental Divide,
down to the Pacific via the Columbia River, and then back,
lasted from May 1804 to September 1806. They kept crude track of
their position using dead reckoning based on hand-held compass read-
ings of their headings and visual estimates of distances covered. Their
journals contain course and distance records such as:
S. 40 W ½ [mile] to some trees in a Stard. [starboard] bend
followed by the entry
S. 20 E 1 [mile] to some willows on the Lard. [larboard] Side
On 2 June 1805, the date of these entries, they made a total of
nineteen course and distance notes.
1
Similar notes exist for every twist
and turn of their route during all those months. Estimates of distance
and compass headings were usually to the nearest quarter mile and the
nearest five degrees, and were sometimes even more precise. Working
with such notes and preliminary sketches, Clark made a series of route
maps covering the whole trip. He, as well as others, later used these
route maps and other information to create large-scale maps that
included many important features of what is now the western United
States (ref. 1, vol. 1).
However, it was well understood in advance that because of the
unavoidable crudeness of the dead reckoning, sizable errors in the posi-
tions of points on the route maps would accrue as the journey pro-

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lewis and clark
169
ceeded. Therefore, in accordance with their instructions from President
Thomas Jefferson, who anticipated this problem, they carried naviga-
tional instruments with them, including a chronometer, a sextant, and
an octant, all of the best quality available. If the true geographical posi-
tions—the latitudes and longitudes—could be determined for a succes-
sion of locations along the route, it would be possible to correct the
route maps for much of the accumulated dead-reckoning error in both
scale and orientation. This would improve the accuracy of the final
large-scale maps.
Among Jefferson’s instructions to Lewis were the following:
Beginning at the mouth of the Missouri, you will take observations of
latitude and longitude, at all remarkable points on the river. . . .
and
Your observations are to be taken with great pains and accuracy;
to be entered distinctly and intelligibly for others as well as yourself;
to comprehend all the elements necessary, with the aid of the usual
tables, to fix the latitude and longitude of the places at which they
were taken; and are to be rendered to the war-office, for the purpose
of having the calculations made concurrently by proper persons
within the United States.
2
The “usual tables” that Lewis and Clark took with them were
three volumes of the Nautical Almanac and Astronomical Ephemeris,
probably for the years 1803, 1804, and 1805,
3
and Maskelyne’s Tables
requisite to be used with the Nautical Ephemeris for finding the lati-
tude and longitude at sea.
4
Lewis did complete one calculation of latitude and longitude from
his own measurements at the mouth of the Missouri River, and
attempted another at Fort Mandan. At Fort Mandan he also availed
himself of a fortuitous lunar eclipse to try to establish the longitude of
the place using measurements of a sort different from the usual ones.
But aside from these efforts, surprisingly, the only recorded use he and
Clark ever made of their many astronomical measurements was to
determine latitudes. And until lately, apparently, no one else has ever
used the Lewis and Clark data to calculate longitudes. Recently Robert
Bergantino, who combines a long-standing interest in the Lewis and
Clark expedition with considerable expertise in the theory and practice
of celestial navigation, successfully calculated a longitude from Lewis
and Clark observations.
5
Traditionally, historians have supposed that this failure was due to
the poor quality of the Lewis and Clark observations, or to the failure
of Lewis and Clark to make all the required measurements at each

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richard s. preston
location, and have agreed with Donald Jackson’s offhand dismissal of
their astronomical observations as “highly suspect” due to “inept-
ness.”
6
Ineptness, however, is far from the explanation. The problem is
that the data were taken in a somewhat unorthodox way to be used in
calculations that some experts considered improper. Until Bergantino,
apparently, no one who understood the unorthodox procedure ever
tried to make the calculations.
How Were Astronomical Observations To Be Used
To Determine Position?
In the early nineteenth century, the principal tools of the navigator
were a sextant (or an octant, which is a similar instrument), a clock or
watch with a second hand, a copy of the Nautical Almanac for the cur-
rent year, and a copy of the Tables requisite.
The sextant was used for two purposes. One was to measure the
altitude of a celestial body, which is its height, in degrees, above the
horizontal (Fig. 1). The other was to measure the angular separation
(the lunar distance) between the moon and some other body such as
the sun, a star, or a planet (Fig. 2).
The clock or watch was used to determine the time at which a sex-
tant reading was taken. Unlike the chronometers used in more modern
navigation, this timepiece was not expected to keep Greenwich time,
but it was expected to gain or lose time at a known, fairly steady rate
so that the time interval between measurements could be determined
accurately.
The Nautical Almanac contained tables that allowed the navigator
to determine the locations in the sky of the sun, moon, planets, and
many stars for any instant of any day of the year for which that edition
was valid.
The Tables requisite is a collection of tables of logarithms and trig-
onometric functions and some other specially devised tables designed
to simplify the navigator’s calculations.
At the time of the Lewis and Clark expedition, as at present, lati-
tude was easily determined by measuring the altitude of the sun at its
highest point in the sky—that is, at noon. By combining this value of
the altitude with information tabulated in the Nautical Almanac for
that year and day, the latitude was found by a simple calculation.
Lewis and Clark made such measurements frequently and usually cal-
culated the latitude soon afterward.
If the navigator knows the Greenwich time at the instant of noon,
he can also determine his longitude from a simple calculation. The
immediate difficulty is that it is hard to identify the instant when the

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171
sun is at its maximum altitude because the vertical component of the
sun’s apparent motion is smallest near noon, and the sun stays near its
highest altitude for quite a while. This is not a problem for determining
latitude, but it produces a considerable uncertainty in the calculated
longitude.
A surveyor or explorer on land can use an indirect method, known
as the equal-altitudes method, to determine what his clock or watch
would have read at noon if he had been able to identify that instant.
Lewis and Clark often made equal-altitudes measurements, and they
Figure 1. The horizontal plane containing the sextant is depicted here. The
altitude of a celestial body is defined as the angle between this plane and
the direction of the celestial body. An observer at sea actually measures the angle
between the horizon and the direction of the celestial body. To obtain
this
measured angle has to be corrected for the fact that the horizon is slightly below
the horizontal, depending on how high the sextant is above the water. Lewis and
Clark, however, were on land, so they used an artificial horizon, which obviated
this problem.
Figure 2. The angles and are the altitudes of a star and the moon. The lunar
distance measured by the sextant is the angle between the moon and the star.

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richard s. preston
did so in the following way. About four hours before noon, one of
them measured the sun’s altitude and recorded it along with the watch
time. About eight hours later, as the sun was going down toward the
horizon, the observer noted the watch reading at the instant the sun
came back down to exactly the same altitude as at the morning read-
ing. The time halfway between the watch times of the morning and
afternoon equal-altitude observations was then taken to be the watch
time of local noon (after a small correction required because the sun
itself would have drifted slightly to the north or the south—depending
on the time of year—between the two readings).
A further difficulty in determining the Greenwich time of noon was
that chronometers that would keep Greenwich time reliably over
months and years were scarce and expensive. This would not have
been a serious problem if the navigator had had a fairly good watch
and could check it against Greenwich time every so often, as one can
do now with a radio time signal. But even though a time check actually
was available, as will be described, it was important to know how fast
or slow the watch was running (its “rate”) so that the Greenwich time
at local noon could be calculated correctly from the difference in watch
readings between noon and the time check. The rate at which the
watch was gaining or losing time was found by making equal-altitudes
measurements on successive days and noting the elapsed watch time
between two successive noons. (The time from noon to noon varies
throughout the year, but the Nautical Almanac contains the informa-
tion necessary to allow for this.) Typically the rate was some number
of seconds per day, either gained or lost. Once the rate was known, the
Greenwich time of local noon could be determined from the Green-
wich time of the time check and the difference between the watch read-
ing at the instant of the time check and the watch reading at noon.
The only remaining question is how to obtain a time check.
The Time Check
Before there were chronometers that could keep Greenwich time accu-
rately enough over the months or years that a mariner or explorer
might be away from civilization, the time check required for finding
longitude had to be obtained from other astronomical measurements.
One of the best astronomical clocks available on a nearly daily basis
for at least part of each year consists of the principal satellites of Jupi-
ter. The disappearances and reappearances of Jupiter’s satellites, as
their orbital motions carry them around that planet and into and out
of view, occur rather often and are predictable. A good astronomical
telescope and the appropriate astronomical tables provided frequent

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173
opportunities to determine Greenwich time fairly accurately by the sat-
ellites of Jupiter. Unfortunately for mariners, the chaotic motions of
ships on the high seas prevented the use of astronomical telescopes,
which have to be held very steady for these observations. Inland
explorers on foot or in small boats could use dry land as a steady plat-
form, but few of them carried good observing telescopes, which are
bulky and fragile.
Other occasional but predictable events, such as eclipses of the
moon or sun, can also be used to determine Greenwich time.
The Method of Lunar Distances (the “Method of Lunars”)
A more convenient astronomical clock, although difficult to read with
sufficient accuracy, is the moon. As viewed from the earth, the moon
does not move through the sky at the same rate as the fixed stars. It
falls behind by about 13 per day, and, moreover, the path it follows
across the sky drifts northward or southward by noticeable amounts
from day to day. The seemingly irregular motion of the moon against
the background of the fixed stars has, in fact, become very predictable,
thanks to Isaac Newton and his successors. This means that at any
given instant the angular separation of the moon from any star or
planet or from the sun has a definite, predictable value. By measuring
this separation (the lunar distance) and using appropriate astronomical
tables, the Greenwich time of the lunar-distance measurement may be
determined. Because of this, for many years each of the annual editions
of the Nautical Almanac had tables showing lunar distances from the
moon to a selection of stars (including the sun) for every third hour of
each day of the year. With this information the navigator could deter-
mine by interpolation the Greenwich time that corresponded to his
measured lunar distance. This was his time check.
As outlined above, a typical day of observations for latitude and
longitude would consist of the following steps:
About four hours before noon, measure and record the altitude of
the sun and record the watch time.
At noon, measure and record the altitude of the sun.
Calculate the latitude from the noon altitude.
About four hours after noon, observe the altitude of the sun and
record the watch time when the sun reaches exactly the same alti-
tude as recorded at the morning measurement.
Calculate the watch time of noon as the time halfway between the
watch times of the morning and afternoon equal-altitudes measure-
ments, corrected as necessary for the rate of the watch, and for the

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richard s. preston
slight drift of the sun north or south (depending on the season)
between the two measurements.
Whenever it becomes possible (before, between, or after the equal-
altitudes measurements), measure the lunar distance to a chosen
celestial body. Record the lunar distance along with the watch time
of the measurement.
Determine the Greenwich time of the lunar-distance measurement
by comparing the lunar distance with data derived from the Nauti-
cal Almanac.
Use the Greenwich time of the lunar-distance measurement with
the difference between the watch time of the lunar-distance mea-
surement and the watch time of noon (corrected for the rate of the
watch) to determine the Greenwich time of noon.
The Greenwich time of noon leads directly to the longitude, as stated
earlier.
We now come to an important point that will explain why the Lewis
and Clark observations were considered useless by some. The observed
lunar distance cannot be used directly with the Nautical Almanac to
determine the Greenwich time of the reading. The reason for this is
refraction and parallax.
Parallax is a perspective effect. The moon is much closer to the
earth than the fixed stars so at any instant its apparent position among
the fixed stars depends slightly but significantly on where the observer
is on earth. All the data in the tables of the Nautical Almanac are pre-
sented as if measured by an observer at the center of a transparent
earth. For a fictitious observer at that location, by convention, the par-
allax is taken to be zero. Real observers, however, are necessarily at the
surface of the earth, so all measurements involving the position of the
moon have to be adjusted for parallax before being compared with
numbers in the Nautical Almanac. Position measurements of the sun
and the planets, and even the fixed stars, are also subject to parallax,
but those bodies are so far away that neglect of this fact is a relatively
minor source of error and can be ignored in determinations of longi-
tude by the lunar-distance method.
The refraction effect is the downward bending of light rays coming
to the earth from outer space as they pass through the atmosphere.
Refraction makes any celestial object appear to be higher in the sky
than it would if the earth had no atmosphere. The tables in the Nauti-
cal Almanac are calculated as if there were no refraction. Therefore the
sextant altitude
7
of any celestial body has to be corrected for refraction
and possibly for parallax before the tables can be used. Only the sex-
tant altitude needs such corrections, and these corrections depend only

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175
on the sextant altitude itself. The sextant altitude of the moon will
always be significantly too low, and the sextant altitude of anything
else will always be slightly too high.
These vertical shifts in observed altitudes, being of different
amounts and in different directions, change the apparent angular sepa-
ration of the moon and the other body. Obviously, then, the sextant
lunar distance
8
has to be corrected for both refraction and parallax
(Fig. 3).
In the early nineteenth century there were widely known proce-
dures for calculating the corrected lunar distance,
4,9–12
but for any such
calculation the sextant altitudes of the moon and the other body at
the instant of the lunar-distance measurement are needed, as well as
the sextant lunar distance itself. This is an extra complication because,
in principle, the lunar distance and the two altitudes must be measured
simultaneously, or as nearly so as possible. Thus, either two more ob-
servers and two more sextants must be present, or the single observer
must measure both altitudes in succession just before and again just
after the lunar-distance measurement and then take averages of the
before and after readings. Then, finally, the observed lunar distance
can be corrected and the lunar-distance tables in the Nautical Almanac
used to determine Greenwich time.
Figure 3. Using the values of , (from Fig. 1), along with small corrections to
and for refraction and parallax, the corrected lunar distance is calculated by
spherical trigonometry. The corrected lunar distance is then compared with values
in the Nautical Almanac to determine Greenwich time at the instant of the
measurement. Because Lewis and Clark never measured
and
when they
measured , Ferdinand Hassler was unable to calculate any
values and could
not go on to determine the Greenwich times needed for computing any of their
longitudes (see text).

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richard s. preston
Thus, for observers on land in the early nineteenth century, the
most common and most trusted version of the method of lunars for
determining longitude required
Morning and afternoon equal altitudes of the sun
Noon altitude of the sun
A lunar distance and two altitudes, all measured simultaneously, if
possible
Why Weren’t the Lewis and Clark Longitudes Calculated?
At first glance it is difficult to understand the neglect of the Lewis and
Clark astronomical observations. They always made their own deter-
minations of latitude from their noon altitude measurements, calculat-
ing and recording their results on the same day. This was a relatively
simple calculation. It is the measurements for longitude that have, until
recently, languished completely unused. Certainly Lewis and Clark had
more pressing things to do than to spend time on long, finicky calcula-
tions of longitude while on their journey. In fact, in view of Jefferson’s
instruction to turn over their data to the war office for calculation by
“proper persons,” it seems clear that Jefferson did not expect them to
bother with longitude calculations at all while en route. In this he was
following the advice of one of his navigation experts.
13
Inspection of the astronomical data recorded in the Lewis and Clark
journals
14
reveals that in no case did they make (or at least record) the
required altitude measurements. In his journal, Lewis described the
various artificial horizons he used for measuring the altitudes of the sun,
moon, and stars,
15
but although the two men made many sets of mea-
surements of lunar distances from such bodies as the sun, Spica, etc.,
not once did they record the altitude of the moon or of the other body
at the time of the lunar-distance measurement. It was the total absence
of these altitude measurements in the Lewis and Clark data that pre-
vented the calculation of their longitudes by the person who was
appointed to carry out this task.
Why did Lewis and Clark fail to take and record the altitudes some
considered essential for the method of lunars? Before setting out for
the west, Lewis took a cram course on celestial navigation with the
mathematician and astronomer Robert Patterson and with the astrono-
mer and surveyor Andrew Ellicott, both of whom were prominent
members of the American Philosophical Society. Patterson had created
an astronomical notebook, which gives detailed instructions for the
standard method of lunars much as I have described it. However,
according to his instructions,
16
“As it may frequently happen that the

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177
alts. [altitudes] of one or both of the bodies cannot well be taken you
must then compute the app. [apparent] alts by prob. 4th, and this
method is generally to be preferd [sic] on land [emphasis added].”
Patterson’s Problem 4th, which is also a part of his notebook,
describes a method of calculating the altitude of a body at any instant
using the known latitude of the observer; the estimated longitude; a
Greenwich time for that instant calculated from the estimated longi-
tude and the difference between the watch time of the instant and the
watch time of noon (with a correction for watch rate).
Although this is a reasonable procedure and gives excellent results,
it seems to have been controversial at the time. Indeed, at first sight
there appears to be an element of circularity in a procedure that starts
with an estimated longitude and a corresponding estimate of the
Greenwich time, and derives from them first the actual Greenwich time
and then the actual longitude using no observational data except the
latitude, an observed lunar distance, and two watch readings whose
relation to Greenwich time is unknown.
Some works on navigation from that period show how to calculate
the required altitudes, and at least one does not. Maskelyne’s Tables
requisite,
4
Problem VII, describes the calculation of an altitude using
an estimated longitude without explicitly recommending the use of cal-
culated altitudes for correcting lunar distances. Somewhat further on
in the text, in Problem XI, the correction of lunar distances is described
but, again, without mention of the possibility of using calculated alti-
tudes for this purpose. John Robertson’s Elements of Navigation
11
gives a procedure for correcting lunar distances using calculated alti-
tudes, as does John Moore’s New Practical Navigator
12
as well as
Nathaniel Bowditch’s New American Practical Navigator,
9
which used
Moore’s work as a model.
Bowditch advises against calculating the altitudes, suggesting that
the results are not accurate enough, but none of the other authors dis-
cusses the validity of the results. As late as the fifth edition (1822), P.
Kelly’s Practical Introduction to Spherics and Nautical Astronomy
10
does not even mention calculation of the required altitudes. On the
other hand, Edward Riddle’s 1824 Treatise on Navigation and Nauti-
cal Astronomy
17
does recommend calculation of the required altitudes,
but only for night observations. For his comprehensive History of
Nautical Astronomy (1968),
18
Charles Cotter must have read widely in
the literature on lunar distances. That he barely mentions the possibil-
ity of calculating the altitudes suggests that he believed the method was
used rarely, if at all, and that he was himself skeptical of the validity of
the method, as many navigators would be on first acquaintance with it.
Clearly, Robert Patterson understood and believed in the method.

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Likewise, Andrew Ellicott, who was Lewis’s other navigation instruc-
tor, had demonstrated his use of it in a self-congratulatory entry for 22
September 1799, in the appendix of his Journal of Andrew Ellicott.
19
Likewise, Philip Turnor
20
and David Thompson,
21
who were exploring
western Canada in the years before, during, and after the Lewis and
Clark expedition, used this very method of supplying the missing alti-
tudes, and their results have turned out to be very reliable.
Lewis seems to have learned and understood the method very well.
In a notebook entry (Codex O) for 22 July 1804, he mentioned a series
of lunar-distance measurements he made while at the mouth of the
Missouri River in December 1803. He calculated both the latitude and
longitude, but, uncharacteristically, he entered the results of the calcu-
lation but not the measurements. As he did not have the three sextants
or octants needed for the required three simultaneous measurements, it
is likely that he used the Problem 4th method recommended by Patter-
son, or a similar method recommended by Ellicott. In a later field
note
22
Lewis initiated a lunar-distance calculation based on his sun-
moon observations of 23 February 1805, while at Fort Mandan. His
calculation breaks off at the point where he has calculated the altitude
of the sun’s center “by Mr. Elicot’s formula” and “by Mr. Patterson’s,”
but before he has calculated the moon’s altitude. (It is interesting that
the two calculated altitudes differ by more than a degree, but it is prob-
able that if both required altitudes were calculated using the same for-
mula, the final results for the longitude would have been practically the
same for both formulas.)
In any case, it appears that all the rest of the Lewis and Clark mea-
surements were made with the understanding that calculated altitudes
would be used to calculate longitudes from their data. In the next sec-
tion I describe Patterson’s Problem 4th method. Unfortunately, while
calculation of the required altitudes simplified the measurements, it
made the calculations themselves harder. Lewis and Clark were wise to
adopt this method of measurement and wise not to try to do the calcu-
lations themselves while en route.
What Was the Problem 4th Method?
As previously mentioned, the Problem 4th calculation required (1) pre-
vious knowledge of the latitude as determined in the usual way by a
noon altitude measurement, (2) the watch time of noon as determined
from the equal-altitudes measurements, and (3) the watch time of the
lunar-distance measurement. The procedure was then as follows:
Adopt a reasonable estimate for the longitude.

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lewis and clark
179
Using this estimated longitude, calculate the corresponding esti-
mate of the Greenwich time of the lunar-distance measurement
using this longitude and the difference between the watch time of
noon and the watch time of the lunar-distance measurement (cor-
rected for watch rate).
Use the estimated longitude, the corresponding estimate of the
Greenwich time of the lunar-distance measurement, the measured
latitude, and the Nautical Almanac tables to calculate the altitudes
of the moon and the other body at the instant of the lunar-distance
measurement. (These altitudes are only approximate, of course,
because they are based on an estimate of the longitude, but they
happen to be extremely good approximations.)
23
Also, these alti-
tudes are estimates of the altitudes after correction for refraction
and parallax.
“De-correct” these altitudes for refraction and parallax to obtain
the sextant altitudes—the altitudes before corrections for refraction
and parallax.
Use these approximate altitudes in the usual way, as if they were
exact, to calculate the corrected lunar distance. (Although based on
an estimated longitude, this corrected value will be surprisingly
close to the properly corrected value if the estimate used for the
longitude is fairly reasonable.)
Use this corrected lunar distance and the Nautical Almanac to
obtain a final and much better estimate of Greenwich time at the
instant of the lunar-distance measurement.
Use this improved value of the Greenwich time of the lunar-
distance measurement with the difference between the watch time
of noon and the watch time of the lunar-distance measurement
(corrected for the rate of the watch) to work back to the Greenwich
time at noon.
Use the Greenwich time at noon to calculate the actual longitude
by the usual method. (This is the final value of the longitude.)
This looks very much like a single iteration of a procedure recently
used by Bergantino
5
to calculate a Lewis and Clark longitude. Bergan-
tino, however, suggests using these results as initial guesses for a sec-
ond round of calculations to obtain even better estimates of the
Greenwich time and the longitude. Such successive approximations can
be repeated until there is no discernible difference between the input
and output values of the longitude and time.
I have written a computer program that carries out Bergantino’s
iterative procedure, and I find that the results converge rapidly. Even
when the initial estimate of the longitude is off by several degrees, the

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richard s. preston
final value is reached after three iterations, while the longitude calcu-
lated in the first iteration is usually within a few minutes of the final
value. Therefore, in practice, the longitude obtained from the first iter-
ation should be acceptable as “good enough,” considering that thirty-
minute errors in the calculated longitude are not unusual for the lunar-
distance method.
Like Bergantino, I find that the longitudes I calculate from Lewis
and Clark data agree quite well with the modern values for the longi-
tudes of their respective locations. Thus, Bowditch’s reservations about
this method do not seem well founded, and the use of the method by
Robert Patterson, Andrew Ellicott, Philip Turnor, David Thompson,
Lewis and Clark, and probably many others, was quite appropriate.
Not content with this departure from the standard method of
lunars, Andrew Ellicott developed an advanced version of it. In a letter
to Jefferson dated 6 March 1803,
13
he stated that he could dispense
with the noon altitude measurement to determine the latitude and that
he could get both the latitude and the longitude from just the morning
and afternoon equal-altitudes measurements and the lunar-distance
measurement.
A navigator who understood the modern “lines of position”
method of determining both latitude and longitude from two altitude
measurements would see immediately how this could be done. But
since the position-line method was not discovered until 1837,
24
it is
unlikely that Ellicott had any such thing in mind.
After I read Charles Cotter’s History of Nautical Astronomy, it
occurred to me (ref. 18, p. 271) that Ellicott was resorting to the
method of double altitudes, which is a means of determining the lati-
tude directly using altitude measurements on the sun made at two dif-
ferent times (not necessarily equal-altitudes measurements) and the
elapsed time between them. No noon altitude measurement is neces-
sary. When two such readings have been made (the two equal-altitudes
measurements can be used for this second purpose as well as for finding
the watch reading at noon), the results can be used with information
from the Nautical Almanac to determine the latitude. This, however,
requires tedious trigonometric calculations, or a repetitious successive-
approximations procedure, or a complicated graphical procedure
involving stereographic projections. The difficult double-altitudes cal-
culation for the latitude was used only infrequently and reluctantly at
that period. This is why, following standard practice, Lewis and Clark
usually made high-noon measurements of the sun’s altitude, as well as
equal-altitudes measurements. The latitude could then be determined
without having to make the formidable double-altitudes calculation. But
if, for any reason, the noon altitude of the sun could not be measured,

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181
the latitude could still be found using the double-altitude method as a
last resort.
I have used computer programs to calculate latitudes as well as lon-
gitudes from twenty-three different sets of Lewis and Clark observa-
tions using the Problem 4th method both by itself and with Ellicott’s
1803 extension of it. As can be seen from the accompanying table,
both methods work well. All but three of the longitudes obtained from
moon-to-sun lunar distances agree with the modern values to within a
half a degree, which is about what is expected from good lunar-distance
determinations.
Before the advent of computers, any of my calculations would have
had to be done by hand. These calculations are ideal for solution by a
desktop computer, which can do repetitive calculations very quickly, so
that one complete calculation takes only a few seconds. I have not tried
to do any of these computations by hand, but I estimate that, with
practice, a single iteration of either of the two methods would take an
experienced navigator at least an hour using only such old-fashioned
tools as tables of logarithms and trigonometric functions.
In his letter to Jefferson, Ellicott had recommended that only
moon-sun distances were to be used. Lewis and Clark often ignored
this recommendation. They were forced to do so because useful moon-
sun measurements are not possible on every day of the month; at other
times moon-star measurements are the only alternative. Also, they occa-
sionally made lunar-distance measurements without making any noon
or equal-altitudes measurements at all. Such measurements could not
be used to determine longitude. They did, however, provide a way to
determine the rate of the watch that did not require the observer to remain
at a fixed location, although this was not as accurate as the equal-altitudes
method if the observer could remain at the same longitude. In some
cases they probably intended to make the required equal-altitudes mea-
surements the next day, but were prevented by poor weather condi-
tions or by the need to tend to more urgent matters.
Jefferson had specified that Lewis and Clark were to hand over
their astronomical measurements to the war office for calculation of
the longitudes. In May 1807, after his return to the east coast, Lewis
paid Ferdinand Hassler a hundred dollars in advance to make the cal-
culations. Hassler, an astronomer and surveyor, was then the mathe-
matics professor at West Point, and presumably represented the war
office. Hassler was the principal organizer and first superintendent of
the United States Coastal Survey, so we can assume he was well
acquainted with the method of lunars. But he may not have known
that he could calculate the missing altitudes or, if he did, he may have
mistrusted results obtained by this method. By 1810 he was a professor

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richard s. preston
at Union College, and was having a terrible time dealing with the
Lewis and Clark data. In a letter to Robert Patterson
25
he seems to be
describing an effort to cope with the missing altitudes by using approx-
imations, when he writes of “calculating backwards from the circum-
stances & times given [to obtain] what ought to have been observed.”
He complains that he has only one of the journals, and that even this is
apparently a copy of the original and may well contain errors. After
comparing Lewis’s position for the mouth of the Missouri with previ-
ously published maps and other information, as well as Lewis’s own
data taken farther up the Missouri, Hassler concludes that Lewis’s lon-
gitude for the mouth of the Missouri is off by about two degrees. Actu-
ally, Lewis’s value agrees better with the modern value than Hassler’s
“corrected” version of Lewis’s value. (The custodian of most of the
original journals, Nicholas Biddle, realized that Hassler had been given
an inadequate map of the Mississippi to work with and, according to a
better map in his possession, Lewis had the position about right. This
opinion was relayed to Hassler.)
26
By 1817 Hassler had abandoned the calculations “in despair.”
27
The reason for Hassler’s despair is not clear. As previously noted,
Patterson’s Problem 4th described a procedure that was widely known
and was, in fact, a valid procedure yielding reliable results. Why did no
one convince Hassler that many of the Lewis and Clark data were com-
pletely adequate? Robert Patterson and Andrew Ellicott, who were Lewis’s
instructors in celestial navigation and fellow members with Hassler of the
American Philosophical Society, were in a position to do so.
Throughout his career Hassler made many influential friends and
many influential enemies.
28
Patterson was one of the friends. Not only
did he recommend Hassler for the Coastal Survey position, but even as
late as 1815 he wrote warm letters of recommendation for him to the
former president, Thomas Jefferson, and to President James Madi-
son.
29
Patterson knew from Hassler’s 1810 letter about his trouble
with the calculations. Perhaps he gave Hassler advice that Hassler was
unable or unwilling to use, or perhaps Patterson was convinced by
Hassler’s letter that the Lewis and Clark data were useless and thought
it best to let the matter drop.
As to whether Ellicott could or would have helped Hassler, one can
only speculate. On several occasions in later years Hassler published
disparaging remarks about Ellicott’s professional ability and accused
Ellicott of trying to damage his reputation,
29,30
so it may be there was
already some animosity between them at this time. Ellicott, as an army
major and Lewis’s mentor, might have wanted to do the longitude cal-
culations for the war office, which is where Jefferson had directed the
calculations be performed. But it was Hassler, as a professor at West

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lewis and clark
183
Point and a civilian, but also an employee of the war office, who got
the job. Thus, Ellicott might not have been inclined to help Hassler. In
fact, Ellicott may even have resented Hassler’s appointment to the West
Point professorship: at about this time Hassler was forced to resign his
West Point position because Congress had supplied no funding for it;
31
when funding eventually was provided, it was Ellicott who was ap-
pointed to Hassler’s old position. But any imputation of malice to Elli-
cott is pure conjecture, to which the rejoinder must be that Ellicott
would have had an interest in seeing a useful outcome of his work with
Lewis, and this would have motivated him to be helpful if he had been
aware of Hassler’s difficulties.
Jefferson’s Disappointment
The idea for the Lewis and Clark expedition originated with President
Thomas Jefferson, who was personally involved in all phases of the
planning for it. Jefferson was interested in everything, and he instructed
the explorers to return information on a wide variety of subjects: bot-
any, zoology, geography, geology, meteorology, Indian customs and
languages, and anything else. Lewis and Clark obliged him by keeping
voluminous and detailed journals that take up a foot of shelf space in a
printed edition. As late as 1817, with Hassler out of the picture, Jeffer-
son, still hopeful, was waiting for a secretary of war to be appointed so
that he could “propose to him to have made, at the public expence, the
requisite calculations, to have the map corrected in its latitudes and
longitudes, engraved and published on a proper scale.”
32
As we have
seen, the corrections never became available and Jefferson remained
forever disappointed. The considerable effort that Lewis and Clark put
into making usable, high-quality astronomical measurements was
totally wasted.
Computer Calculations of Position From the
Lewis and Clark Data
For each set of lunar-distance readings the table shows
the date of the observation and the celestial partner for the lunar-
distance measurement
the modern values of latitude and longitude for the point of
observation
the latitude calculated from the measured altitude of the sun at
high noon and the average of the corresponding longitudes calcu-
lated using Bergantino’s version of the Problem 4th method

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184
richard s. preston
the average values of latitude and longitude calculated by Ellicott’s
1803 method, using only the lunar-distance and equal-altitudes
measurements
the number of lunar-distance readings in the set
the error of the average longitude calculated for the set
the probable error of the average longitude calculated for the set
Calculation of a longitude requires equal-altitudes measurements
and lunar-distance measurements, all made at the same location. Some
of Lewis’s data meet these requirements, but, as it happens, none of
Clark’s do. The table gives results calculated from observations made
from June 1804 to August 1805. Lewis usually made a succession of
measurements of the lunar distance to a particular celestial body,
spaced a few minutes apart. Often he made as many as twenty mea-
surements, and on one occasion forty-eight measurements in one set.
My computer programs for both the 1799 and the 1803 methods
make a complete calculation for each lunar-distance measurement
within one set, and then take an unweighted average. Only this average
for each set is presented in the table. (At the time of Lewis and Clark,
only one calculation would have been made for each set using the aver-
age of the lunar distances and the average of the times for the set.)
Although the results calculated from moon-star distances are usually
wide of the mark, the fourteen moon-sun results are much better, with
an rms deviation from the modern values of 35 minutes of longitude
(about 30 miles at latitude 40 ). This is about standard for the method,
and refutes the accepted opinion that the Lewis and Clark astronomi-
cal observations were of poor quality.
The probable error of the mean longitude is an estimate of the con-
sistency of the longitudes calculated from that set of measurements.
This is not a measure of the validity of the longitude calculated for the
set because it makes no allowance for any systematic errors due to such
factors as improper calibration of the sextant, incorrect watch rate,
and observer bias in measuring with the sextant. Except for one set of
only two measurements (10 July 1805, to Pegasi), the average of the
probable errors of the longitude is only 5 . This indicates that if the
systematic errors in Lewis’s measurements could have been eliminated,
or could be allowed for in the calculations, the calculated longitudes
would be much nearer to the correct values.
As mentioned earlier, Lewis made several measurements at Fort
Mandan to determine the longitude of the Mandan villages. Usable
lunar distances were measured on 6 February and 23 February 1805 in
the usual fashion, but earlier a lunar eclipse provided a different possi-
bility for determining Greenwich time and the longitude. Arlen J.

Page 18
lewis and clark
185
Latitude & longitude
Number
of lunar
distance
readings
Probable
error
of the
mean
Observation
Modern
values
a
Lewis &
Clark/
Bergantino
b
Ellicott
c
Error
in
longitude
2 June 1804
38 36 0 N
38 31 N
38 35 N
Moon to sun
91 57 0 W
91 47 W
91 47 W
26
10
4
3 June 1804
38 36 0 N
38 31 N
39 1 N
Moon to sun
91 57 0 W
91 30 W
91 36 W
36
27
5
29 June 1804
39 7 12 N
39 5 N
39 7 N
Moon to sun
94 36 0 W
94 8 W
94 8 W
48
28
2
30 June 1804
d
39 10 48 N
39 7 N
39 7 N
Moon to sun
94 40 12 W 94 18 W
94 18 W
20
22
5
11 July 1804
40 1 48 N
39 56 N
39 24 N
Moon to Spica
95 22 48 W 93 40 W
93 52 W
14
102
5
12 July 1804
40 1 48 N
39 56 N
39 26 N
Moon to sun
95 22 48 W 94 51 W
94 57 W
16
32
4
17 July 1804
40 32 24 N
40 27 N
41 1 N
Moon to Spica
95 39 0 W
93 11 W
93 6 W
8
148
7
22 July 1804
41 10 48 N
41 3 N
41 10 N
Moon to
Antares
95 51 36 W 94 30 W
94 31 W
6
81
5
27 July 1804
41 10 48 N
41 6 N
41 46 N
Moon to
Aquilae (Altair) 95 51 36 W 96 24 W
96 24 W
6
33
3
13 Aug. 1804
42 19 48 N
No noon
42 18 N
Moon to sun
96 21 36 W observation 95 19 W
26
62
4
31 Aug. 1804
42 51 0 N
No noon
42 51 N
Moon to sun
97 27 36 W observation 98 43 W
6
76
11
6 Feb. 1805
47 16 54 N
47 21 N
47 14 N
12
39
3
Moon to sun
101 16 24 W 100 37 W 100 34 W
23 Feb. 1805
47 16 54 N
47 21 N
47 18 N
Moon to sun
101 16 24 W 101 46 W 101 46 W
12
30
1
3 April 1805
e
47 16 54 N
47 21 N
47 31 N
Moon to sun
101 16 24 W 101 0 W
100 59 W
7
16
6
22 April 1805
48 6 22 N
No noon
48 7 N
Moon to sun
103 39 0 W observation 103 36 W
16
3
3
9 June 1805
47 55 45 N
47 29 N
47 37 N
Moon to
Aquilae (Altair) 110 29 4 W 108 31 W 108 32 W
6
118
10
9 June 1805
47 55 45 N
47 29 N
47 37 N
Moon to Spica
110 29 4 W 108 25 W 108 24 W
6
125
6
10 July 1805
f
47 27 23 N
47 3 N
47 17 N
Moon to Antares 111 17 36 W 108 12 W 108 14 W
8
185
2
(Continued)

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richard s. preston
Latitude & longitude
Number
of lunar
distance
readings
Probable
error
of the
mean
Observation
Modern
values
a
Lewis &
Clark/
Bergantino
b
Ellicott
c
Error
in
longitude
10 July 1805
f,g
47 27 23 N 47 3 N
47 17 N
Moon to
Pegasi (Markab) 111 17 36 W 108 27 W 108 28 W
2
171
41
21 July 1805
46 42 15 N No noon
46 52 N
Moon to sun
111 48 10 W observation 112 3 W
8
15
6
29 July 1805
45 55 27 N 45 24 N
45 56 N
Moon to sun
111 31 10 W 111 19 W 111 12 W
20
12
1
29 July 1805
45 55 27 N 45 24 N
45 59 N
Moon to Antares 111 31 10 N 109 59 W 109 45 W
10
92
11
20 Aug. 1805
h
44 59 25 N 44 40 N
44 52 N
Moon to Sun
112 51 52 W 112 56 W 112 56 W
15
4
2
a. From a list compiled by Robert Bergantino (private communication).
b. Latitude from Lewis & Clark noon observation; longitude by Bergantino’s iterative method.
c. Latitude and longitude by Ellicott’s method of 1803.
d. According to the journal entries, the location appears to be unchanged from 29 June, so the latitude and
equal-altitudes measurements from that day were used in the calculations for this date.
e. The recorded moon-to-sun distances for 3 April 1805 all lie between 43 and 44 . According to the Nautical
Almanac for that year no such values were possible at any time on that day. When I tried changing 43 and 44
to 53 and 54 in the recorded lunar distances, the result was so close to the modern value that I included it in
the table, despite the doubtful propriety of such an emendation of the data.
f. The equal-altitudes data for 10 July 1805, as listed in Lewis’s journal, are in bad shape. The altitude and some
of the time readings are identical with the altitude and the corresponding times listed for 3 July.
g. On 10 July 1805 Lewis made lunar-distance measurements on two stars, either of which might have been
Pegasi. He wrote that the star in his second set of measurements was the more probable candidate. In agreement
with his opinion, the longitude I calculate from the second set of measurements is reasonable, but that from the
first set is meaningless. Only the results for the second star are given in the table.
h. For the calculations of the longitude for the moon-Antares observations of 20 August 1805, I followed
Bergantino in discarding all of the recorded distance readings except for readings 4, 5, and 6. For the latitude
calculation I used the value he derived from the Lewis and Clark noon altitude of the sun.
Large
33
has given a comprehensive discussion of the significance of the
lunar eclipse observations. Briefly stated, David Thompson had deter-
mined the longitude of this place some years earlier, and had reported
his result as 101 25 W. Lewis’s lunar eclipse result was 99 26 45 W,
and the correct longitude is 101 16 24 W. Thus, in hindsight, we see
that Thompson’s lunar-distance measurement was better than Lewis’s
lunar eclipse measurement. But Lewis’s two lunar-distance measure-
ments yield longitudes of 100 37 W and 101 46 W. These are much
closer to the mark, and their average is very close to the correct value.
There do appear to be significant systematic errors in Lewis’s data.
Except for the first three readings in 1805, the latitudes he calculated
from the noon altitudes of the sun (third column) are consistently low
by about 5 . Either he was using the wrong value of the index error of
the octant (which was always used for these measurements) or there

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lewis and clark
187
was a systematic personal bias in his measurements. Fortunately, in
most cases, the resulting value of the longitude is rather insensitive to
the choice of latitude used for the calculation, as can be seen from the
fairly consistent agreement between the longitudes calculated with and
without use of latitudes obtained from noon measurements (third
and fourth columns). Similarly, there is a pronounced tendency for
the calculated longitudes to be too far east. It is especially conspicu-
ous that all but one of the moon-star longitudes are too far east by
more than a degree. Of course the reason for this cannot be deter-
mined with certainty, but two possible explanations are use of an
incorrect value of the index error of the sextant, and a personal bias
in making the measurements.
The observer does not measure distances to the center of the moon,
but to the nearest or farthest point (the nearer or farther “limb”) of the
moon. The distance of a star to the center of the moon is then found by
adding or subtracting the known radius of the moon, as appropriate.
While looking through the sextant, the observer sees a direct image of
the star and, superimposed on it, a view of the moon that originates as
a reflection from a tiltable mirror. As the tilt of the mirror is adjusted,
the reading on the sextant scale changes. When the tilt of the mirror is
adjusted to bring the image of the star just into contact with the limb
of the moon’s image, the reading on the scale (after correction for
index error) is the angular distance from the star to the limb of the moon.
To this the moon’s radius must be added or subtracted, and the result
corrected for parallax and refraction.
First, consider the effects of improperly correcting for index error.
If a star is west of the moon and the sextant reading after correction
for index error is too small because of a wrong value of the index error,
the Greenwich time read from the Nautical Almanac will be too early
and the calculated longitude will be too far east. Conversely, if the star
is east of the moon and the corrected sextant reading is again too
small, the calculated longitude will be too far west. For the longitudes I
have calculated from moon-star distances, the star was west of the
moon in six cases for which the average error in the longitude is about
110 east, and east of the moon in three cases for which the average
error in longitude is again about 110 east. Thus, improper correction
for index error is not the main problem.
Next, consider a personal bias such that the observer always thinks
that, as seen in the sextant, a star is exactly touching the limb of the
moon when, in fact, there is a small gap between the two. If the mea-
surement is to the west side of the moon, which will be the case in the
early evening, the calculated Greenwich time will be too early and
the longitude too far east. If the measurement is to the east side of the

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richard s. preston
moon, which will be the case in the early morning, the calculated
Greenwich time will be too late and the calculated longitude too far
west. These results do not depend on whether the star itself is east or
west of the moon. And, indeed, the eight evening moon-star longitudes
are all too far east, by an average of about 110 , while the one morning
moon-star distance (27 July 1804) yields a longitude that is too far
west by 33 . Examination of the results from the moon-sun measure-
ments reveals similar tendencies, but of much smaller magnitude. This
indicates that just such a personal bias has influenced Lewis’s data.
For Lewis’s observations it would take an error of about 3 minutes
of arc in the observed lunar distance to produce a shift as big as 110 in
the calculated longitude. Considering that the diameter of the moon is
about 30 , this is an error equivalent to 10 percent of the moon’s diam-
eter. It may be difficult to understand how a gap this large between star
image and moon image could be mistaken for no gap at all. Part of the
problem may be that, in lunar distance measurements, it is impossible
to watch the moon and star images moving slowly and steadily toward
or away from each other along the line separating their centers. The
observer must superimpose an additional small, vibratory twisting
motion to his sextant to make the two images swing back and forth in
opposite directions along a line perpendicular to the line of centers.
The object of this is to note the instant at which the two images just
brush each other during one of these oscillations. That the sextant
often has to be held in a very awkward position compounds the diffi-
culty of this process.
The rather poor results from these moon-star measurements jus-
tify Ellicott’s 1803 recommendation that only moon-sun distances be
measured. There are other arguments supporting this recommenda-
tion: not only is it easier to judge when the moon and sun images are
just brushing past each other, but moon-sun distance measurements
are made in the daytime when it is easier to read the watch and the
sextant scale and to record the readings than at night by the light of a
lantern.
The low values of latitude obtained from noon altitude measure-
ments, mentioned previously, cannot be explained by an observer bias
of the type just described. If the observer judges that the lower limb of
the sun is just touching the upper limb of the reflection of the sun in the
artificial horizon when, in fact, there is actually a gap, the altitude of
the sun obtained from this measurement will be too low, and the
resulting calculated latitude too high. Since the actual case is just the
opposite, the best explanation of the low latitude values appears to be
an incorrect value of the index error of the octant that Lewis and Clark
always used for their noon measurements for latitude.

Page 22
lewis and clark
189
Acknowledgments
In order to make my calculations I had to have information from the 1804
and 1805 editions of the Nautical Almanac. This was difficult to get because
existing copies for those years are scarce and are treated as rare books. Fortu-
nately I have been provided with photocopies of relevant pages from several
editions of the Nautical Almanac for those years by Robert Bergantino of
Butte, Montana, and by Brenda Corbin of the U.S. Naval Observatory
Library, Washington, D.C., who also provided copies of important pages from
an early edition of Bowditch’s New American Practical Navigator, Maske-
lyne’s Tables requisite, and Kelly’s Practical Introduction to Spherics and Nau-
tical Astronomy, as well as from other sources important for understanding
early nineteenth-century navigational astronomy. With the understanding I
have gained and with the Nautical Almanac tables available I have been able
to write computer programs and enter the requisite astronomical data for par-
ticular dates into computer files for use by the programs. Guy Benson, of
Raleigh, N.C., provided me with useful documents including a copy of Robert
Patterson’s astronomical notebook. With permission from the Smithsonian
Institute, the Library of the American Philosophical Society provided a photo-
graphic copy of Andrew Ellicott’s astronomical journal. I have also benefited
from other information supplied by Robert Bergantino, Bruce Stark of
Eugene, Oregon, and Gary Moulton, editor of The Journals of the Lewis and
Clark Expedition.
References
1. Gary E. Moulton, ed., The Journals of the Lewis and Clark Expedition (Lincoln:
University of Nebraska Press), 4:244.
2. Thomas Jefferson, “The Life of Captain Lewis,” in History of the Expedition
under the Command of Captains Lewis and Clark, ed. Paul Allen (Philadelphia:
Bradford and Inskeep, 1814).
3. The Nautical Almanac and Astronomical Ephemeris, published by the Commis-
sioners of Longitude, London. The Nautical Almanac is still published each year,
although the format has changed over the years. In those days it was published
several years in advance for the benefit of travelers expecting to be gone for a long
time.
4. Tables requisite to be used with the Nautical Ephemeris for finding the latitude
and longitude at sea, published by the Commissioners of Longitude, 2d ed., 1781.
Lewis and Clark had some edition of this work with them.
5. Robert Bergantino, private communication.
6. Donald Jackson, Thomas Jefferson and the Stony Mountains (Urbana: University
of Illinois Press, 1981), 176–77. For a similar opinion, see also Silvio A. Bedini,
“The Scientific Instruments of the Lewis and Clark Expedition,” in Mapping the
North American Plains: essays in the history of cartography, ed. Frederick C.
Luebke, Francis W. Kaye, and Gary E. Moulton (University of Oklahoma Press,
1987).
7. “Sextant altitude,” as used here, signifies the sextant reading of the altitude after
correction for index error and semidiameter, but not for parallax and refraction.
No dip correction is necessary for the Lewis and Clark data, because they used an
artificial horizon for their altitude measurements.

Page 23
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richard s. preston
8. “Sextant lunar distance,” as used here, signifies the sextant reading of the lunar
distance after correction for index error and semidiameters.
9. Nathaniel Bowditch, The New American Practical Navigator (Newburyport,
Mass., 1802).
10. P. Kelly, A Practical Introduction to Spherics and Nautical Astronomy, 1st ed.
(London, 1796). Lewis and Clark had some edition of this work with them.
11. John Robertson, The Elements of Navigation, 6th ed. (London, 1796).
12. John Hamilton Moore, The New Practical Navigator, 13th ed. (London, 1798).
13. Donald Jackson, ed., Letters of the Lewis and Clark Expedition 1783–1854 with
Related Documents (Urbana: University of Illinois Press, 1978), item 19, Andrew
Ellicott to Jefferson, 6 March 1803.
14. There have been several editions of the Lewis and Clark journals, the latest and
most comprehensive being that of Gary E. Moulton (The Journals of the Lewis
and Clark Expedition [Lincoln: University of Nebraska Press]).
15.
Moulton, Journals of the Lewis and Clark Expedition 1:410–13.
16. Meriwether Lewis, Astronomy Notebook, Missouri State Historical Society,
Columbia, C1074.
17. Edward Riddle, Treatise on Navigation and Nautical Astronomy (London: Bald-
win, Cradock and Joy, 1824).
18. Charles H. Cotter, A History of Nautical Astronomy (New York: American
Elsevier, 1968).
19. Andrew Ellicott, The Journal of Andrew Ellicott (1803; reprinted Chicago: Quad-
rangle Books, 1962), appendix, 39–42.
20. The Journals of Samuel Hearne and Philip Turnor Between the Years 1774 and
1792, ed. J. B. Tyrrell (New York: The Greenwood Press, 1968). Philip Turnor
was using this same method as early as 1790 as can be inferred from the tabulated
data shown on p. 352. The “required” altitudes of the moon and the other body
are missing.
21. David Thompson, Journals of David Thompson, No. 14, Ontario Provincial
Archives. The “required” altitudes are missing from Thompson’s handwritten
data sheets. It is clear from marginal notes that he calculated these altitudes,
although not all of the details are given. Thompson learned navigation from Philip
Turnor (ref. 20). See also Jeffrey Gottfred, “David Thompson’s Navigation,”
Northwest Journal 9 (December 1996).
22. Moulton, Journals of the Lewis and Clark Expedition 3:301.
23. To see that this is so one may picture an observer at a certain latitude who mea-
sures the altitude of a star exactly seven hours after his local noon. A second
observer 15 (one hour) west of the first observer but at the same latitude will see
the same star at almost the same altitude one hour after the first observer does.
This will be the same length of time, seven hours, after his own local noon. A third
observer at the same latitude but 45 (three hours) west of the first will see the
same star at nearly the same altitude three hours after the first observer does,
which will again be seven hours after his own local noon. The conclusion from
this is that at a given latitude the observed altitude of a given star depends more
on the time from noon than on the observer’s longitude. Thus, in calculating the
approximate values of the missing altitudes, the important quantity is the elapsed
time since local noon, which the navigator has from his watch readings. Approxi-
mate altitudes calculated this way are best for the stars and planets, and worst for
the moon.
24. Thomas H. Sumner, A New and Accurate Method of Finding a Ship’s Position at
Sea (Boston, 1834). A modification of Sumner’s method was published by A.
Marcq-St. Hilaire, Revue Maritime et Coloniale (August 1875), and is the version
now in common use.
25. Jackson, Letters, item 332, Ferdinand Hassler to Robert Patterson, 12 August
1810.
26. Jackson, Letters, item 334, Nicholas Biddle to John Vaughan, c. 13 October 1810.

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191
27. Reuben Thwaites, ed., Original Journals of the Lewis and Clark Expedition
1804–1806 (New York: Dodd, Meade and Co., 1905), 7:405.
28. Florian Cajori, The Chequered Career of Ferdinand Hassler (New York: Arno
Press, 1980).
29. Catharine VanCortlandt Mathews, The Life and Letters of Andrew Ellicott (New
York: Grafton Press, 1908; reprint Alexander, N.C.: WorldComm, 1997).
30. R. L. Hassler Norris, Translation from the German of the memoirs of Ferdinand
Rudolph Hassler, by Emil Zschokke, published in Aarau Switzerland, 1877. With
supporting documents published 1882 (Nice: V.-Eug. Gather and Co., 1882), 73–
74, 76.
31. Correspondence of Ferdinand Hassler, New York Public Library. Although several
authors have asserted that Hassler lost his West Point position because he was not
an army officer, letters in this collection written in 1809 to and from W. Eustis,
secretary of war, mention only that there was no money for the position itself
because it was not authorized by law.
32. Jackson, Letters, item 399, Thomas Jefferson to Peter S. DuPonceau, 7 November
1817.
33. Arlen J. Large, “Fort Mandan’s Dancing Longitude,” We Proceeded On 13.1
(February 1987).