[p. 18:] To find the Longitude by Lunar observation
Directions

At any time when the sun and moon, or the moon and any of the stars from which her distance is calculated in the Nautical Almanac^{1} on the given day, are both visible, and neither of the bodies less than 5° high; your instrument being previously adjusted, or the index error ascertained, and also the error of your watch found by a previous observation as directed [in] problem 3d, take a set of three or more observations of the apparent angular distance of the sun and moon's nearest limbs, or of the star from the moon's nearest or farthest limb, viz. that which is fully enlightened, noting the corresponding times per watch; and at the time that you measure the distance, let two [p. 19:] assistants take the altitudes of the bodies, respectively. Or the distances and the altitudes may all be taken by the same observer, at small equal intervals of time in the following order, viz.,
 Of these distances, times and altitudes, take means by dividing their respective sums by the number of observations.
 Let the mean altitudes be so far corrected as respects semidiameter, so as to obtain the apparent altitudes of the centres above the true horizon.

To the apparent distance of sun and moon's nearest limbs add the sun's semidiameter, Nautical Almanac page III, [p. 20:] of the month,
Light Touch
Lewis must have smiled at Patterson's whimsical embellishments of an astronomer's shorthand symbols.  If at the time of taking the above observations, one of the bodies (but especially the sun) be not less than 3 or 4 points of the compass [33¾° 45°] from the meridian, the time at place observation may be computed from the altitude of the body, as in Problem III independently of the time per watch, and this should always be done when circumstances admit.
 As it may frequently happen that the altitudes [p. 21:] of one or both of the bodies cannot well be taken you must then compute the apparent altitudes by problem 4th, and this method is generally to be preferred on land.
From the above data, the longitude of place observation may be computed as in the following example.
Example
Suppose the apparent angular distance of the sun and the moon's nearest limbs (by taking the mean of a set of observations) to be 83°30'04". The apparent altitude of the sun's lower limb measuring 52°44' and that of the moon's lower limb 24°32', Greenwich time September 22, 1799 about 9 hours p.m. Time at place observation (allowing for error of watch from true apparent solar time) 15 hours p.m.
Required the longitude of place observation from the meridian of Greenwich?
[p. 22:]
Form V
[p. 23:]
 Add when C is greater than B, otherwise subtract.
 Subtract when C is greater than B; otherwise add.
 Subtract when either H or I exceeds 90°; or when H is greater than I.
 Add when either H or I exceeds 90°; or when H is less than I.
 In Table VIII find the correction of moon's altitude, then in Table XIII under the nearest degree to Q at the top, find two numbers; one opposite to the nearest minute to moon's correction of altitude found as above, and the other opposite the nearest minute to first correction (N) and the difference of these two numbers will be the third correction. This correction may, without sensible [error], be generally omitted.
 Add when Q is less than 90°, otherwise subtract.
 These are to be found in Nautical Almanac from page 8th to page 11th of the month opposite the day of the month and the sun or the star from which the moon's distance was observed; taking out the two distances, which are next greater and next less than the true distance (S) calling that the preceding distance which comes first in the order of time and the other the following distance.[p. 24:]
 The Greenwich time and time at place observation must both be reckoned from the noon of the same day.
 When the true Greenwich time is greater than the time at place observation the longitude is west; otherwise it is east. When a longitude comes out more than 12 hours or 180°, subtract it from 24 hours or 360° and change its name. ––
Example 2nd
Suppose the apparent angular distance of the star Regulus from the moon's farthest limb to measure 54°18', apparent altitude of moon's lower limb 20°10', apparent altitude of star 58° 32', Greenwich time November 23d about 17 hours p.m. Time at place observation allowing for error of watch 17h 20m 25sec p.m. Required the longitude?
[p. 25:]
Thus ends the fivepart instruction manual in celestial observation that mathematicianastronomer Robert Patterson prepared especially for Meriwether Lewis. On the following page, perhaps at Patterson's suggestion, Lewis laid out a table in which data from his own observations could be entered in an orderly manner. However, he never used it, nor did he ever follow any similar plan for recording his field observations; had he done so, it might have been easier for another person to calculate, or recalculate, the desired latitudes, longitudes, declinations, chronometer errors, and so on.
1. A Nautical Almanac is a yearbook for navigators and astronomers, containing the predicted positions of the sun, moon, and planets with respect to Earth, every day of the year. The first one was published in London in 1766. Three items listed as "Nautical Ephemeris" appear in the summary of purchases Lewis made in preparation for the expedition.
Funded in part by a grant from the NPS ChallengeCost Share Program