While Meriwether Lewis was at Camp Island near the Three Forks of the Missouri he spent many hours preparing for and taking celestial observations. From all those observations, however, the only calculations he made were for latitude from the sun's noon altitude, July 28 and 29. Lewis's calculated latitudes, unfortunately, are nearly 30 miles farther south than the true latitude of his Camp Island location. This difference in latitude results from his using the wrong index error for the octant. The other observations were not calculated until nearly two hundred years after Lewis took them; those observations were:
- Two sets of a.m and p.m. Equal Altitudes of the sun to check the chronometer's time
- Three sets of Lunar Distance observations for longitude (two with the sun, one with Antares)
- Three sets of observations (two with the sun, one with Polaris) to determine the variation of the compass needle or magnetic declination
Before the advent of electronic calculators and computers all the mathematical operations would have been done "long hand." Multiplication, division, powers and roots would have been done by logarithms. All operations using trigonometric functions also would have been performed using logarithms. In addition, the procedures for making the calculations usually were set out in work-sheet forms in books that had been published by mathematicians trained in making the calculations. Thus, the person doing the calculations merely filled in the observational data in the proper places and followed the outlined procedure, step by step–often without understanding the why or wherefore of the mathematical operations.
All the calculations that follow were made using an electronic calculator in a non-program mode, and the operations are set out in step-by-step fashion to help the reader follow the complex operations.
The mathematician of the Lewis-and-Clark era would have used the Nautical Almanac for the year of observation. I also used the Nautical Almanacs for 1803–1806 even though there are several excellent computer programs that can recalculate–with greater accuracy and precision than often exists in the almanacs of the time–the needed celestial information. The mathematician of the Lewis-and-Clark era also would have used standard tables such as Tables Requisite to determine the corrections for refraction and parallax and would have used numerous other tables in them to facilitate the tedious long-hand operations. I did not have access to Tables Requisite for the years of the expedition, but used modern formulae to obtain refraction and parallax and other needed parameters. In addition, because most of the navigation tables were designed for use at sea, they do not provide values for refraction at altitudes greater than about 2000 feet.
The calculations made from Lewis's celestial observations are given below in the following sequence regardless of the date or time at which each was made made:
- Latitude from noon observation of the sun
- Chronometer error at noon from Equal Altitudes observations
- Chronometer time of an observation from any other observation for which the chronometer's time and sun's altitude is given
- Longitude from Lunar Distance observations
- Magnetic declination1
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- 1. For a complete explanation of calculating geographic location from Lewis's observations at the Three Forks, see "Calculations for the Celestial Observations that Meriwether Lewis made at the Three Forks of the Missouri" (PDF).