# Longitude

Page 4 of 5

Aboard merchant and naval ships from the late 1700s to the late 1800s, when several observers and several sextants were available, Lunar Distance observations were well orchestrated. The method was as follows: One observer measured the angular distances betwween the moon and sun or star and another observer recorded the time of the measurements. A third observer (and sometimes a fourth) measured the altitude of the two bodies at the beginning, end and sometimes the midpoint of the Lunar Distance observations. The times for these observations also were recorded. The altitudes obtained then were averaged to the middle time of the measurements for angular distance. The altitude of each body thus derived was called its sextant altitude and had to be corrected for index error, dip of the horizon, and semidiameter (except for a star). This gave the apparent altitude of the center of the body at the middle time of the observation. The apparent altitude then was corrected for refraction and parallax to give the true altitude of the body observed. Both the true altitude and the apparent altitude of the bodies were needed for the longitude calculations.

Lewis and Clark, however, had just one sextant and they observed only the angular distance between the moon and the sun or star. Thus the true altitude of the moon and the sun or star had to be calculated by a method such as that outlined in the Fourth Problem of Robert Patterson's Astronomy Notebook. The calculation involved using a) the latitude of the place of observation, b) the hour angle of the moon and sun or star, and c) the declination of the moon and sun or star. The apparent altitude then had to be determined in reverse from the calculated true altitude.

1. Latitude: The latitude usually was calculated from one or more Meridian Observations of the sun.
2. Hour Angle: The hour angle of the sun came from the corrected Local Apparent Time of the observation (see Local Time) and was the number of hours, minutes and seconds of the observation before or after solar noon.
The hour angle of the moon or star was calculated from 1) the corrected Local Apparent Time of the observation, 2) the right ascension of the sun and 3) from the right ascension of the moon or star.
3. Right Ascension and Declination: The right ascension and declination of the moon or sun were obtained by proportioning their right ascension and declination (given at Greenwich Apparent Noon in the Nautical Almanacs of the era) to the estimated Greenwich Time of the observation. The Greenwich Time of the observation was obtained from the longitude, which in turn was derived by dead-reckoning or by estimate of distance traveled from a point of known longitude.

The right ascension and declination of the star observed was calculated from the coordinates of that star as listed in Tables Requisite or in a similar book of tables. These coordinates usually were for the beginning of some year, say 1805. It was necessary to apply the given annual rate of change of these coordinates to the date (fraction of a year) of the observation.

Each step described above required lengthy calculations using logarithms. Logarithms for numbers and trigonometric functions of angles were tabulated in special books. Many of these tables gave logarithms only to four decimal places, which reduced the precision of the calculations. Mathematicians hoping to obtain higher precision for their calculations had to make interpolations for each number used. Occasions for making errors abounded. For these reasons a longitude derived from Lunar-Distance observation easily could be in error by ±15 arc minutes and not infrequently by as much as half a degree.

## The Observations

On July 29 about 4 p.m., when Lewis began the first of his two observations of the Lunar Distance from the sun, the moon was little more than a pale white sliver, three days old and about 40° above the horizon. It stood a little west of south nearly over the Madison River where the river emerges from the mountains. By the time Lewis finished his observations of the moon's distance from Antares—the bright, reddish star in the constellation Scorpio—the moon had circled in its daily orbit just to the northwest of the Tobacco Root Mountains and stood only about 5 degrees above the horizon.

The captains made no calculations for longitude from these observations. The longitudes given below from the captains' observation were calculated using an adaptation of the method devised by Chevalier Jean Borda in 1787.1

Table III, Longitude of Three Forks Camp
Date With Calculated Longitude Longitude from map interpretation
1805 July 29 Sun obs 1 111˚13' west
1805 July 29 Sun obs 2 111˚10' west
1805 July 29 Antares 110˚47' west
Average   111˚03' west
Sun only   111˚11½' west 111˚30'40" west
Error
Error in arc minutes Error in statute miles
All 3 observations 0˚27.7' east 22¼ too far east
Sun observations only 0˚19.2' east 15½ too far east

On Clark's map of 1805, made at Fort Mandan, two possible "three forks" are shown, that farther to the northeast is at a longitude of 115°07' W, that farther to the southwest is at 116°50' W. The longitude for the Three Forks is shown at about 110°50' W on the Lewis and Clark map of 1806, at 109°57' W on Clark's 1810 map and at 110°02' W on the Lewis and Clark map of 1814.

1. Jean-Charles de Borda (1733-1799) was a French mariner, mathematician, physicist, and nautical astronomer who developed, among many other scientific and mathematical innovations, an important new method for measuring an arc of a meridian. He also devised a series of new trigonometric tables in conjunction with his own innovative surveying techniques. During the American Revolution, as an officer in the French Navy he commanded a fleet of six ships that saw action against the British in behalf of the United States.

Funded in part by a grant from the National Park Service, Challenge-Cost Share Program.