Approximating Longitude

Page 3 of 6

Calculating Latitude: The Steps Continued

Step 7a . . . Lewis would have had several means by which to estimate the longitude of the junction of the Kansas and Missouri Rivers. From his own observations he had determined Camp Dubois to be at or near 38°55'20"N, and about 90° west (89°57'45" by astronomical observations and 90°00'20" by chronometer). Lewis also had information that Thomas Jefferson had obtained from British sources regarding the Mandan villages. These villages, as determined through celestial observations made by the British explorer David Thompson in 1798, were at or near 47°17'22" North Latitude, 101°14'24" West Longitude.

The coordinates of Camp Dubois and Fort Mandan, thus, could be used as anchor points between which Clark could fit the maps of the Missouri that he had plotted from the Expedition's river survey. If Clark had used this method (which he clearly did not), the mouth of Kansas River should have been plotted at about 94° West Longitude.1

The King map of 1803 that Lewis carried, however, showed the mouth of Kansas River at 94°55' West Longitude. Lewis, for his first approximation of longitude, could pick an average longitude of 94°30' for his calculations.

The nautical almanac gives the sun's declination for apparent noon at Greenwich. Lewis, using an estimated longitude of 94°30' W for the mouth of Kansas River, must determine what the sun's declination would be at 6:18 p.m. Greenwich Apparent Time for each of his noon observations. On 27 June, for example, the sun's noon declination at Greenwich by the 1804 Nautical Almanac was 2°20' 59" north. On the 28th at noon it was 23°18' 23". The difference in 24 hours apparent time is 2'36". In the 6h18m since noon at Greenwich, then, the sun's declination would have decreased by 41 seconds, making the sun's declination at noon at the mouth of Kansas River 23°20' 18":

Steps 8 and 9

After all three meridian observations have been recalculated, the results are:

Recalculated Latitude

The average difference of four seconds between the recalculated latitude and that from the calculations made at the mouth of Kansas River stems from the estimated longitude and from a possible error on the 28th. The octant, it should be remembered, could be read only to the nearest 30 arc seconds, so both values actually become 39°05½' North Latitude. After having calculated the chronometer's error for 27, 28 and 29 June 1804, Lewis probably also would attempt to calculate the latitude of the mouth of Kansas River from the time and altitude of the sun obtained during the Equal Altitudes observations or from the observations for magnetic declination. After checking these observations he would find that, for most of them, the sun had been too far from the meridian or else was too close to bearing either east or west.

Nevertheless, Observation No. 3, the second p.m. observation for Magnetic Declination with the Sun taken on 27 June might provide a useful latitude. The calculations yield a latitude of 39°14'59", but the sun likely was too far west of the meridian at the time, and accuracy deteriorates with time and angular distance from the meridian. On the other hand, the result—derived from the magnetic declination—suggests that latitude derived from the average of his three meridian altitudes of the sun is good.

Suppose that, after having calculated the longitude for the mouth of Kansas River from his Lunar Distance observations on 29 June 1804, Lewis had found that the difference between calculated longitude and dead-reckoned longitude was a degree or slightly more. Would he use the longitude from his Lunar Distance observation to recalculate the latitudes? Probably not, because at this time of year the sun's declination is changing only about 3 arc minutes per day, and even several degrees difference between longitudes would make little change in the recalculated latitude.

Comparisons and Conclusions

By comparing the Expedition's river survey with the oldest detailed maps and with modern maps and aerial photos, the mouth of the Kansas River in 1804 was at or near 39°07'N, 94°36½'W. The expedition's river survey, made by using compass bearings and estimated distances, even when corrected to True North (see Magnetic Declination) puts the mouth of the river at 39°29½"N (and 94°54"W)—30 miles too far north of its actual latitude. The average latitude of this junction, recalculated from Lewis's observations of the sun's Meridian Altitude with the octant, is 39°05½'N. This latitude is 1½' (just 1¾ miles) south of the redetermined latitude of 39°07'N. The positional accuracy (at least as to latitude) that could be obtained even by an instrument as imprecise as the octant justifies Jefferson's instructions to take celestial observations to correct the ground survey—the Courses and Distances—the captains made with compass bearings and estimated distances.


1. See Clark's map of 1805, Moulton, ed., Atlas, Map 32a. Clark probably prepared this map either by re-plotting their river survey at a smaller scale (1 inch = 100 miles) or redrafting the separate maps he had made of the Missouri from the Expedition's river survey (1 inch = 6 miles). Despite often being hailed as a "cartographer," Clark's mapping abilities, though adequate for most tasks, often were compromised by time, equipment and training. On his map of 1805, the mouth of Kansas River is shown at 93° West Longitude, not 94° West as he should have shown it had he done his plotting carefully.

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