## Converting Observational Data into Geographical Data

Navigational Mathematics

After returning to St. Louis in September 1806, Lewis sent some of the Expedition's celestial observations and a payment in advance to mathematician Rudolph Hassler in Schenectady, New York, for work Jefferson had asked him to undertake. It is probable that Hassler was expected to make only the calculations that would yield longitudes from those observations. This interpretation comes from a letter Hassler wrote to astronomer-mathematician Robert Patterson in 1810,^{1} Hassler does not mention latitudes or magnetic declination in that letter, but he specifically describes problems with the longitude information and also describes ambiguities, uncertainties and mis-calculations in the original data. One of Hassler's principal impediments was the lack of a detailed map showing where the observations were taken; another is the absence of adequate information about the equipment used and the captain's method of taking and recording their observations. Undoubtedly, Hassler made a sincere effort at calculating some of the longitudes, but Clark never received any information from him for the observations that he and Lewis took—coordinates that would have helped place key points on the map that ultimately accompanied the 1814 Biddle edition of their journals.

Lewis calculated (and sometimes miscalculated) most of the latitudes from observations of the sun's noon altitude in 1804 and 1805, but no one—within the lifetime of Lewis, Jefferson or Clark—is known to have successfully calculated all geographic data from the other observations the captains made. Only with the onset of the Lewis and Clark Bicentennial were any of these other calculations made.^{2}

## A Conjecture

While in Pennsylvania during the spring of 1803, Lewis received a hasty education in taking and recording the celestial observations Jefferson ordered him to make on the expedition. (The only observation he knew how to take before his meeting with Andrew Ellicott was that for the meridian observation of the sun, which he learned from Jefferson.) Undoubtedly, Lewis also received some training on how to make the necessary calculations to turn those observational data into latitude, longitude, time, altitude and direction. Suppose, however, that instead of just being taught the basics, his training had been thorough, giving him the competency to correctly calculate all his celestial observations, converting them into geographical information with an accuracy comparable to that of the better maritime navigators of his day. Suppose further that Lewis, when he was at Fort Mandan, had found the time to calculate or re-calculate all the observations he had taken up to that point.

These conjectural calculations of Lewis's would have been made using five-place logarithms and "note book" methodology following procedures devised by persons such as Robert Patterson or Andrew Ellicott. In addition, his initial calculations, which required only an estimation of his longitude, would have depended upon the geographical information he had available. Consequently, the latitude, longitude and magnetic declination obtained by his calculations would, in most cases, have differed somewhat from what might be obtained nowadays. The object here is to discuss and evaluate the geographic data that might have been derived at the time of the Expedition.

## Calculting Latitude: The First Steps

Of all the celestial observations the captains took, the simplest to calculate is the latitude from a meridian (noon) observation of the sun. Let's assume these calculations would be made first. There were a number of different ways to make them but, assuming the mathematics were correct throughout, the results should be essentially the same. Lewis's method appears to have been as follows: Example from Lewis's Meridian Altitude observation, 27 June 1804, mouth of Kansas River:

At this point Lewis needs to know his approximate longitude; a good estimate will do—call it a "dead-reckoned" longitude. From it he can determine the approximate Greenwich time of the observation, and with that he can determine the sun's approximate declincation^{3} at the moment ofhis observation. This determination is necessary because the Nautical Almanacs of his day gave the sun's declination only for Local Apparent (solar) Noon at Greenwich, and the sun's declination, except at the solstices, is constantly changing.

Suppose, for example, Lewis had taken an observation of the sun's altitude at noon on 1 May at Camp Dubois, which was at about 90° West, or 6 hours' difference in time from Greenwich. Suppose further that he had used the sun's declination for Greenwich noon instead of adjusting that declination for the six-hours' difference in time. Had he done so, his calculated latitude would be too far north by 5 miles.

1. Ferdinand Rudolph Hassler to Robert Patterson, 12 August 1810, in Donald Jackson,

Letters of the Lewis and Clark Expedition with related documents 1783-1854(Urbana: University of Illinois Press, 1978), 2:556-559.2. See, for example: Richard Preston, "The Accuracy of the Astronomical Observations of Lewis and Clark," Proceedings of the American Philosophical Society, vol. 144, no. 2 (June 2000), 168-191; Robert Bergantino, "Revisiting Fort Mandan's longitude,"

We Proceeded On, November 2001, 19-26.; Eileen Starr, 2001, "How the captains found latitude and (sometimes) longitude,"We Proceeded On ,November 2001, 12-18; L. A. Rudner and Hans Heynau, "Revisiting Fort Mandan's Latitude,"We Proceeded On,November 2001, 27-30; Bruce Stark, letter to the editor,We Proceeded On,August 2002.3. Except at the summer and winter solstice, the sun's declination—that is, its angular distance north or south of the celestial equator—is constantly changing. The Nautical Almanacs of Lewis and Clark's day gave the sun's declination only for Greenwich Apparent Noon. For an observation made at any other time or longitude, the sun's declination had to be calculated.

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