To Thomas Jefferson from William Dunbar, 8 October 1805
Natchez 8th. October 1805
Dear Sir
I have by this mail written to the Secretary at War, and given him the reasons of our tardy progress respecting the red river expedition.
In your last you mentioned the name of Colo. Freeman as a proper assistant to the principal Conductor of the expedition: not knowing any person of that Name but the Officer commanding the troops at New Orleans, I concluded that he had expressed a desire to go upon the expedition, in this I find that I have committed a mistake as that Gentleman knows nothing of the matter. I am therefore at a loss to know who was intended: as it appears that we shall still suffer some delay, I should be very glad if a qualified person could be sent on either as principal or Second: it would seem that we must give up the idea of finding persons qualified in any other department of Science but merely the geographical part; a good disposition to observe and record such new objects as may present themselves must supply the rest.
I mentioned in my last that one very simple method had occurred to me of ascertaining in certain circumstances the Longitude of places, which is much better calculated for travellers by land than Voyagers by sea; the method is such that a single observer with a good altitude instrument, altho’ deprived of the use of a time keeper, may still make useful observations for the advancemint of geographical knowledge. I shall now just mention the principles & shall hereafter send you some examples of the Calculation. The excellence of the usual lunar method of determining the Longitude depends (supposing her theory to be perfect) upon her quick change of place from West to east; but it cannot be denied that it requires great dexterity to make good observations, which is evident from the disproportion of the times to the distances in the hands of the best observers, and this arises from the slow progress of the moon which Causes the Contact to appear to be continued for many seconds of time; were this observation similar to a meridian altitude, it might certainly be taken to any desireable accuracy, that is, were the motion of the moon from North to South in place of from West to east, the moon’s altitude when brought upon the meridian by the rotation of the earth would furnish an easy & very Correct mode of ascertaining the Longitude: Now altho’ the proper motion of the moon is from west to East, yet her orbit makes so considerable an angle with the equinoctial circle, that there are two portions of each lunation when the moon’s change of declination is very rapid, exceeding 6° in 24 hours, that is 5" of a degree in one minute of time; if therefore under favorable circumstances we take the Moon’s greatest altitude near the meridian, we shall thence be enabled to ascertain the Moon’s declination at the moment of her passing our meridian; we must then find the time at Greenwich when the Moon had that declination and also the time when the Moon passed the meridian of Greenwich, from which data the Longitude is easily found: this method will require the use of some interpolations and an equation for the Correction of the Moon’s altitude on the meridian, because her greatest altitude will not be on the meridian, but to the East or West according as She is increasing or diminishing her north polar distance. I have communicated this method to my Worthy friend Mr. Briggs who is pleased with the idea & intends giving it consideration.
I have the honor to be with high respect and attachment Your most Obedient Servant
William Dunbar
DLC: Papers of Thomas Jefferson.
Enclosure
8 Oct. 1805
Of finding the Longitude from the Moon’s meridian altitude.
The usual mode of making the lunar observation for the purpose of ascertaining the Longitude requires the aid of a Chronometer or good Watch to a single observer, and as time-pieces of delicate construction are liable to derangement, the discovery of A method, by which one observer without a knowledge of the precise time, may be enabled to make useful observations, becomes a desideratum of value.
There are two portions of time during each lunation, when the Moon’s change of declination is sufficiently rapid to afford us the means of solving this useful problem: those times are when the Moon is on or near the Celestial Equator, and may be extended to four or five days, i.e. two before and two after the day on which the Moon crosses the Equator: the Moon’s change of declination in the most favorable position exceeds 6° in 24 hours or 15" in 1’ of time, and altho’ this is scarcely half the Moon’s motion in longitude, yet it must be remembered that this new method is no other than a meridian altitude, which may be taken to a degree of precision, never to be attained in the usual manner of taking the Moon’s distance from a Star. The accuracy of this method depends upon the precision with which the Latitude of the place of observation has been ascertained and the correctness of the lunar meridian altitude; and as probable errors are now greatly reduced by the perfection of modern instruments, it may be presumed that the error in Longitude resulting from this method ought generally to fall within 1’ of time or a quarter of a degree, and even much less, with instruments of accurate adjustment minutely divided, in the hands of experienced observers.
At sea this method cannot always be used to advantage, on account of the change of place of the ship, which might render the latitude doubtful to several minutes, and would cause an error of an equal number of degrees in Longitude.
The Moon’s greatest altitude being taken, a correction becomes necessary, because the greatest altitude is not upon the meridian, but to the East or West, according as the Moon is increasing or diminishing (by change of declination) her Zenith distance, and which may be calculated as follows: Having cleared the moon’s apparent altitude of the effects of parallax and refraction, the difference between it & the Colatitude will give the moon’s declination nearly, from which the time at Greenwich may be found nearly by simple proportion; for this time find the Moon’s change of declination for any unit of time, which may conveniently be one minute; then find the Moon’s depression from the meridian in zenith distance for 1’ of time, caused by the diurnal rotation of the Earth; (for which a convenient formula will be given below) those two effects may be considered as operating in the same line and in opposite directions without any sensible error in the result when very near the meridian. Put x to represent the time from the meridian when the Moon will have the same altitude as upon the meridian, a for the change of declination & b for the depression in 1’ of time, then use the first increases in the direct and the second (for small quantities) in the duplicate ratio of the times hence we shall have bx2 = ax…x = a/b; at half this time from the meridian the change of declination will be double the depression and will be the maximum or point of greatest altitude; hence the correction will be represented by a2/4b. i.e. the square of the change of declination divided by 4 times the depression will be the meridian correction to be subtracted from the altitude already cleared of the effects of parallax and refraction; the remainder will be the true altitude when on the meridian: the difference between the corrected meridian altitude of the Moon’s Center and the Colatitude will be the Moon’s true declination when on the meridian: the time at Greenwich when the Moon had that declination being found and also the time of the Moon’s passage over the meridian at Greenwich, take their difference and find the increase of the Moon’s AR above that of the sun during the interval of the times of the D’s passing the 2 meridians, which increase subtracted from the difference of time in passing the two meridians will be the Longitude in time.
In order to calculate with sufficient accuracy the time corresponding to the Moon’s declination and also the time of her passing the Meridian of Greenwich, it will be necessary to prepare 4. right ascensions and 4 declinations of the Moon to seconds, which in the Nautical Almanac are set down to minutes only; with the aid of those, we can by the method of interpolation find very correctly the times which are sought, and in regard to the Moon’s declination, the effect of nutation is not to be disregarded, because an error of every second in the calculated declination may produce an error of 4" of time in the Longitude; the same care should be taken to correct the latitudes found by the sun’s meridian altitude from the effect of the same cause.
The following is a convenient formula for finding the Correction to be applied to the altitude cleared of the effects of parallax and refraction, expressed in logarithmics language.