Agreeably to my promise, I have investigated thy Problem for finding the longitude by lunar observation. In reducing the operation to a practical formula, in every modification which I have been able to give it, a knowledge of the time of observation appears essential.

Without a knowledge of the time, the Moon’s right ascension, or longitude may be found, and, from the Nautical Almanac, the time at Greenwich, of that observation; but these are not sufficient data for finding the terrestrial longitude. Assuming, therefore, the time, as one of the data, I have labored faithfully, for two days, in order to simplify the process; yet, in every view, it appears to me too intricate and troublesome, for any but an adept in Astronomy, and one who has, by practice, acquired considerable expertness in making observations.

In the course of my attention to this subject a very simple method of finding the longitude has occurred to me. It requires no instruments but a good watch giving seconds, and a Portable Transit such as our friend Andrew Ellicott can easily furnish. The observations require but little skill, and, without unnecessary inattention, this method admits not of an error, in longitude, greater than 6½ miles when the moon is in her Apogee, and when in her Perigee not greater than 4 miles. By taking a mean of several observations, and by a regular course of similar observations made in the City of Washington, the probability of error will be greatly diminished. On account of the uncertainty which still exists in the best lunar calculations, a regular course of observations made at the Zero of American longitude, as a standard of comparison for similar observations made in different places, would, in my opinion, furnish incomparably the best means of improving the Topography of America.

A Method of finding the Longitude, by the Moon’s Culmination.

Observations.

Note the instant of time when the moon’s enlightened limb is on the meridian. Note also the instant of culmination of several known fixed Stars, as near the time of the moon’s culmination as can be obtained, either before, or after it, or both, for ascertaining the correctness of the watch.

A scientific traveller should multiply these observations as much as possible, keep an accurate list of them and of the places to which they correspond, and defer the calculations to his leisure, or the skill of Astronomers.

Formula.

1. By the estimated longitude, reduce the time of observation to Greenwich time; which call, G.

2. Take, from the Nautical Almanac, the difference between the moon’s right ascension for the time next after and next before G. which difference call D.

3. Take the Moon’s Semidiameter answering to G. which call S.

4. As D : 12 hours :: S : C.

5. When the moon’s enlightened limb is west, add C to, when east, subtract it from, the time of observation, and the sum or remainder will be the time of her centre passing the meridian, which call T.

6. Take the time of the moon’s passage over the meridian of Greenwich, for the same day (M) and for the following day (N.)

7. N—M

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: 360° :: : 24h ::

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T—M

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: The Longitude, in degrees &c. : The Longitude, in Time.

N.B. This formula is adapted solely to West longitude.

Example.

Admit that on the 4th. of the 3rd. Month 1803, the appulse of the Moon’s enlightened limb to the meridian was observed at 9 h. 9 m. 12 sec. apparent time,—Longititude estimated to be 5 hours West.

	h m s
	9. 9.12	Moon’s R. Asc.	Mar. 5. Noon		134.°	37’
Est. Long. +	5		Mar. 4. Midnight		128.	34
G =	14. 9.12			D =	6.	3
S =	14.’ 57"	Time of observation			9. 9.	12
	6.° 3’ :	12 hours :: 14.’ 57" :		C =	+ 29.	38
N =	10.h 15m			T =	9.38.	50
M =	9. 29			M =	9.29.	—
	46				9.	50

46m

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: 360° :: : 24h ::

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9.m 50s

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: 77° :  5h 8m

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Longitude West of Greenwich. This is the longitude of the Capitol.

The practice of this method would be greatly facilitated by two tables, one, of the moon’s passage over the meridian of Washington, true to a second; and the other, the time of her semidiameter passing the meridian, also true to a second, for every noon and midnight. These would, perhaps, if calculated for a few years to come, furnish the best substitute for a course of actual observations at that place.