Some time since, I addressed a letter to you on the subject of a first meridian for the United States at the permanent seat of their government, to be effected by ascertaining the longitude of the Capitol in this city from Greenwich observatory, in England, being the spot from which many, if not the whole of our mariners are in the habit of reckoning their departure. It is proper that the result of this undertaking should be communicated to such scientific characters in this country as are supposed to feel any desire that we should shake off that kind of dependence which we have too long had on a foreign country; and as you are to be considered among the most distinguished of those characters, permit me to send you an abstract of the calculations for the purpose, founded on the occultation of η. Pleiadum, (Alcyone) by the Moon, observed near the President’s house, on the evening of the 20th of October, 1804.

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		° ′ ″
Latitude of the Capitol, in Washington, by observation,		38.52.57.N.
		dec
〃 reduced, (334 to 333) ☞		38.42.52.939.
Right ascension of η. Pleiadum, allowing aberration and nutation,		53.59. 6.273.
Declination do allowing do do		23.29.45.143.N.
Obliquity of the Ecliptic, October 20. 1804,		23.27.54.250.
Longitude of the star, by computation,		57.16.35.925.
Latitude do		4. 1.59.809.N.
Estimated longitude from Greenwich, in time, 5. h. 7. m. 36. sec. = 76.° 54.′ 0.″ West.
		h. m. sec. dec.
Time, by watch, of the immersion, reduced to the Capitol,		9.30. 9.32.
Error of the watch,		– 7.32.75.
Apparent time of immersion at the Capitol,		9.22.36.57.
Sun’s right ascension, then,		13.42. 5.28626.
Right ascension of medium cœli, in time,		23. 4.41.85626.
		° ′ ″
Equal to	♓︎︎.	16.10.27.8439.
or 76.° 10.′ 27.″ 8439. dec. from the beginning of ♑︎, the nearest solstitial point.
		°
Altitude of the nonagesimal,		49.36.41.464.
Longitude of the nonagesimal,	♈︎.	5.52.41.926.
☽’s true longitude,	♉︎.	26.26.10.399.
☽’s true distance from the nonagesimal, (East)		50.33.28.473.
〃 horizontal parallax, reduced,		1.[ ].59.0134.
Parallax in longitude,		36.17.580.
☽’s apparent or visible longitude,		57. 2.27.979.
〃 true latitude, North,		4.30.25.399.
Parallax in latitude,		37.31.516.
☽’s apparent latitude at immersion, North,		3.52.53.883.
For the Emersion.
		h. m. sec. dec.
Time by watch, of the emersion, reduced to the Capitol,		10.24.47.32.
Error of the watch,		– 7.32.75.
Apparent time of emersion at the Capitol,		10.17.14.57.
Sun’s right ascension, then,		13.42.13.82697.
Right ascension of medium cœli, in time,		23.59.28.39697.
		°
equal to,	♓︎.	29.52. 5.9545.
or 89.° 52.′ 5.″ 9545. dec. from the beginning of ♑︎.
		° ′ ″ dec.
Altitude of the nonagesimal,		54.56.23.787.
Longitude of the nonagesimal,	♈︎.	17.35.28.761.
☽’s true longitude,	♉︎.	27. 0.26.933.
〃 true distance from the nonagesimal, (East)		39.24.58.172.
〃 horizontal parallax, reduced,		1. 0.58.4066.
Parallax in longitude,		32. 8.867.
☽’s apparent longitude,		57.32.35.800.
〃 true latitude, North		4.29. 6.143.
Parallax in latitude,		32.22.518.
☽’s apparent latitude at emersion,		3.56.43.625.
〃 motion in apparent longitude, during the transit,		30. 7.821.
〃 motion in apparent latitude do		3.49.742.
〃 center south of the ★, at immersion,		9. 5.926.
〃 , at emersion,		5.16.184.

The difference of apparent latitude of the ★ and ☽’s center, was therefore greater at the immersion than at the emersion.

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In occultations, the Moon’s motion in apparent longitude should be multiplied by the co-sine of the star’s latitude, or of the Moon’s apparent latitude at the middle time between the immersion and emersion, to reduce the motion in apparent longitude to a parallel to the Ecliptic: the former is to be preferred, when the Star’s latitude has been obtained with due precision.

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The remaining part of the process, to find the difference of apparent longitude of the ☽’s center and the points of occultation, (or those parts of the Eastern and Western limbs of the Moon at which the Star immerged and emerged) will be explained by the following figure, and the annexed remarks.

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FSG, represents a parallel to the Ecliptic, passing through the star S.

A, the apparent place of the ☽’s center at the immersion. D, at the emersion.

ABC, the Moon’s motion in apparent longitude, × co-sine of ★’s latitude.

CD, the Moon’s motion in apparent latitude. AF, DG, the difference of apparent latitude between the ★ and ☽’s center. AS, DS, the Moon’s semidiameter at immersion and emersion, (corrected) SE, the nearest approach of the centers of ★ and ☽. CAD, the angle of inclination of the Moon’s apparent orbit. AD, the chord of transit, or ☽’s apparent path. AE and ED, segments of the base, or Moon’s apparent path. EAS, EDS, angles of conjunction, and ASF, DSG, the central angles at the immersion and emersion, whence FS, GS, the differences of apparent longitude of the ★ and ☽’s center are to be found, from which, by applying the parallaxes with a contrary sign, the true differences of longitude, as they would be seen by a spectator placed at the center of the Earth under the meridian of the Capitol, in Washington, will be obtained. The intervals of time between the beginning and end of the transit and the ecliptical conjunction, may then be determined from the Moon’s motion in longitude reduced to the ecliptic.

It is customary in occultations to apply the inflexion of the Moon’s light to the augmented semidiameter, both at the immersion and emersion: the quantity of that element having been variously estimated by astronomical writers, it is thought advisable to use a mean of the following—

		″
	Dr Mackay,	–3.	5
	〃 Vince,	3.	–
	Mr Garnett, (American req: tables)	2.	977.
	〃 Ferrer, ditto,	2.	18.
Mean	Inflexion of the Moon’s light.	–2.	914.

The Moon’s augmented semidiameter arising from a change of altitude, may be thus found—

ar: comp: log. cosine true altitude, + log. cosine apparent altitude, + log. sine horizontal semidiameter, – radius, = log. sine ☽’s augmented semidiamr from which the inflexion of light is always to be subtracted.

	° ′ ″ dec.
Log. cosine true altitude, at imm:	32.13.45.342.	arith: comp:	0.0726702.54
〃 co-sine apparent altitude,	31.23.14.461		9.9312879.18.
〃 sine ☽’s horiz: semidiameter,	16.38.375		7.6848668.59
〃 sine ☽’s augmented semidiam:	16.47.511.		7.6888250.31
Inflexion of light,	–2.914
☽’s corrected semidiameter, AS,	16.44.597.
For the ☽’s corrected semidiameter at the emersion.
	° ′ ″ dec.
Log. cosine true altitude,	42.25.22.181	arith: comp:	0.1318338.61.
〃 cosine apparent alt:	41.40.52.804		9.8732361.93.
〃 sine ☽’s horiz: semid:	16.38.147		7.6847679.09.
〃 sine ☽’s augm: semida	16.49.868		7.6898379.63.
Inflexion of light,	–2.914.
☽’s corrected semidiameter, DS,	16.46.954.

Log. ☽’s motion in apparent longitude,	1807.″ 821 dec.	3.2571554.40
〃 cosine ★’s latitude,	4.° 1.′ 59.″ 809	9.9989230.28.
☽’s motion in a parallel to the ecliptic, ABC,	1803.″ 3434	3.2560784.68.
	″ dec
☽’s motion in apparent latitude, CD,	229.742  log	2.3612403.80.
ar: comp: log. ABC,	1803.3434	6.7439215.32.
log. tangent angle of inclination, CAD,	7.° 15.′ 36.″ 890 dec.	9.1051619.12.
Log. ABC,	1803.″ 3434 dec.	3.2560784.68.
ar: comp: log. cosine angle of inclination,	7.° 15.′ 36.″ 890 dec.	0.0034960.98
Chord of transit, AD,	1817.″ 9191 dec.	3.2595745.66

In the oblique plane triangle, ASD, we have now the three sides, AD, 1817.″ 9191 dec., AS, 1004.″ 597 dec., and DS, 1006.″ 954 dec., to find the segments of the base, AE and ED.

	″ dec.
arith: comp: log. AD,	1817.9191.	6.7404254.34.
Sum of AS and DS,	2011.551. log.	3.3035310.16.
difference,	2.357. log	0.3723596
(x)	2.6080	0.4163160.50.
	″ dec.	″ dec.
AD, ∓ (x) = 2 AE,	1815.3111.½	907.65555. =	AE.
2 ED,	1820.5271.½	910.26355. =	ED.
	″ dec.
Log. of Segment AE,	907.65555	2.9579210.66.
arith. comp: log. AS,	1004.597	6.9980080.99.
Log. cosine: angle of conjunction, EAS,	25.° 22.′ 39.″ 735 dec	9.9559291.65.

When the difference of apparent latitude of the ★ and ☽’s center is greater at the immersion than at the emersion, the sum of the angle of inclination and angle of conjunction is equal to the central angle. The contrary at the emersion.

		° ′ ″ dec.
Angle of conjunction,	EAS,	25.22.39.735.
Angle of inclination,	CAD,	+7.15.36.890.
Central angle,	ASF,	32.38.16.625.

arith: comp: log. cosine ★’s latitude,		4.° 1.′ 59.″ 809. dec.	0.0010769.72.
Log. ☽’s semidiameter, AS,		1004.″ 597. dec	3.0019919.01.
〃 co sine central angle ASF,		32.° 38.′ 16.″ 625.	9.9253646.17.
	′ ″
〃 diff. of apparent long. FS.	14. 8.073. =	848.″ 073	2.9284334.90.
☽’s apparent long. at immersion,	57. 2.27.979
app: longitude point of occultation,	57.16.36.052.
★’s longitude,	57.16.35.925.
point occultation East of ★,	0. 0. 0.127.

	′ ″ dec.
Parallax in longitude at the emersion, corrected,	32. 8.881.
diff: of apparent longitude,	–15.59.777.
True diff: of longitude of ★ and ☽’s center at the emersion,	+16. 9.104.

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The Moon’s true motion in longitude for 12 hours, reduced to the ecliptic, at the middle time between the immersion and ecliptical conjunction at Washington, was

	° ′ ″
	7.31.58.554.
At a middle time between the emers: and ecl: conj:	7.31.55.875.

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For the intervals of time.

As 7.° 31.′ 58.″ 554. dec is to 12 hours, so is 50.′ 25.″ 670. dec to 1. h. 20. m. 19. Sec. 909 dec., which added to 9. h. 22. m. 36. sec 339 dec., the corrected time of immersion, gives 10. h. 42. m. 56. sec. 248. dec. the time of ecliptical conjunction of ☽ and ★, at the Capitol in Washington, by the immersion.

As 7.° 31.′ 55.″ 875 dec., to 12 hours, so is 16.′ 9.″ 104. dec to 25. m. 43. sec. 940 dec., which added to the time of emersion corrected, = 10. h. 17. m. 14. sec. 392. dec gives 10. h. 42. m. 58. sec. 332 dec., the time of ecliptical conjunction, by the emersion.

	h. m. sec. dec.
Apparent time of ecliptical conjunction, by the immersion,	10.42.56.248.
, by the emersion,	10.42.58.332
Mean. Time of true conj: ☽ and ★, at the capitol,	10.42.57.290.

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Apparent time			☽’s longitude.
at Greenwich.
h.	m.	° ′ ″ dec.
15.	30	57. 3.41.018	A.		1st diff:
					′ ″ dec.		2d diff.
				+	6.16.319.	a1.
15.	40	57. 9.57.337.	B.				– 029. a2.
				+	6.16.290.	b1.
15.	50	57.16.13.627.	C.				– 029. b2.
	>			+	6.16.261.	c1.
16.	00	57.22.29.888.	D.				– 029. c2.
				+	6.16.232.	d1.
16.	10	57.28.46.120.	E.

	° ′ ″ dec
Star’s longitude,	57.16.35.925.
Moon’s longitude at 15. h. 50 m (C)	57.16.13.627.
difference,	22.298.

As c1 6.′ 16.″ 261 dec, to 10 minutes, so is 22.″ 298. dec to 35 Sec. 557 dec. the time, nearly approximated.

The equation arising from 35. Sec. 557 dec., and the second difference –029. amounts to ,0035, which added to 6.′ 16.″ 261 dec, gives 6.′ 16.″ 2645 dec., the Moon’s motion in ten minutes at the approximate time; then,

As 6.′ 16.″ 2645. dec to 10 minutes, so is 22.″ 298 dec, to 35. sec. 556 dec, which added to 15. h. 50 m., gives 15. h. 50. m. 35. Sec. 556 dec., the apparent [time]1 of true conjunction of ☽ and ★ at Greenwich.

		h. m. Sec. dec.
Apparent time of true conjunction	at Greenwich,	15.50.35.556
do	at Washington,	10.42.57.290.
Longitude in time, West,		5. 7.38.266	= 76.° 54.′ 33.″ 990 dec.

By the emersion.

″ dec.
Log. of segment ED,  910.26355.			2.9591821.85.
〃 ☽’s semid: DS, 1006.954 ar. comp			6.9969903.72.
		° ′ ″ dec.
〃 co-sine angle conjunct:		25.18.35.662	9.9561725.57.
EDS
Angle of inclination, CAD,		–7.15.36.890.
Central angle, DSG,		18. 2.58.772.

Log. co-sine ★’s latitude, 4.° 1.′ 59.″ 809. dec. arith. comp.						0.0010769.72.
〃 ☽’s Semidiam: DS, 1006.″ 954 dec lo						3.0030096.28.
〃 co-sine central angle, DSG, 18.° 2.′ 58.″ 772 dec.						9.9780838.38.
		′	″	dec.
diff. apparent long. GS,	–15.		59.	777.	= 959.″ 777 dec.	2.9821704.38.
☽’s apparent long. at emers:	57.	32.	35.	800.
app: long. point of occult.	57.	16.	36.	023.
★’s longitude;	57.	16.	35.	925.
point occult: east of ★,	0.	0.	0.	098.

The apparent longitude of the points of occultation not agreeing exactly with the Star’s longitude, a correction is necessary, which is thus made.—

The interval of apparent time between the immersion and emersion, is = 54. m. 38 sec, or 3278 Sec, and the ☽’s motion in apparent longitude reduced to a parallel to the ecliptic, ABC, 1803.″ 3434 dec.,—the excess at the immersion, 3.″ 127 dec, and at the emersion, 0.″ 098. dec.—then,

		″ dec.		″ dec.
As 1803.″ 3434 dec ∶ 3278 Sec. ∷		0.127.		0.231.
		0.098.		0.178.

which subtracted respectively, from the apparent times of immersion and emersion, gives 9. h. 22. m. 36. sec. 339. dec. for the apparent time of immersion, and 10. h. 17. m. 14. sec 392 dec, for the time of emersion, corrected.

			° ′ ″ dec.
At the corrected time of immersion, the Moon’s true longitude was			56.26.10.255.
Parallax in longitude, then,		+	36.17.597
Moon’s apparent longitude,			57. 2.27.852
diff: of apparent longitude,		+	14. 8.073.
apparent longitude of the point of occultation, agreeing			57.16.35.925.
with the ★’s longitude,
	° ′ ″ dec
At the corrected time of emersion, the Moon’s true long: was			57. 0.26.821.
Parallax in longitude, then,		+	32. 8.881.
Moon’s apparent longitude,			57.32.35.702.
diff: of apparent longitude, as above,		–	15.59.777.
Apparent longitude of the point of occultation, agreeing			57.16.35.925.
with the Star’s longitude,
			′ ″ dec.
Parallax in longitude at the immersion, corrected,			36.17.597.
apparent difference of longitude,		+	14. 8.073.
True difference of long. of the ★ and ☽’s center, at immersion,			50.25.670.

In recomputing the angles of conjunction and central angles, (which has been done in a similar communication to bishop Madison,) a small error has been discovered, which makes the excess of the difference of apparent longitude between the star and point of occultation to be the same at the emersion as at the immersion, viz: 0.″ 121. dec.—The mean of times of ecliptical conjunction of the Moon and Star at the Capitol, in Washington, is found to be 10. h. 42. m. 57. sec. 562 dec., and the time at Greenwich, which has also been re-computed, = 15. h. 50. m. 35. sec. 557 dec., differing only ^{¹⁄₁₀₀₀} part of a second from the former; —from which the difference of longitude, in time, between the meridians, is = 5. h. 7. m. 37. sec. 995. dec. or 76.° 54.′ 29.″ 975 dec.; hence, without a sensible error, the longitude [of] the Capitol, in Washington, admitting the ratio of the equatorial to the polar axis of the Earth to be as 334 to 333, may be estimated at 5. h. 7. m. 38 sec., or 76.° 54.′ 30.″ west of Greenwich.—

As some of the essential elements used in the operation have been calculated again, and brought to minute exactness, and great care taken to have all of them correct, the accuracy of the result will not, it is presumed, be questioned by those who are capable of understanding the process; and if I were not myself satisfied in this respect, I should not have ventured to submit with the confidence I shall do, to yourself and other scientific gentlemen, particularly those who are natives of the United States, an undertaking which has for its object an entire abolition of one of the sources of dependence on a foreign nation, of whose conduct to us for a series of years, let every American attached to the constitution, laws and soil of his own country, be an impartial judge.

It will be remembered, that a variance must necessarily exist between the longitude of any two places on the Earth, considered as a spheroid, and when reduced by any assumed ratio of the equatorial to the polar diameter, to a sphere, as referred to it’s center; this difference in the distance between this place and Greenwich, will probably amount to 2 or 3 minutes of longitude: but it has always been customary in a determination of the longitude from solar eclipses or occultations (the best methods hitherto discovered) to make an allowance for the spheroidical form of the Earth: the ratio used in this computation makes the figure approach more towards a sphere than the proportion of 230 to 229.

If you can find leisure amidst the large mass of letters which no doubt, you are daily in the habit of receiving from all quarters, and the more pleasing avocations of domestic life in your retirement at Monticello, to favor me with your opinion on the subject of this communication, it will be gratefully acknowledged.