I transmit to the American philosophical Society, an abstract of such calculations as I have hitherto made to determine the longitude of the Capitol in this city from Greenwich observatory, in England. The variation in the results will be found, on examination, to arise more from probable errors in the data, or in the lunar tables from which the positions in the Nautical Almanac have been computed, than from any mistake in the process or principles of calculation.

The first result is founded on an occultation of a Tauri, (Aldebaran) by the Moon, observed by Mr Andrew Ellicott, on the 21st of January, 1793;the apparent times of contacts, and the latitude of the place of observation, as extracted from the fourth volume of transactions of the American philosophical society, and furnished by their Secretary, are as follow:—

		h. m. Sec
January 21st 1793.	Immersion,	at 7.55.49½.		P.M.
	Emersion,	at 9.25.21½

 Latitude of the place, 38.° 52.′ 40.″ North.

Estimated longitude, (assumed for the computation, 5. h. 7. m. 35.8 Sec. = 76.° 53.′ 57.″

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The British impression of the Nautical Almanac, gives the following calculated times at Greenwich—

	h. m.			′
January 21st 1793	14.32½	Im:	of a ♉︎, ★,	12½	N: of ☽’s center.
	14.57½	Em:		14–

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The Moon’s positions at noon and midnight, by the meridian of Greenwich, are taken from the Nautical Almanac, and set down, with their successive differences, as follow:—

1793			° ′ ″
January	20th	Midn:	53.46.59. A.	° ′ ″
				+ 6.12.35.	a1.	′ ″
	21st	Noon,	59.59.34. B.			– 3.14.	a2.	″
				+ 6. 9.21.	b1.			+ 24.	a3.	″
	〃	Midn.	66. 8.55. C.			– 2.50.	b2.			0. a4.
				+ 6. 6.31.	c1.			+ 24.	b3.
	22.	Noon,	72.15.26. D.			– 2.26	c2.
				+ 6. 4. 5	d1.
	〃	Midn:	78.19.31. E.

Moon’s Latitude (South.)

1793.			° ′ ″
January	20th	Midn.	4.46. 3. A	′ ″
				+ 10.56.	a1.	′ ″
	21.	Noon,	4.56.59. B.			– 3.31.	a2	′ ″
				+  7.25.	b1.			– 0.2	a3	″
	〃	Midn:	5. 4.24. C.			– 3.33.	b2.			+5.a4.
				+  3.52.	c1.			+ 0.3.	b3
	22.	Noon,	5. 8.16. D.			– 3.30.	c2
				+  0.22.	d1.
	〃	Midn.	5. 8.38. E.

Supposing the positions in longitude and latitude stated in the Nautical Almanac, to be strictly correct, and admitting the ratio of the equatorial diameter to the polar axis of the Earth, to be as 320 to 319, the Star’s longitude and latitude, and the true conjunction of the Moon and Star at Greenwich, have been found by calculation, of which the following is an abstract—

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For Greenwich.

By the immersion.—		° ′ ″ dec.
Latitude by observation,		51.28.40.000.	N.
〃  reduced, (320 to 319.)		51.18.10.469.
Obliquity of the Ecliptic, January 21st 1793		23.27.48.324.
Constant log. to reduce the Moon’s equat. hor. par:		9.9991690.
for lat. and ratio
		° ′ ″ dec.
apparent time of immersion, (Naut. Alm.) 14. h. 32½ m. =		218. 7.30.000.
Sun’s right ascension,		304.56.14.484.
Right ascension of the meridian from beginning of ♈︎,		163. 3.44.484.
ditto, from beginning of ♑︎, (West)		106.56.15.516.
Altitude of the nonagesimal,		49.57.12.984.
Longitude of the nonagesimal from beg: of ♈︎,		141.22.37.928.
Moon’s true longitude, by the tables,		67.26.46.197.
〃   true distance from the nonagesimal (West)		73.55.51.731.
〃   true latitude, (South)		5. 5.30.891.
〃  equatorial horizontal parallax,		0.55. 6.035
〃  horizontal parallax, reduced, (320 to 319)		0.54.59.715
〃  parallax in longitude,		0.40.45.036.
〃  apparent dist: from nonag: (West)		74.36.36.767.
〃  parallax in latitude		0.36.20.466
〃  apparent latitude, (South)		5.41.51.357.
〃  augmented Semidiameter, arising from appt alt.		0.15.  3.634.

By the Emersion.

	° ′ ″ dec.
apparent time of emersion, (Naut. Alm.) 14. h. 57½ m. =	224.22.30.000
Sun’s right ascension,	304.57.20.165.
Right ascension of the meridian, from beg: of ♈︎,	169.19.50.165.
ditto, from the beginning of ♑︎. (West)	100.40. 9.835.
Altitude of the nonagesimal,	47.56.42.656.
Longitude of the nonagesimal from beginning of ♈︎,	145.50.20.270.
Moon’s true longitude,	67.39.31.255
〃 true distance from the nonag: (West)	78.10.49.015
〃 true latitude, (South)	5. 5.40.946.
〃 Equatorial horizontal parallax,	0.55. 5.549.
〃 horizontal parallax, reduced, (320 to 319)	0.54.59.230.
〃 parallax in longitude,	0.40.12.942.
〃 apparent dist: from the nonag. (west)	78.51. 1.957.
〃 parallax in latitude	0.37.27.504.
〃 apparent latitude, South,	5.43. 8.450.
〃 augmented Semidr arising from appt alt:	0.15. 2.126.

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The difference of apparent longitude and latitude of the Moon and Aldebaran, may be found accurately by the following method, using an assumed latitude of the Star, near the truth.—

			° ′ ″
Moon’s apparent lat. at imm: South			5.41.51.357
diff: of lat. ☽ and ★, (Naut. Alm.)			– 12.30 –––
	° ′ ″		5.29.21.357
Moon’ apparent lat. at emersion,	5.43.8.450.
diff: of lat. ☽ and ★, (Naut. Alm)	– 14.0 –––
			5.29. 8.450.
Mean, assumed latitude of the Star,			5.29.14.903.

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	° ′ ″
Moon’s apparent longitude at emers:	66.59.18.313.
ditto, at imm:	66.46. 1.161.
〃 motion in apparent longitude	〃 13.17.152.

Moon’s motion in apparent long:		797.″ 152 log.		2.9015412.
Star’s assumed lat. 5.° 29.′ 14.″ 903		cosine		9.9980051.
Moon’s motion in apparent long. reduced		793.″ 499. dec.		2.8995463.
to same parallel to ecliptic as ★,

 

Moon’s motion in appt lat. during transit, 77.″ 093. log.	1.8870149.
〃 motion in app: long. reduced, 793.″ 499. ar. comp. log.	7.1004537.
Angle of inclination, 5.° 32.′ 47.″ 116. tang.	8.9874686.
Moon’s motion in apparent long. reduced 793.″ 499. dec log.	2.8995463.
Angle of inclination, 5.° 32.′ 47.″ 116. dec ar. comp: cosine	0.0020380.
Chord of transit, 797.″ 231. log.	2.9015843

Messrs Ferrer and Garnett, in their calculation of the longitude of Kinderhook, in the state of New York, from the solar eclipse of June 16th 1806, found the inflexion of the Moon’s light to be –2.″ 977; that quantity will be applied in this case.—

	″	″	″
Moon’s augmented Semr at imm:	903.634	– 2.977, =	900.657
do do at emer:	902.126.	– 2.977. =	899.149.
Sum of the Moon’s Semidiameters, corrected			1799.806.

	″
Chord of transit,	797.231	ar. comp. log.	7.0984157.
Sum of Semidrs, corrected,	1799.806	log.	3.2552256.
diff: of Semidiameters,	1.508	log.	0.1784013.
	(x) 3.404	log.	0.5320426.
″	″
Chord of transit, 797.231. ± (x)	800.635. (a)
	793.827. (b)

(a) = 800.″ 635. dec	log.			2.9034346.
☽’s cor. semr at imm.	900.″ 657. × 2 =	1801.″ 314		6.7444106.
	° ′ ″	ar. com. log.
angle of conjuncn	63.36.37.388.	cosine		9.6478452
angle of inclinat:	–5.32.47.116.
Central angle	58. 3.50.272.	at the immersion.

(b) = 793.″ 827		log.	2.8997258.
☽’s corr: semr at emers.	× 2 = 1798.″ 298.	ar. co. log.	6.7451384.
	° ′ ″
angle of conjunction,	63.48.16.892.	cos:	9.6448642.
angle of inclination,	+5.32.47.116.
Central angle	69.21.  4.008.	at the emersion.

Star’s assumed lat: 5.° 29.′ 14.″ 903. ar. co. cosine,	0.0019949.
☽’s corrected Semr at immersion, 900.″ 657. log.	2.9545595.	log.	2.9545595.
Central angle at ditto  58.° 3.′ 50.″ 272. cos.	9.7234329.	Sine	9.9287231.
diff: of app: longitude, 7.′ 58.″ 614. = 478.″ 614. log.	2.6799873.	log.	2.8832826.
		764.″ 333. dec
		=12.′ 44.″ 333. diff. of
		apparent lat.
Star’s assumed lat. 5.° 29.′ 14.″ 903 ar. co. cosine,	0.0019949.
☽’s corrected Semr at emersion, 899.″ 149 log.	2.9538316	log.	2.9538316.
Central angle, at ditto 69.° 21.′ 4.″ 008. cosine	9.5473318.	Sine	9.9711640
diff: of app. long. 5.′ 18.″ 536 = 318.″ 536. log.	2.5031583.	log.	2.9249956.
		841.″ 387. =
		14.′ 1.″ 387. diff. of
		apparent lat.

		° ′ ″
Moon’s apparent latitude, at immersion, (South)		5.41.51.357.
diff: of apparent lat. found above		– 12.44.333.
Star’s lat. by computation from imm. (South)	° ′ ″	5.29. 7.024
Moon’s apparent lat. at emersion,	5.43.8.450.
diff: of apparent lat. as above,	– 14.1.387
		5.29.7.063.
Mean result Star’s lat. (South) by comput:		5.29.7.043.
		′ ″ dec.
Moon’s parallax in longitude, at the immersion		– 40.45.036
diff: of apparent longitude ☽ and ★,		+  7.58.614.
true difference of longitude ☽ and ★,		– 32.46.422.
Moon’s true longitude, immersion at Greenwch		67.26.46.197.
Star’s longitude, found by the immersion,		66.53.59.775.
parallax in longitude at the emersion,		– 40.12.942.
diff: of apparent long. ☽ and ★,		–  5.18.536.
true diff: of longitude ☽ and ★,		– 45.31.478.
Moon’s true longitude at emersion,		67.39.31.255.
Star’s longitude, found by the emersion,		66.53.59.777.
〃 do by the immersion,		66.53.59.775.
Mean result Long. of the &, by compn		66.53.59.776.
Supposing the ☽’s positions to be correct.

The difference of apparent longitude of ☽ and ★, may also be obtained by the following process—

		″
Moon’s corrected Semidr at imm: in seconds, &c.		900.657
diff: of apparent lat. ☽ and ★,		764.333.
	sum,	1664.990. log.	3.2214116.
	diff.	136.324. log.	2.1345723.
			2)5.3559839
			2.6779919.
Star’s lat. 5.° 29.′ 7.″ 043. dec. ar. comp. cosine			0.0019933.
diff: of apparent longitude, 7.′ 58.″ 614 = 478.″ 614 =			2.6799852
		″
Moon’s corrected Semidr at emersion,		899.149
diff: of apparent lat.		841.387.
	Sum,	1740.536. log.	3.2406830.
	diff.	57.762. log.	1.7616422
			2)5.0023252
			2.5011626.
Star’s lat. 5.° 29.′ 7.″ 043. ar. co. cosine,			0.0019933.
diff: of apparent long.  5.′ 18.″ 534. = 318.″ 534. log.			2.5031559

The true conjunction of Moon and Star at Greenwich happened when the Moon had the same longitude as the Star, which, by accurate calculation, is found to have been at 13. h. 28. m. 16. Sec. 358. dec.

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For the place of observation at Washington.

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	° ′ ″
Latitude, by Mr Ellicott,	38.52.40.000. N.
reduced, (320 to 319)	38.42.09.512.
Const. log. to reduce ☽’s equat. hor. par. for lat: & ratio,	9.9994655
	° ′ ″
Estimated long. from Greenwich, 5. h. 7. m. 35.8 Sec. =	76.53.57. West.
	° ′ ″
Apparent time of immersion, 7. h. 55. m. 49½ Sec =	118.57.22.500.
Sun’s right ascension correspg time at Greenwch	304.52.20.420
Right ascension of meridn from beg. of ♈︎,	63.49.42.920.
ditto, from beginning of ♑︎ (East)	153.49.42.920.
Altitude of the nonagesimal,	72.51.36.375.
Longitude of the nonagesimal, from beg. of ♈︎,	68.53.15.150.

	° ′ ″
☽’s true longitude, (Naut. Alm:)	66.41.18.569.
〃 true distance from the nonagesimal, (West)	2.11.56.581.
〃 true latitude, (South)	5. 4.52.981.
〃 equat. horizontal parallax,	0.55. 7.767.
〃 horizontal parallax, reduced	0.55. 3.699.
〃 parallax in longitude,	0. 2. 3.507.
〃 apparent distance from the nonag: (west)	2.14. 0.088.
〃 parallax in latitude,	0.21. 7.919.
〃 apparent latitude, (South)	5.26. 0.900.
〃 augmented Semidr arising from apparent alt	0.15.15.190.

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	° ′ ″ dec.
Apparent time of emersion, 9. h. 25. m. 21½ Sec. =	141.20.22.500
Sun’s right ascension correspg time at Greenwch	304.56.15.680.
Right ascension of the meridian from ♈︎,	86.16.38.180.
ditto from beg. of ♑︎, (East)	176.16.38.180.
Altitude of the nonagesimal	74.43.18.614.
Longitude of the nonagesimal from ♈︎,	86.59.20.658
Moon’s true longitude, (Naut. Alm.)	67.27. 0.123.
〃 true distance from the nonag. (West)	19.32.20.535.
〃 true latitude, (South)	5. 5.31.078.
〃 Equat. horizontal parallax	0.55. 6.024.
〃 horizontal parallax, reduced,	0.55. 1.957
〃 parallax in longitude,	0.18. 5.329.
〃 apparent distance from the nonag.	19.50.25.864.
〃 parallax in latitude.	0.19. 9.004.
〃 apparent latitude (South)	5.24.40.082
〃 augmented Semidr arising from appt alt.	0.15.14.021.
Star’s latitude, by computation, (South)	5.29. 7.043
Moon’s apparent lat. at immersion,	5.26. 0.900.
diff: of apparent lat. ★ south of ☽’s center,	– 3. 6.143.
Star’s latitude (South)	5.29. 7.043.
Moon’s apparent lat. at emersion,	5.24.40.082.
diff: of apparent lat. ★ South of ☽’s center,	– 4.26.961.

For the difference of apparent long. of ☽ and ★.

		″	″	″
Moon’s augmented Semr	at imm:	915.190 –	2.977  =	912.213.
ditto	at emers:	914.021 –	2.977. =	911.044.

		″
Moon’s corrected Semr at immersn		912.213
diff: of apparent lat. do		186.143.
		1098.356	3.0407431.
		726.070	2.8609785.
			2)5.9017216
			2.9508608.
Star’s lat.	5.° 29.’ 7.” 043	ar. comp. cos.	0.0019933.
diff: of app: long.	+ 14.′ 57.″ 127	= 897.″ 127	2.9528541.
		″
Moon’s augmented Semr at emers:		911.044
diff: of apparent lat. at do		266.961
sum,		1178.005. log.	3.0711471.
diff.		644.083. log.	2.8089418
			2)5.8800889
			2.9400444
Star’s lat. 5.° 29.′ 7.″ 043 ar. comp. cos.			0.0019933.
diff. of app. long. –14.′ 35.″ 060 = 875.″ 060 dec.			2.9420377.

	′ ″
difference of apparent longitude at imm.	+ 14.57.127.
parallax in longitude,	–  2. 3.507.
true difference longitude at the immersn	+ 12.53.620.
difference of apparent long. at emersion,	– 14.35.060.
parallax in longitude,	– 18. 5.329.
true diff. of longitude, at emersion,	– 32.40.389.

The Moon’s hourly velocity in longitude, at a middle time between the immersion and estimated time of true conjn at Washington, was 30.′ 37.″ 842, and between the true conjunction and emersion, 30.′ 36.″ 981

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As hourly velocity, = 30.′ 37.″ 842, to one hour or 60 min: so is true difference of longitude, 12.′ 53.″ 620, to the interval of time, = 25. m. 15. Sec 382 , which added to 7. h. 55. m. 49. Sec. 500. dec gives 8. h 21. m. 4. Sec. 882 dec, the time of true conjunction at Washington, found by the immersion.

As hourly velocity, = 30.′ 36.″ 981 dec, to one hour, or 60. min So is true difference of longitude, 32.′ 40.″ 389, to the interval of time, = 1. h. 4. m. 1. Sec. 847 dec, which subtracted from 9. h. 25. m. 21½. Sec gives 8. h. 21. m. 19. Sec 653, the time of true conjunction at Washington found by the emersion.

	h. m. Sec. dec
By the immersion,	8.21. 4.882.
〃 the emersion,	8.21.19.653.
Mean result true conjunction at Washn	8.21.12.267.
ditto at Greenwich,	13.28.16.358.
Longitude in time	5. 7. 4.091.
	° ′ ″
equal to	76.46. 0.136.

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On the foregoing principles, an occultation of n Pleiadum, (Alcyone) by the Moon, on the 20th of October, 1804, has been computed, the apparent times, reduced to the Capitol, are as follow:—

			h. m. Sec. dec
	October 20th 1804	Immersion at	9.22.28.33		P.M.
		Emersion, at	10.17.22.72.

The British impression of the Nautical Almanac, gives the following calculated times at Greenwich:—

		h. m
	October 20th 1804,	16. 8½. Imm.	of n Pleiad: +	1.⅖ N.	of ☽’s center
		17.15¼. Em:		1 ⅕ S.

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		h. m. Sec. dec
True Conjunction at Greenwich	at	15.50.28.105.
Do at Washington,		10.43. 0.066.
Longitude in time, West,		5. 7.28.039.
		° ′ ″
	equal to	76.52. 0.585.

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Annular eclipse of the Sun, observed at a place N. 71.° W. one mile, ^⅜, American measure, from the Capitol, on the 17th September, 1811; the apparent times, as follow:—

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		h. m. Sec
	Beginning of the eclipse, at	0.22. 9.		P.M.
	Annulus formed at	2. 2. 6
	do broken, at	2. 6.53
	End of the eclipse at	3.36.53.

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		h. m. Sec. dec
True Conjunction at Greenwich		6.57.14.915.
At Washington,
By first external contact		1.49.20.078.
〃 Second ditto		1.48.32.111.
〃 first internal contact,		1.48.42.823
〃 Second do		1.48.28.692.
Rule 1st		5. 7.54.837.
2		5. 8.42.804.
3		5. 8.32.092.
4		5. 8.46.223.
Mean Long. of the place of observn		5. 8.28.989
	equal to	77. 7.14.835.
diff: of longitude to the Capitol,		– 1.26.978.
By Solar Eclipse, September, 17th 1811		77. 5.47.857.
〃 occultation of n Pleiad: Oct: 20. 1804		76.52. 0.585.
〃 occultation of a Tauri, Jany 21. 1793.		76.46. 0.136.
Longitude of the Capitol W. of Greenwch		76.54.36.193.

This is supposed to be a near approximation to the truth; the errors in the lunar tables being much diminished in their effect, by several observations made at considerable intervals of time.