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William Lambert to Thomas Jefferson, 23 April 1812

From William Lambert

City of Washington, april 23d 1812.

Sir,

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.″

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–

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—

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.

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.
° ′ ″ 
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.

For the place of observation at Washington.

 ° ′ ″
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.
° ′ ″ 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

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.

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.
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.

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:—

  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.
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.

I am, Sir, with great respect, Your most obedt servant.

William Lambert.

RC (PPAmP: APS description begins American Philosophical Society description ends Archives, Manuscript Communications); on five folio sheets; addressed: “Thomas Jefferson late president of the U. States, and president of the American philosophical Society, Monticello, Virginia”; franked; postmarked Washington, 24 Apr.; endorsed by TJ as received 29 Apr. 1812 and so recorded in SJL; endorsed at APS description begins American Philosophical Society description ends : “on the Longitude of Washington read 19th June 1812.” Enclosed in TJ to Robert Patterson, 29 May 1812.

For the action taken by the american philosophical society, see note to Lambert to TJ, 20 Apr. 1812.

Index Entries

  • American Philosophical Society; and W. Lambert search
  • American Philosophical Society; Transactions search
  • astronomy; and calculations of prime meridian search
  • Capitol, U.S.; longitude measurement search
  • Ellicott, Andrew (1754–1820); astronomical observations of search
  • Ferrer, José J. de; calculates longitude search
  • Garnett, John; calculates longitude search
  • Kinderhook, N.Y.; longitude of search
  • Lambert, William; calculates longitude of U.S. Capitol search
  • Lambert, William; letters from search
  • Nautical Almanac and Astronomical Ephemeris (London edition); and W. Lambert’s calculations search
  • New York (state); longitude of Kinderhook search
  • Washington (D.C.); longitude of search

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Jefferson Papers
Title
William Lambert to Thomas Jefferson, 23 April 1812
Author
Lambert, William
Recipient
Jefferson, Thomas
Date
23 April 1812

Reference

Cite as
“William Lambert to Thomas Jefferson, 23 April 1812,” Founders Online, National Archives, https://founders.archives.gov/documents/Jefferson/03-04-02-0540. [Original source: The Papers of Thomas Jefferson, Retirement Series, vol. 4, 18 June 1811 to 30 April 1812, ed. J. Jefferson Looney. Princeton: Princeton University Press, 2007, pp. 651–660.]

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