Thomas Jefferson Papers
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William Lambert to Thomas Jefferson, 8 January 1812

From William Lambert

City of Washington, January 8th 1812.

Sir,

I have the honor to transmit an abstract of the calculation of the longitude of Monticello west of Greenwich, founded on the apparent times of the internal contacts of Sun and Moon on the 17th of September last, as contained in your letter of the 29th of December; and having ascertained the elements with scrupulous exactness, tested by various rules, the accuracy of the result, according to the data furnished, may be confidently relied on.



°   
 Lat. of Monticello, by observation 38.° 8.′ 0.″ N. reduced, (320 to 319) 37.57.33.341.
Constant log. to reduce the Moon’s equat. hor. parallax, for the lat. and ratio 9.9994827.
°   
 Obliquity of the Ecliptic, Sept. 17th 1811 23.27.42.690 
h. m. Sec. °   
 Estimated longitude of Monticello, supposed near the truth 5.15.20 = 78.50.0.W.


h. m. S.   °   dec. 
 Annulus formed 1.53. 0   = 28.15. 0.000 
 Estimated long. from Greenwich +5.15.20 
 Corresponding time at Greenwich 7. 8.20. ☉’s R.A. 174.26.55.519.
 Right ascension of the meridian, ♎,  22.41.55.519.
 Altitude of the nonagesimal, 46.44. 3.732 
 Longitude of the nonagesimal, ♎.  2.41.31.560 
 Moon’s true longitude ♍.  24. 2. 7.711 
  true distance à nonagesimal, (West) °   8.39.23.849.
  hor. parallax, reduced, (320 to 319) 0.54.5.916
 Sun’s hor. parallax,  0.8.700
 hor. parallax ☽ à ☉, 0.53.57.216 
 ☽’s parallax in longitude, (correct) 0. 5.58.862 
  apparent distance à nonagesimal, 8.45.22.711 
  true latitude, north ascending 0.37.20.676.
  apparent longitude ♍.  23.56. 8.849.
 Sun’s longitude, ♍.  23.57. 7.341.
 diff. of apparent longitude, ☽ west of ☉, +  0. 0.58.942 
 ☽’s parallax in latitude (correct) 0.36.58.430 
  apparent latitude, north,   dec   0. 0.22.246.
  horizontal Semidiameter 14.45.595  }   
 Augmentation, + 0.10.185  Semidiam. corrected. 14.52.803
 Inflexion of light – 0. 2.977.
 ☉’s Semidiameter, 15.57.246.
 Irradiation of light, – 1.623  Semidiam. corrected. 15.55.623
 difference of Sun and Moon’s Semidiameters, corrected  1. 2.820.


h. m. Sec °   dec. 
 Annulus broken, 1.59.25 =  29.51.15.000 
 Sun’s right ascension, (corresponding time at Greenwich) 174.27. 9.923.
 Right ascension of the meridian, ♎, 24.18.24.923.
 altitude of the nonagesimal, 46. 5.42.940 
 Longitude of the nonagesimal, ♎, 4.12.14.333.
 Moon’s true longitude, ♍, 24. 5.17.322.
  true dist. à nonagesimal, (West) °     10. 6.57.011 
  hor. parallax, reduced 0.54.5.942
 Sun’s hor. parallax, – 0.8.700
 hor. parallax, ☽ à ☉, 0.53.57.242 
°   dec.
☽’s Parallax in longitude (correct) 0. 6.54.272
 apparent distance à nonagesimal 10.13.51.283.
 true latitude, north ascending 0.37.38.905 
 apparent longitude ♍. 23.58.23.050 
Sun’s longitude, ♍, 23.57.23.018.
diff. of apparent longitude, ☽ East of ☉,  0. 1. 0.032.
☽’s parallax in latitude, (correct) 0.37.24.643.
 apparent latitude (north) 0. 0.14.262.
  dec. 
 horiz. Semidiameter, 14.45.604. }
 Augmentation, +10.028. Semid. corrected, 0.14.52.655 
 Inflexion of light, – 2.977 
Sun’s Semidiameter, 15.57.247. Semid. corrected. 15.55.622 
Irradiation of light – 1.623 
difference of Sun and Moon’s Semidiameters, corrected. 0. 1. 2.967 


 1st internal contact 2d
   
diff: of Semids 62.820 diff. of Semids   62.967 
☽’s apparent lat. 22.246 14.262 
Sum,  85.066 log. 1.9297560  77.229. log.   1.8877804 
diff. 40.574 log. 1.6082478  48.705. log. 1.6875735 
2)3.5380038  2)3.5753539 
1.7690019. 1.7876769.5
0.0000000 
 ☽’s apparent lat. co.sine, ar. comp. +0.0000000  diff. of app. long. 1.7876769.5
     
diff: of apparent ☉ and ☽, +  58.749  1.7690019. – 1. 1.330.
 Parallax in longitude –5.58.862. Parallax in long. – 6.54.272.
true diff. of long. ☉ & ☽, –5. 0.113. true diff. long. – 7.55.602.

The Moon’s hourly velocity of the Moon from the Sun, at a middle time between the formation of the annulus and the true conjunction of the Sun and Moon at Monticello, was 27.′ 6.″ 0328; and between the breaking of the annulus and the true conjunction, 27.′ 6.″ 0505. dec.



As 27.′ 6.″ 0328 to one hour, or 60 minutes, so is true diff. of long. ☉ and ☽, 5.′ 0.″ 113. to the interval of apparent time, which subtracted from 11. m. 4. Sec. 443. dec which subtracted from 1. h. 53. m. 0, S. the time of the formation of the annulus, gives 1. h. 41. m. 55. Sec. 557. dec the time of true conjunction of Sun and Moon at Monticello, by the first internal contact.

As 27.′ 6.″ 0505. to one hour, or 60 minutes, so is true difference ☉ and ☽, 7.′ 55.″ 602, to 17. m. 32. Sec. 960, which subtracted from 1. h. 59. m. 25. Sec gives 1. h. 41. m. 52. Sec. 040 dec., the time of true conjunction, by the second internal contact.

h. m. Sec. dec 
 1st 1.41.55.557.
 2 1.41.52.040.
Mean true conjunction at Mont. 1.41.53.798 
 ditto at Greenwich, 6.57.14.915.
Longitude in time, West, 5.15.21.117.  =  78.° 50.′ 16.″ 755. dec.

Another method.



   dec }
Moon’s apparent motion in lat. during the annular appearance,  7.984. log + 10 =
 dec   10.9022205 
  apparent motion in longitude, 118.974  log.  2.0754521.
°    
tangent, angle inclination, 3.50.21.108 8.8267684 
Moon’s apparent motion in longitude  log. 2.0754521.
angle of inclination, ar. co. cosine + 0.0009757 
 
Chord of transit, 119.242  log. 2.0764278.
 
diff. of Semidiameters, 62.820   (t)
 & 62.967   (u)
 Sum, 125.787.  (v)
 diff. 0.147   (w)


 
 As chord of transit, 119.242.  log. co. ar.  7.9235722
 To (v) 125.787.  log 2.0996358
 So (w) 0.147   log 9.1673173
 To (w) 0.155.  log 9.1905253
 
Chord of transit, –x, =  119.087. half   59.5435 (r)
do +x, =  119.397. half, 59.6985 (s)
 
 Log. (r) 59.5435 + 10 11.7748343 
 Log. (t) 62.820 °      1.7980979  
Angle of conjunction 18.35.11.714   9.9767364  
  of inclination, +3.50.21.108 
Central angle 22.25.32.822   cosine 9.9658480.  sine. 9.5814791.
 
 diff. of Semidiameters   62.820   log. 1.7980979.  log. 1.7980979.
 diff. of apparent longitude, +58.069  log 1.7639459   log. 1.3795770 
app. lat.
 
   23.965.
 (s) 59.6985  log. + 10 11.7759634
 (u) 62.967  log.  1.7991061
°    
angle of conjunction, 18.32.20.714.  Cosine  9.9768573
Angle of inclination –3.50.21.108  
Central angle, 14.41.59.606    cosine 9.9855469  sine  9.4044164
 (u)  62.967  log.  1.7991061  log. 1.7991061
diff. of apparent long. –60.905.  log.  1.7846530  log. 1.2035225
15.″ 978. app. latitude.
   dec.   
 Parallax in long. –5.58.862 } Parallax in long. –6.54.272
 diff. of app. long. +0.58.069  diff: of app. long. –1. 0.905
true diff. long. ☉ and ☽,  –5. 0.793. true diff. long. ☉ & ☽,  –7.55.177.

As hourly velocity ☽ à ☉, 27.′ 6.″ 0328 to one hour, or 60 minutes, so is true diff. long. –5.′ 0.″ 793. to 11. m. 5. s. 949, which subtracted from 1. h. 53. m. 0. S gives 1. h 41. m. 54. Sec. 051. dec. the time of true conjunction of Sun and Moon at Monticello, by the formation of the annulus.

As hourly velocity ☽ à ☉, 27.′ 6.″ 0505. to one hour, or 60 minutes, so is true difference of longitude, –7.′ 55.″ 177, to 17. m. 32. Sec. 021 dec, which subtracted from 1. h. 59. m. 25. S, gives 1. h. 41. m. 52. Sec 979 the time of true conjunction, by breaking of annulus.—

h. m. Sec. dec.
 By formation of annulus 1.41.54.051 
 〃 breaking of ditto 1.41.52.979.
True conjunction ☉ & ☽, at Monticello, 1.41.53.515.
  at Greenwich, 6.57.14.915   °   dec.
Longitude in time, West 5.15.21.400   =  78.50.21.000
 By first method 78.50.16.755.
 Mean result 78.50.18.877.

The above may be considered as an accurate determination of the longitude of Monticello, by the internal contacts, supposing the latitude of the place, the apparent times of formation and breaking of the annulus, and the Sun and Moon’s positions in the Nautical almanac, to be correctly given. The last method may be explained by the following figure.—

The line ESC, = FDG, represents a small portion of the ecliptic, passing through the center of the Sun, S, equal to the Moon’s apparent motion in longitude from the Sun, during the appearance of the annulus.

 F, the Moon’s center at the formation, B, at the breaking of the annulus.

 FS, the difference of the Sun and Moon’s semidiameters (corrected) at the beginning, BS, at the end. EF, the Moon’s apparent latitude at the beginning, CB, at the end of the annular appearance. SA, the nearest approach of the centers of ☉ and ☽. GFB, the angle of inclination, FAB, the chord of transit, or the Moon’s motion in the apparent orbit. AFS, the angle of conjunction at the beginning, ABS, at the end. FSE, and BSC, the central angles, from which, the difference of apparent longitude of Sun and Moon, SE, at the formation, and SC, at the breaking of the annulus, may be correctly ascertained, as in the foregoing process.

I am, Sir, with great respect, Your most obedient servant,

William Lambert.

RC (DLC); on two folio sheets; at foot of text: “Thomas Jefferson, late President U.S.” Enclosed in Lambert to TJ, 9 Jan. 1812.

Index Entries

  • astronomy; and calculations of Monticello’s longitude search
  • astronomy; and solar observations search
  • geography.; and TJ’s calculations of latitude search
  • Lambert, William; calculates Monticello’s longitude search
  • Lambert, William; letters from search
  • Monticello (TJ’s estate).; latitude of search
  • Monticello (TJ’s estate).; longitude of search
  • sun; and calculation of longitude search
  • sun; annular eclipse of1811observed search