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.
