Enclosure: William Lambert’s Methods of Calculating the Moon’s Parallaxes, 27–28 June 1822
Enclosure
William Lambert’s Methods of Calculating the Moon’s Parallaxes
To find the Moon’s parallaxes in longitude and latitude, independent of the altitude and longitude of the nonagesimal.
An example will be taken from the report relative to the longitude of the City of Washington, in the case of the immersion of η Pleiadum, (Alcyone) October 20th 1804.
With the Moon’s true longitude, 56.° 26.′ 12.″ 93, latitude 4.° 30.′ 25.″ 30. dec. N. and the obliquity of the Ecliptic, 23.° 27.′ 54.″ 25. dec compute the Moon’s Right Ascension, 52.° 58.′ 27.″ 81, and declination, 23.° 45.′ 25.″ 45. dec North.
Find the angle between the parallels to the ecliptic and equator, = 13.° 55.′ 2.″ 59 dec, which subtracted from 90°, leaves 76.° 4.′ 57.″ 41. dec the angle between the meridian passing thro’ the Moon’s centre, and a parallel to the Ecliptic.
From the Moon’s Right Ascension, increased by 360.° subtract the Right ascension of the meridian, 346.° 8.′ 35.″ 37, the remainder, 66.° 49.′ 52.″ 44 dec, is the Moon’s horary angle, or distance from the meridian, East.
With the Moon’s horary angle, = 66.° 49.′ 52.″ 44. dec E. declination, 23.° 45.′ 25.″ 45. dec North, and the latitude of the place, reduced, 38.° 42.′ 59.″ 44. dec N. compute the Moon’s true altitude, 32.° 12.′ 16.″ 72 dec, and angle of position, 57.° 58.′ 3.″ 67. dec.
| ° | ′ | ″ | dec | ||
| angle between meridn passing thro’ ☽’s centre, and parl to eclip: | 76. | 4. | 57. | 41. | |
| Angle of position, at do | + | 57. | 58. | 3. | 67 |
| Angle between the vertical circle, and a parallel to eclip: | 134. | 3. | 1. | 08. | |
| or the excess above 90°, (in this case) of the parallactic angle, = 44.° 3.′ 1.″ 08. dec |
With the Moon’s horizontal parallax, reduced, 1.° 0.′ 58.″ 82. dec and the true altitude, 32.° 12.′ 16.″ 72. dec compute the Moon’s parallax in altitude, = 0.° 52.′ 5.″ 05 dec, and apparent altitude, exclusive of refraction, = 31.° 20.′ 11.″ 67. dec.
For the parallax in latitude.
| ° | ′ | ″ | dec | |||
| Moon’s hor: parallax, reduced, | 1. | 0. | 58. | 82 | Sine, | 8.2488931. |
| 〃 apparent altitude, | 31. | 20. | 11. | 67 | Cosine, | 9.9315224. |
| angle between vert: cir: and par: to ecl: | 134. | 3. | 1. | 08 | Sine | 9.8565655. |
| Parallax in lat. found nearly. | 0. | 37. | 26. | 03 | Sine | 8.0369810. |
| ° | ′ | ″ | dec | ||
| Moon’s true latitude, North, | 4. | 30. | 25. | 30. | |
| Parall: in lat. found nearly, | − | 0. | 37. | 26. | 03. |
| Moon’s apparent latitude, N: found nearly, | 3. | 52. | 59. | 27. |
For the parallax in longitude.
| ° | ′ | ″ | ||||
| Moon’s hor. parallax, reduced, | 1. | 0. | 58. | 82 | Sine, | 8.2488931. |
| 〃 apparent altitude, | 31. | 20. | 11. | 67 | Cosine, | 9.9315224. |
| Angle between vert: circle and par: to eclip. | 134. | 3. | 1. | 08 | Cosine | 9.8421658. |
| Moon’s apparent lat. found nearly | 3. | 52. | 59. | 27 | ar. co. cosine | 0.0009981. |
| 〃 parallax in longitude, | 0. | 36. | 17. | 77 | Sine | 8.0235794 |
Correction of the Moon’s apparent Latitude.
| Constant log. | 4.7124. | ||||
| Twice sine of par: in longitude, | 16.0471. | ||||
| ° | ′ | ″ | |||
| Sine of twice Moon’s true lat: = | 9. | 0. | 50. | 60 | 9.1950 |
| Correction | − | 0. | 90 | 9.95451 | |
| 3. | 52. | 59. | 27. | ||
| Moon’s app: lat. N. correct | 3. | 52. | 58. | 37. | |
| 〃 true lat. | 4. | 30. | 25. | 30 | |
| Parallax in latitude, | 0. | 37. | 26. | 93. |
The parallactic angle may also be computed by having the altitude of the nonagesimal, the Moon’s true distance therefrom, and the true latitude; for, we have two sides of an oblique angled spherical triangle, viz: the altitude of the nonag: = 49.° 35.′ 51.″ 28 dec, and Moon’s distance from the North pole of the Ecliptic, = 85.° 29.′ 34.″ 70. dec. also, the contained angle, 50.° 35.′ 6.″ 30. = the Moon’s true distance from the nonagesimal given, to find the parallactic angle, = 44.° 3.′ 1.″ 05, and Moon’s true altitude, = 32.° 12.′ 16.″ 74. dec hence the parallax in altitude is found to be 0.° 52.′ 5.″ 05 dec, and the Moon’s apparent altitude, = 31.° 20.′ 11.″ 69. dec exclusive of refraction.
In computing the Moon’s parallaxes in longitude and latitude, we use the log. cosine of the parallactic angle for the parallax in latitude, and the sine for the parallax in longitude;—for the cosine of 44.° 3.′ 1.″ 05 dec, is equal to the sine of 134.° 3.′ 1.″ 05, and the sine of the former, equal to the cosine of the latter. It will be found, on trial, that the small variance in the parallactic angle and Moon’s true altitude, obtained by different methods, will not amount to the 1⁄500 part of a second in the parallaxes.
We shall now proceed to find the parallaxes in longitude and latitude by the method which I generally use for that purpose.
| ° | ′ | ″ | dec | ||||
| Moon’s hor: parallax, reduced, | 1. | 0. | 58. | 82 | Sine, | 8.2488931. | |
| Altitude of the nonagesimal, | 49. | 35. | 51. | 282 | Sine | 9.8816761. | |
| Log. A, | 8.1305692. | log. A. | 8.1305692. | ||||
| Moon’s true dist. à nonag: 50.° 35.′ 6.″ 30 sine. | 9.8879369 | cosine, | 9.8027270. | ||||
| log. B, | 8.0185061 | (C) | 7.9332962 | ||||
| Natural | num: C, | 0085762. | |||||
| Natural cosine Moon’s | true lat. | 9969077. | |||||
| Corresponding log. | 9.9949207. | (D) | 9883315. | ||||
| arith: comp: | 0.0050973. | ||||||
| log. B | + 8.0185061. | ||||||
| tangent, | 8.0236034. | par: in longitude, | |||||
| 0.° 36.′ 17.″ 77. dec | |||||||
which added to the Moon’s true distance, gives 51.° 11.′ 24.″ 07 dec, the Moon’s apparent distance from the nonag: East.
For the Moon’s apparent latitude.
| ° | ′ | ″ | dec | |||
| Moon’s hor: parallax, reduced, | 1. | 0. | 58. | 82 | Sine, | 8.2488931. |
| Altitude of the nonagesimal, | 49. | 35. | 51. | 28 | Cosine | 9.8116769. |
| Corresponding natural number, | − 0114967. | (E) | 8.0605700. | |||
| Natural Sine Moon’s true lat: | 0785814. | |||||
| diff: = Nat. number, (F.) | 0670847 | log. | 8.8266234. | |||
| ° | ′ | ″ | ||||
| Moon’s true dist: à nonag: | 50. | 35. | 6. | 30. | ar. co. s. | 0.1120631. |
| 〃 apparent dist. do | 51. | 11. | 24. | 07 | Sine, | 9.8916650. |
| 〃 true lat. North, | 4. | 30. | 25. | 30. | ar. co. co.s. | 0.0013451. |
| 〃 apparent lat. N. | 3. | 52. | 58. | 36. | tang. | 8.8316966. |
| Parallax in latitude, | 0. | 37. | 26. | 94. | ||
By the first method, we refer to the Equator,—by the latter to the Ecliptic, and the results agree as nearly as might be expected by using seven places of logarithms, besides the index, instead of eight or nine. This process will also shew the care taken to have the Elements correct, in the original calculations, to ascertain the longitude of the Capitol in this City from Greenwich Observatory.
William Lambert.
| City of Washington |  |
| June 27, 1822.3 |
The foregoing example exhibits the process when the Moon had north latitude: the immersion of γ ♉, on the 12th of January, 1813, when the Moon had south latitude will here be taken from page 37. of the printed report.
With the Moon’s longitude, 62.° 31.′ 38.″ 54 dec., latitude, South, 5.° 5.′ 42.″ 58 dec, and the obliquity of the Ecliptic, 23.° 27.′ 43.″ 50 dec, compute the Moon’s Right ascension, = 61.° 29.′ 31.″ 80 dec, and declination, 15.° 41.′ 12.″ 90. dec North.
With the Moon’s lat. Right ascension, and the obliquity of the Ecliptic, find the angle between the parallels to the ecliptic and equator, = 10.° 59.′ 53.″ 64, which subtracted from 90°, gives 79.° 0.′ 6.″ 36 dec, the angle between the meridian passing thro’ the Moon’s centre, and a parallel to the Ecliptic.
From the Moon’s R.A. 61.° 29.′ 31.″ 80 dec, Subtract the R.A. of the meridian, = 20.° 57.′ 45.″ 95 dec, for the Moon’s horary angle, or distance from the meridian, = 40.° 31.′ 45.″ 85. dec. East.
With the Moon’s horary angle, = 40.° 31.′ 45.″ 85. dec E. decl: 15.° 41.′ 12.″ 90. dec N. and the latitude of the place, reduced, = 38.° 42.′ 59.″ 44. dec N. compute the Moon’s true altitude, = 47.° 44.′ 15.″ 23. dec and angle of position, = 48.° 55.′ 54.″ 02. dec
| ° | ′ | ″ | dec | ||
| Angle between the merid. passing thro’ the ☽, and par: to ecl: | 79. | 0. | 6. | 36. | |
| Angle of position, (Moon east of meridian) | + | 48. | 55. | 54. | 02 |
| Angle between the vertical circle, and par: to ecliptic, | 127. | 56. | 0. | 38. |
With the Moon’s hor: parallax, reduced, 0.° 59.′ 24.″ 517 dec, and Moon’s true altitude, = 47.° 44.′ 15.″ 23 dec, compute the parallax in altitude, 0.° 40.′ 28.″ 06, and Moon’s apparent altitude, (exclusive of refraction) = 47.° 3.′ 47.″ 17. dec
For the Moon’s apparent latitude.
| ° | ′ | ″ | |||||
| Moon’s hor. parallx reduced, | 0. | 59. | 24. | 517, | Sine, | 8.2375540. | |
| 〃 apparent Altitude, | 47. | 3. | 47. | 17 | cosine, | 9.8332698. | |
| angle bet. vert. circle and par: to ecl: | 127. | 56. | 0. | 38 | Sine, | 9.8969259. | |
| parallax in lat: found nearly, | 0. | 31. | 55. | 05. | Sine | 7.9677497. | |
| Moon’s true lat. South | + | 5. | 5. | 42. | 58 | ||
| Moon’s apparent lat. nearly | 5. | 37. | 37. | 63. |
For the parallax in longitude.
Correction of the Moon’s apparent latitude.
| Constant log, | 4.7124. | |||||
| Twice sine of parallax in longitude, | ° | ′ | ″ | dec | 15.7232. | |
| Sine of twice Moon’s true latitude, = | 10. | 11. | 25. | 16. | 9.2478. | |
| − | 0. | 484 | 9.6834. | |||
| Moon’s apparent lat. found nearly, | 5. | 37. | 37. | 63. | ||
| do do correct | 5. | 37. | 37. | 15. | ||
| Moon’s true latitude, South, | − | 5. | 5. | 42. | 58 | |
| Parallax in latitude, correct, | 0. | 31. | 54. | 57. |
Method II.
| ° | ′ | ″ | dec | |||||
| Moon’s hor: parallax, reduced, | 0. | 59. | 24. | 517. | Sine, | 8.2375540 | ||
| Altitude of the nonagesimal, | 62. | 26. | 37. | 89 | Sine, | 9.9477072 | ||
| Moon’s true lat. | 5. | 5. | 42. | 585 | ar. comp. cos: | 0.0017195. | ||
| (A,) | 8.1869807. | (A.) | 8.1869807. | |||||
| Moon’s true dist. à nonag. | |
Sine 9.6686983 | Cosine | 9.9467509 | ||||
| 27.° 47.′ 48.″ 04. | ||||||||
| (B.) | 7.8556790. | (C.) | 8.1337316. | 6 | ||||
| Nat. number, | 0136060. | |||||||
| arith: comp. | 9863940. | |||||||
| Corresponding log: | 9.9940505. | ||
| arith: comp: | 0.0059495. | ||
| log. (B) | + | 7.8556790. | |
| tangent, | 7.8616285. | par: in long: 0.° 24.′ 59.″ 84. dec |
For the Moon’s apparent latitude.
| ° | ′ | ″ | dec | |||||
| Moon’s horiz: parallax, reduced, | 0. | 59. | 24. | 517 | Sine, | 8.2375540. | ||
| Altitude of the nonagesimal, | 62. | 26. | 37. | 89 | cosine | 9.6652221. | ||
| Log. (D.) | 7.9027761. | |||||||
| Corresponding natural number, | 0079942. | |||||||
| Natural sine Moon’s true lat. | + | 0888102. | ||||||
| Natural number, (E) | 0968044. | log. | 8.9858951. | |||||
| ° | ′ | ″ | ||||||
| Moon’s true dist. à nonag: | 27. | 47. | 48. | 04. | ar: comp. sine. | 0.3313017. | ||
| 〃 apparent dist: do | 28. | 12. | 47. | 88 | Sine | 9.6746364. | ||
| 〃 true lat. South, | 5. | 5. | 42. | 58. | ar: comp. cosine, | 0.0017195. | ||
| 〃 apparent lat. do | 5. | 37. | 37. | 15 | tangt | 8.9935527. | ||
| diff: = parallax in lat. | 0. | 31. | 54. | 57. | ||||
The agreement of the results obtained by two methods so different in their process, shews the correctness of the principles on which both are founded; and if all the elements are computed with due care, there will seldom be a variance of more than 1⁄100 of a second. To have a perfect coincidence, the logarithms, &c. must be extended to more than seven places of figures, besides the index.
My object in transmitting this communication, is to contribute, in some degree, to the knowledge of practical astronomy at some seminary in my native state: for this purpose, it is respectfully forwarded to you; and if any part requires explanation, it will be promptly attended to.
| William Lambert. |
| June 28, 1822. |
MS (ViU: TJP); written entirely in Lambert’s hand on two sheets, each folded to form four pages, with last page of second sheet blank; at foot of text: “Honble Thomas Jefferson, Monticello, Virginia.”
The nonagesimal is the point of an ecliptic highest above the horizon (). The report relative to the longitude of the city of washington was Lambert, Abstracts of Calculations, to ascertain the Longitude of the Capitol, in the City of Washington, from Greenwich Observatory, in England (Washington, 1817; , 7 [no. 303]; TJ’s copy in MBPLi). The printed report is the enclosure to Lambert to TJ, 18 Mar. 1822.
1. Overscoring, here and below, indicates that the characteristic of the logarithm is negative, while its mantissa remains non-negative.
2. Superfluous degree, minute, and second symbols and “dec” abbreviation above numbers immediately preceding editorially omitted.
3. Page 4 ends short at this point.
4. Superfluous second symbol and “dec” abbreviation above “0” and “48,” respectively, editorially omitted.
5. Superfluous degree, minute, and second symbols and “dec” abbreviation above numbers immediately preceding editorially omitted.
6. Page 6 ends here, with repetition of final line “Brought forward” to head of page 7 editorially omitted.
Index Entries
- Abstracts of Calculations, to ascertain the Longitude of the Capitol, in the City of Washington (W. Lambert) search
- astronomy; and lunar calculations search
- Capitol, U.S.; latitude and longitude measurements at search
- Greenwich Observatory, England; and prime meridian search
- Lambert, William; Abstracts of Calculations, to ascertain the Longitude of the Capitol, in the City of Washington search
- Lambert, William; andMessage from the President of the United States, transmitting a Report of William Lambert, on the subject of the Longitude of the Capitol of the United States. January 9, 1822 search
- Lambert, William; and University of Virginia search
- Lambert, William; astronomical calculations search
- Lambert, William; calculates latitude and longitude of U.S. Capitol search
- Lambert, William; lunar calculations search
- longitude; calculations for U.S. Capitol search
- Message from the President of the United States, transmitting a Report of William Lambert, on the subject of the Longitude of the Capitol of the United States. January 9, 1822 (1822; J. Monroe) search
- moon; and astronomical calculations search
- Virginia, University of; Books and Library; books and manuscripts for search