Thomas Jefferson Papers
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# From William Lambert

City of Washington, April 20th 1812.

Sir,

By a letter from Mr John Garnett, Editor of the American impression of the Nautical Almanac, at N. Brunswick, in New-Jersey, it is stated, that an error has been discovered (probably at Greenwich) in M. de la Place’s computations relating to the true form of the Earth, which being corrected, the ratio of 320 to 319, of the equatorial diameter to the polar axis of the Earth, seems now to be agreed upon as more correct than any of the others formerly used in Astronomical calculations. As this ratio may be considered as a standard, the latitude of any place, north or south, from 0.° to 90°, may be reduced by this Easy process.—

To the constant logarithm 9.9972814, add the log. tangent of the latitude of the place, the sum, (rejecting radius) will be the latitude reduced, according to the above ratio.

But to reduce the Moon’s equatorial horizontal parallax, I have found it convenient in practice, to form a table of fixed logarithms to every degree of latitude from 0.° to 90°, which has been constructed on the following principles—

Let the log. of 320, be called (A), and the log. of 319, (B.) then

 A – B, = C . log. C, + log. cotangent lat. place, by observation, – radius, = log. tangent arch D. – Log. cosine lat. place, by observation, + ar. comp. log. sine arch D, = constant log. for lat: and ratio.
To the constant log. for the lat. add the log. sine of the Moon’s equatorial horizontal parallax, the sum, (rejecting radius) will be the log. sine of the Moon’s equatorial horizontal parallax, reduced.

As the Moon’s equat. hor. parallax never amounts to 1.° 2′—the common log. in seconds and decimal parts, may be substituted for the log. sine; the former is, however, more correct.

Table of logarithms for reducing the Moon’s equatorial horizontal parallax, to every degree of latitude from 0.° to 90°, admitting the ratio of the equatorial diameter to the polar axis of the Earth to be as 320 to 319.

 Lat. Logarithms. Lat. Logarithms Lat. Logarithms. ° ° ° ° 0 10.0000000. 20 9.9998414. 41 9.9994160 62 9.9989411. 1 9.9999996 21 9.9998259 42 9.9993924. 63 9.9989216. 2 9.9999983. 22 9.9998098 43 9.9993688 64 9.9989026 3 9.9999963. 23 9.9997931. 44 9.9993452 65 9.9988842 4 9.9999934. 24 9.9997757. 45 9.9993215 66 9.9988663. 5 9.9999896 25 9.9997578 46 9.9992977. 67 9.9988489. 6 9.9999851. 26 9.9997394. 47 9.9992740 68 9.9988321. 7 9.9999798 27 9.9997205 48 9.9992503. 69 9.9988159. 8 9.9999737. 28 9.9997011. 49 9.9992267. 70 9.9988003. 9 9.9999668 29 9.9996812 50 9.9992033 71 9.9987853. 10 9.9999591. 30 9.9996609. 51 9.9991800 72 9.9987709. 11 9.9999507. 31 9.9996402. 52 9.9991569 73 9.9987572 12 9.9999414. 32 9.9996191. 53 9.9991340. 74 9.9987442. 13 9.9999314. 33 9.9995977. 54 9.9991113. 75 9.9987320. 14 9.9999207. 34 9.9995759 55 9.9990889. 76 9.9987205 15 9.9999092 35 9.9995538 56 9.9990667. 77 9.9987097. 16 9.9998970. 36 9.9995314. 57 9.9990448 78 9.9986997 17 9.9998841. 37 9.9995087. 58 9.9990233 79 9.9986904. 18 9.9998705 38 9.9994858 59 9.9990021. 80 9.9986819. 19 9.9998563. 39 9.9994627. 60 9.9 989813. 81 9.9986742 40 9.9994394. 61 9.9989610. 82 9.9986673. 83 9.9986611. 84 9.9986557. 85 9.9986511. 86 9.9986473. 87 9.9986444. 88 9.9986423. 89 9.9986411. 90 9.9986407.

This table will be found to give the Moon’s horizontal parallax, reduced, for any latitude, with greater accuracy, than any, perhaps, heretofore constructed. The application is easy; and for any intermediate minutes and seconds, take the proportional part of the difference from the preceding logarithm, which, in all cases, will be sufficiently exact.

Examples.

Required the log: for lat. 38.° 53.′ 0″—

The log. for 38.° is 9.9994858, and for 39°, 9.9994627, the difference is 231; the prop. part for 53′, is 204,—which subtracted from 9.9994858, gives 9.9994654, a constant log. for lat. 38.° 53.′—

Required the log. for the lat. of Greenwich,

51.° 28.′ 40.″

The log. for 51°, is 9.9991800, and for 52°, 9.9991569, the difference is 231, and the prop. part for 28.′ 40″, is 110, which subtracted from 9.9991800, gives 9.9991690, the constant log. for the lat. of Greenwich.

I am, Sir, with due respect, Your most obedt servant,

William Lambert.

(: description begins American Philosophical Society description ends Archives, Manuscript Communications); written on one sheet folded to form four pages, with text on first three pages and address on the fourth; first three columns of table of logarithms on p. 2 and fourth column on p. 3, with remainder of text adjacent to it; at head of p. 3: “Table Continued”; addressed: “Thomas Jefferson, late president of the U. States, and president of the American Philosopl Society, Monticello, Virginia”; franked; postmarked Washington City, 24 Apr.; endorsed by TJ as received 29 Apr. 1812 and so recorded in ; endorsed at description begins American Philosophical Society description ends : “Lambert Wm on an error in the Nautical Almanac read June 19. 1812.” Enclosed in TJ to Robert Patterson, 29 May 1812.

Having received Lambert’s astronomical calculations above and his letter of 23 Apr. 1812 “from the president of the Society,” on 19 June 1812 the American Philosophical Society referred them to a committee consisting of “Mr Patterson Garnet, Allison,” which did not return a report ( description begins American Philosophical Society description ends , Minutes [ in ]; John Vaughan, Report to description begins American Philosophical Society description ends on Papers to be Published, 6 Nov. 1812 [: description begins American Philosophical Society description ends Archives]).

# Index Entries

• Allison, Burgess; and W. Lambert’s astronomical calculations search
• American Philosophical Society; and W. Lambert search
• American Philosophical Society; TJ as president of search
• Garnett, John; and W. Lambert’s astronomical calculations search
• Garnett, John; corrects P. S. Laplace’s errors search
• Lambert, William; astronomical calculations search
• Lambert, William; letters from search
• Laplace, Pierre Simon, marquis de; error of detected search
• Patterson, Robert; and W. Lambert’s astronomical calculations search