From William Lambert
City of Washington, April 20th 1812.
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.|
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.
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.
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.
RC (PPAmP: APS 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 SJL; endorsed at APS 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 (APS description begins American Philosophical Society description ends , Minutes [MS in PPAmP]; John Vaughan, Report to APS description begins American Philosophical Society description ends on Papers to be Published, 6 Nov. 1812 [PPAmP: APS description begins American Philosophical Society description ends Archives]).
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