From William Lambert
City of Washington, September 19th 1809.
I inclose a table which has been constructed with great care to every 3 minutes of intermediate time between 0 and 12 hours, for computing the Moon’s motion for 12 hours in longitude, latitude, &c. from which the true place may be easily obtained: it was intended to be as fair, as it is accurate, but some parts of it have been stained by accident, before it was finished. The second series for the construction, is entirely my own, having never seen it in any book or treatise on Astronomy: the principle on which it is formed, is strictly correct, and its’ extension to differences of the fourth order, and to seven places of figures in the decimal fractions, may warrant a belief that it is more full and perfect than any one of the kind hitherto made: it may not be necessary to take out more than four or five places of decimals to obtain the equations arising from the successive differences. To facilitate the operation, the first differences of the respective numbers have been annexed. I have endeavored to shew the method of using it by a few examples; and nothing more is required than a knowledge of algebraical addition, subtraction and multiplication to apply the table for ascertaining the Moon’s positions correctly at any time required. As you have been for many years president of the American Philosophical Society, there is a propriety in transmitting it to you, independent of other considerations, which would incline me to pay you such a mark of my respect. There is not a case for a determination of the longitude of places on the Earth in which the Moon’s motions or any of them are employed, but this table will be found useful, because, if the positions at noon and midnight are correct, it will never fail to give them at intermediate times with great precision.
RC (PPAmP: APS Archives, Manuscript Communications); at foot of text: “Thomas Jefferson, late President U.S.”; endorsed by TJ as received 25 Sept. 1809 and so recorded in SJL. Enclosed in TJ to Robert Patterson, 1 Oct. 1809.