In my last communication to you, I stated my intention of entering into astronomical calculations for the purpose of fixing a first meridian for the United States at the permanent seat of their government. It is, perhaps, to be regretted, that a dependence on Great Britain, for a spot of ground from which American navigators and geographers have been hitherto in the habit of estimating their longitud[e] has existed so long as it has done; and it is my wish that this kind of dependence, at least, on that nation, should cease as soon as possible. I have already made my design, known to the public, thro’ the medium of the newspapers, and there appears at this place some degree of anxiety to see a publication of the work: The calculations are now finished, according to the following principles:—

1. The longitude between the Capitol in this City and Greenwic[h] Observatory, has been ascertained on the Spheroid.—

2. The same, reducing the form of the Earth to a sphere, as referred to it’s center, admitting the equatorial to the polar axis, to be as 334. to 333.

3. Supposing the ratio to be as 230 to 229.

The computation has been made from the data afforded by an occultation of x. Pleiadum (Alcyone) by the Moon on the 20th of October, 180[4] which is known to be among the best methods hitherto devised for an accurate determination of the longitude,—and is, in fact, a revision of former calculations for the same object, made in the year 1805, in which the requisite elements have been found with greater precision.—The parallaxes in longitude and latitude have been ascertained by different rules, viz: from the Star’s true altitude, angle of position, and the angle formed between the vertical circle and a parallel to the Ecliptic; also from the altitude and longitude of the nonagesima; (or highest point of the ecliptic) and the Star’s distance therefrom;—and [the] results have been compared with those obtained by the difference of time between the ecliptical conjunction of the Moon and Star at Was[h] and Greenwich: the greatest variance is not more than ^⅓ of an Americ[an] mile, and this has been lessened considerably by taking a mean of three quantities, so that as far as small errors which may then [have?] existed in the lunar tables, will admit, we have a result, if not strictly correct, yet very nearly approximated t[o the] truth.

With respect to the different ratios that have been assumed, it will be recollected, that the proportion of the equatorial to the polar diameter of the Earth, adopted by the celebrated Sir Isaac Newton, is as 692 to 689; that of 230 to 229, (which is nearly the same) has been used by Mr Vince, and several other eminent European mathematicians; but from discoveries of new lunar equations by M. de la Place and M. de Lambre, in France, about the beginning of the present century, the ratio of 334 to 333, seems now to be allowed as more correct. I am disposed to take an arithmetical mean of the two, which will be as 282 to 281; for if the former might have made the figure of the Earth less spherical than it really is, it is possible that the latter may make it more so,—and for this reason, a mean of the two will not be considered improper.

		° ′ ″
The latitude of the Capitol, in Washington, is stated by Mr Ellicott, in the plan of the City, to be		38.53. 0.N.
	° ′ ″
Mr Richard Freeman, a clerk in the Treasury department, by a mean of several observations taken with a good sextant, at Rhode’s hotel, N.70° W. 1⅜ of our miles from the Capitol, has found the latitude of that place to be	38.53.18.
difference of lat. N. to the nearest second.,	— 24.	38.52.54.
Mean Latitude of the Capitol on the Spheroid		38.52.57.

	° ′ ″ dec.
This latitude I have assumed in the calculations, and it’s reduction to a sphere, admitting the ratio to be as 334 to 333, is	38.42.52.939.
allowing the ratio of 230 to 229,	38.38.19.465.
The latitude of Greenwich Observatory is known to be	51.28.40. N.
reduced, (334 to 333)	51.18.36.869.
do (230 to 229)	51.14. 3.279.

The longitude of the Capitol from Greenwich, by a mean of the several methods that have been used, taking the Moon’s motion in the relative orbit, instead of the motion reduced to the ecliptic, to ascertain the different intervals of time, is as follows:—

	h. m. s. dec. ° ′ ″ dec.
Spheroid	5.7.53.201 = 76.58.18.015.W.
reduced (334 to 333.)	5.7.42.249 = 76.55.33.735.
〃 (230 to 229)	5.7.36.739 = 76.54.11.085.

From which, if other ratios of the equatorial to the polar diameter of the Earth, be [occult]ated, the result will be—

	h. m. s. dec. ° ′ ″ dec.
334 to 333.	5.7.42. 249. = 76.55.33.735.		very nearly.
308 to 307.	5.7.40.8715. = 76.55.13.072.
282 to 281,	5.7.39. 494. = 76.54.52.410.
256 to 255.	5.7.38.1165. = 76.54.31.747.
230 to 229.	5.7.36. 739. = 76.54.11.085.

The1 care taken to find all the elements correctly to three decimal places of a second, and the small variance discovered in the result by different methods, will, it is presumed, be sufficient to fix our first meridian with as much, if not greater accuracy, than some in Europe have been settled. I have seen no computation of the kind, in which such strict attention has been paid to ascertain the necessary elements with due precision, as I have used throughout the work.—I have also annexed the most useful cases in oblique angled spherical trigonometry, with rules for their solution in full, unclog’d with algebraical process or formulæ, which have been too often adopted by mathematical writers, and serve only to render the subject more abstruse to the student, without the smallest advantage to be derived therefrom.—As the Moon’s place in longitude, latitude, &c. and the hourly velocity, at intermediate times between noon and midnight, and midnight and noon, are often required in astronomical calculations to great exactness, I have given new2 series and formulæ, by which those elements may be obtained with due precision, extending the equations of differences to the fourth order.

I inclose a newspaper, containing the toasts and proceedings of the Tammany society of Washington, at their anniversary meeting on the 12th of May last. Having been a short time before, appointed secretary of that society, I was the first named of three persons to prepare toasts for the occasion; and, in fact, drafted the whole number of 17, which were agreed to with slight alterations in a few of them—I was happy to avail myself of the opportunity to shew that respect for your character, talents and services, to which I have always considered you entitled; and surely, it ought not to be termed adulation, when you were in your retirement at the time, and divested of presidential powers.

Permit me to hope, that you will be good enough to answer such parts of my former and this letter, as relate to the object herein mentioned.