Your communication of the 29th ulto was received this morning. The Equatorial is an instrument with which I am unacquainted; but if it enables an observer to ascertain the horary angle, or time from noon, with sufficient accuracy; or if it be so Constructed, as to show the true azimuth from the north or south points, and the latitude of the place, together with the Sun’s declination be known, the horary angle, or distance of the observed Object from the meridian, may be correctly found; but if the azimuth be not shewn with due precision, there will, of course, be a proportionate error in the apparent time. There is, independent of the angular distances between the Moon and Sun, or the Moon and a fixed Star, commonly called “Lunar Observations,” another method to determine the longitude of a place, little used, or perhaps, known in this Country, which is, Let the observed altitude of the Moon’s upper or lower limb, be taken, from which compute, in the usual manner, the true altitude of her center,—find the correct apparent time of this observation (or set of observations, which is to be preferred). With the Moon’s declination found correctly, by interpolation from successive differences, according to the assumed longitude from Greenwich, Paris, &c—the true altitude, and the latitude of the place, find the Moon’s true distance from the meridian, at that time, which, if the Moon be East of the meridian, added to, or if west, subtracted from, the apparent time, reduced to degrees, &c. will give the difference of the Sun and Moon’s right ascensions at the time and place of Observation. By the help of the Nautical Almanac, or Connaissance de Tems, find, by interpolation, the apparent time at Greenwich, Paris, &c. when the Sun and Moon had that difference of Right Ascensions, and the difference of times turned into degrees, &c will give the difference of meridians in longitude, considering the form of the Earth to be that of a perfect Sphere. This method does not need any comparison of the Moon with a fixed star, whose nutation and aberration in R.A. ought to be computed and allowed for, if great accuracy be required. I have reason to believe that this method is equal to, if not preferable to the angular distances. I have not, however, by any actual experiment, or observation at this place, been able to put it into practice. The principle is the same as transits of the Moon over the meridian, except the computation of the Moon’s distance at the time and place of observation.

Being desirous of answering your favor promptly, I have not yet bestowed that consideration upon it, which it deserves. I have some doubts whether the subdivisions of the Equatorial can shew the apparent time as accurately as could be wished; if not, the most to be expected from its use, would be an approximation wide of, or near the truth, according to the deviation.—If this reply, which is written without due reflection on the subject, be not satisfactory, another opportunity, when I have more leisure, will be taken; and if you wish an example of this method to find the longitude, I shall take a pleasure in transmitting it in my next, or compute any that you may propose, when proper data shall have been furnished for that purpose.