If two persons, at different points of the same hemisphere, (as Greenwich and Washington, for example,) observe the same celestial phenomenon, at the same instant of time, the difference of the times marked by their respective clocks is the difference of their longitudes, or the distance between their meridians. To catch with precision the same instant of time for these simultaneous observations, the moon's motion in her orbit is the best element; her change of place (about a half second of space in a second of time) is rapid enough to be ascertained by a good instrument with sufficient precision for the object. But suppose the observer at Washington, or in a desert, to be without a time-keeper; the equatorial is the instrument to be used in that case. Again, we have supposed a contemporaneous observer at Greenwich. But his functions may be supplied by the nautical almanac, adapted to that place, and enabling us to calculate for any instant of time the meridian distances there of the heavenly bodies necessary to be observed for this purpose. The observer at Washington, choosing the time when their position is suitable, is to adjust his equatorial to his meridian, to his latitude, and to the plane of his horizon; or if he is in a desert where neither meridian nor latitude is yet ascertained, the advantages of this noble instrument are, that it enables him to find both in the course of a few hours. Thus prepared, let him ascertain by observation the right ascension of the moon from that of a known star, or their horary distance; and, at the same instant, her horary distance from his meridian. Her right ascension at the instant thus ascertained, enter with that of the nautical almanac, and calculate, by its tables, what was her horary distance from the meridian of Greenwich at the instant she had attained that point of right ascension, or that horary distance from the same star. The addition of these meridian distances, if the moon was between the two meridians, or the subtraction of the lesser from the greater, if she was on the same side of both, is the differences of their longitudes. This general theory admits different cases, of which the observer may avail himself, according to the particular position of the heavenly bodies at the moment of observation. Case 1st. When the moon is on his meridian, or on that of Greenwich. Second. When the star is on either meridian. Third. When the moon and star are on the same side of his meridian. Fourth. When they are on different sides.