equator, which measures the difference of their longitudes, the circumference of the circle representing 360 degrees or 24 hours; making 15 degrees of longitude one hour of time. To find the difference of longitude then between any two places, only requires us to be able to determine exactly the local time at each place, at the same instant; for which purpose chronometers whose rates are known, and which have been set to, or compared with, Greenwich mean time, are used, particularly at sea where other means more to be depended upon, cannot, from the motion of the ship, and the constant change of place, be always resorted to. From these explanations it will easily be seen that of the five following quantities, any three being given, the other two can be found by the solution of a spherical triangle, viz.: 1. The latitude of the place. 2. The declination of the celestial object observed. 3. Its hour angle east or west from the meridian. 4. Its altitude. 5. Its azimuth. Thus in the triangle PZS, named from its universal application the astronomical triangle P is the elevated pole, Z the zenith, and S the star or object observed; and the five quantities above mentioned, or their com . H E SV plements, constitute the sides and angles of the spherical triangle ZPS, PZ being the co-latitude, PS the co-declination, or north polar distance, ZS the co-altitude or zenith distance, the angle ZPS the hour angle, and PZS the azimuth. The further application of this triangle will be seen in the astronomical problems. In all the ordinary observations made for the determination of the latitude, local time, &c., the object observed may be either the sun, or a star whose declination and right ascension are known: the latter is, wherever practicable, to be preferred, as the use of the sun always involves corrections for semi-diameter and parallax; also in many observations of the sun, such as those of equal altitudes for time, or for determining the direction of a meridian line, or circum meridian altitudes for finding the latitude,—still further corrections are requisite on account of the change of the sun's declination during the period embraced by the observations; which corrections are avoided by using a star. The bisection of a star is likewise more to be depended upon than the observed tangent of the sun's limb. At sea, where minute accuracy is neither sought, nor to be obtained; and where at night the horizon is generally obscured, and often not to be discerned at all, this advantage is either not material, or not often to be taken advantage of; but on shore an artificial horizon is always used with reflecting instruments, and upon this the darkness of the night has no effect. In all observations of a star, the clock or chronometer, if not already so regulated, must be reduced to sidereal time; with the sun, on the contrary, the timekeeper must be brought to mean solar time, whether the local or Greenwich time be required. PROBLEMS. PROBLEM I. TO CONVERT SIDEREAL TIME INTO MEAN SOLAR TIME, AND THE REVERSE. THIS problem is of constant use wherever the periods of solar observations are noted by a clock regulated to sidereal time, or those of the stars by a chronometer showing mean time. A simple method of solution is given in the "explanation" at the end of the Nautical Almanac, which has the advantage of not requiring a reference to any other work, and also of all the quantities being additive. The three tables used in this method are those of equivalents; the transit of the first point of Aries in the 22nd; and the sidereal time at mean noon, in the 2nd page of each month. To convert sidereal into mean solar time: To the mean time at the preceding sidereal noon, i. e. the transit of the first point of Aries, in table 22, add the mean interval corresponding to the given sidereal time, taken from the table of equivalents. To convert mean solar into sidereal time:· To the sidereal time at the preceding mean noon, found in table 2, add the sidereal interval corresponding to the given mean time also from the table of equivalents. The mean right ascension of the meridian, or the sidereal time at mean noon given in the Nautical Almanac, is calculated for the meridian of Greenwich, and must, therefore, be corrected for the difference of longitudes between that place and the meridian of the observer. One of Mr. Baily's formulæ for the solution of the same problem is Where M represents the mean solar time at the place of observation, S the corresponding sidereal time, Æ the mean right ascension of the meridian at the preceding mean noon, found under the head of " sidereal time" in page 2 of each month; a, the acceleration of the fixed stars given in Baily's table 6 for the interval denoted by (S-R); and A the acceleration shown in his 7th table for the time denoted by M. Examples. Convert 8h 1m 10s sidereal time, March 6, 1838, longitude 2m 21.5s east, into mean solar time. Mean time at preceding sidereal noon Green H. M. S. 1 4 44.19 2-1507564 3.4362422 Again, to convert 9h 4m 35.748s mean solar into sidereal time. O right ascension at mean noon Greenwich, under head of "Sidereal Time," table 2. Correction for Longitude E: 22h 55m 6.18s * .0027305 is the change in time of sidereal noon in one second; and ·0027379 is the charge in the sun's mean right ascension in one second of time, or 9·8565 in one hour. TO DETERMINE THE AMOUNT OF THE CORRECTIONS TO BE APPLIED TO OBSERVATIONS FOR ALTITUDE, ON ACCOUNT OF THE EFFECTS OF ATMOSPHERIC REFRACTION, PARALLAX, SEMI-DIAMETER, DIP OF THE HORIZON, AND INDEX ERROR. THE formula given by Bradley for computing the value of atmospheric refraction is ra. tan (Z-br), where Z represents the zenith distance of the object, and a and b constants determined by observation; a, the average amount of refraction at an apparent zenith distance of 45°, being assumed = 57′′; and b = 3′′2. The formula of Laplace is •99918827 × c tan Z-001105603 × c tan 3Z, where c is assumed =60" 66. The tables constructed from these formulæ are of course not exactly similar, on account of the difference of the constants, which are also slightly varied in the tables of Bessel, Groom |