bridge, &c. They all suppose a mean temperature, and mean pressure of the atmosphere, corrections being in all cases required on account of the deviation of the thermometer and barometer from these assumed standards. These corrections are however rendered perfectly simple in operation, by the use of any of the numerous tables of refraction; those by Dr. Young being given in table 4 in this volume. The rate of the increase of refraction is evidently, from the above formula, nearly as the tangent of the apparent angular distance of the object from the zenith in moderate altitudes. In very low altitudes (which should always be avoided on this account) the refraction increases rapidly and irregularly, being at the horizon as much as 33'-more than the diameter of the sun or moon. The next correction is for parallax, the explanation of which term has been given in page 165. The sine of its value in any altitude decreases as the cosine of that altitude; but the parallax in altitude may be obtained from the horizontal parallax without computation, by the aid of tables. The parallax given in any ephemeris is the equatorial, which has been shown in page 166 to be always the greatest. The first correction, where great accuracy is required, is on account of the latitude of the place of observation, but this is seldom necessary except in altitudes of the moon. The mean horizontal parallax of the sun is assumed 8"-6; but as our distance from this luminary is always varying in different parts of the earth's orbit, this value must be corrected for the period of the year. In table 8, the sun's horizontal parallax is given for the first day of every month which will facilitate this reduction, the proportional parts being found for any intermediate day. In the Nautical Almanac, however, this quantity is given more correctly for every tenth day. The parallax in altitude, corresponding to this horizontal parallax, can also be ascertained by inspection, from the same general table. The parallaxes of the planets are given for every fifth day, in the Nautical Almanac; but of those likely ever to be found useful in observation, Venus and Mars are the only planets to whose parallaxes any correction need be applied in observing with small instruments. The horizontal equatorial parallax of the moon is to be found for mean noon and midnight of every day in the year, in the third page of each month, in the Nautical Almanac. The corrections for its reduction for the latitude of the place, and the moon's altitude, require, from their magnitude, more care than those of any other celestial body; but in observations at sea the former correction is generally neglected, and the latter is much facilitated by the use of tables giving the reduction for every 10′ of the moon's altitude*. The example given in this case will explain the method of making these corrections. The semidiameter † of the sun is given for mean noon on every day of the year, in the second page of every month of the Nautical Almanac; that of the moon in the third page of each month for both mean noon and midnight; and those of the planets (which are seldom required) in the same table as their parallaxes. The correction for semidiameter is obviously to be applied, additive or subtractive, wherever the lower or upper limb of any object has been observed, to obtain the apparent altitude of its centre;—the moon's semidiameter increasing with her altitude, from the observer being brought nearer to her as she approaches his meridian, must be corrected for altitude, which can be done by the aid of table 7+. The dip of the horizon is a correction only to be applied at sea, and is necessary on account of the height of the eye above the * See Table 8 of Lunar Tables, page 188 of Dr. Pearson's " Astronomy." Riddle's Table, page 154, includes the corrections both for Parallax and Refraction, and is useful for "clearing the lunar distance" to be hereafter explained. + All quantities in the Nautical Almanac are calculated for Greenwich time; allowance must therefore be made, where necessary, for difference of longitude, which is the same as difference of time. The augmentation of the moon's semidiameter for every degree of altitude is given in Table 7 of Dr. Pearson's "Lunar Tables." Altitudes taken with an artificial horizon are obviously double those observed above the sensible horizon. N sensible horizon (on shore an artificial horizon is always used). A larger angle is evidently always observed; and this correction, which can be taken from the 11th table, is always subtractive. The correction for the index error has already been explained. EXAMPLE I. On March 15, 1838, the observed double altitude of the sun's upper limb, taken with a sextant, was 42° 37' 15", the thermometer at the time standing at 42* and the barometer at 29-98 inches. Required the altitude, corrected for semidiameter, refraction, and parallax. In rough altitudes, such as those taken at sea for latitude, no correction is made on ac count of the state of the thermometer or barometer. EXAMPLE 11. On April 6, 1838, at 9 P.M., Greenwich time, in latitude 51° 30', the double altitude of the moon's lower limb was observed 97° 21′ 50′′. Index error of sextant, 50′′. Barometer, 30.1 inc. Thermometer, 54°. * This might have been obtained at once by inspection, by using the tables of Parallax. In these examples no allowance has been made for the dip of the horizon, as the observations were made with an artificial horizon: with the fixed stars no correction is required for semidiameter or parallax. PROBLEM III. TO DETERMINE THE LATITUDE. Method 1st.-By observations of a circumpolar star at the time of its upper and lower culminations. This method is independent of the declination of the star observed the altitudes are observed with any instrument fixed in the plane of the meridian, or (not so accurately, of course) with a sextant or other reflecting instrument, at the moments of both the upper and lower transits of the star; or a number of altitudes may be taken immediately before and after its culminations, and reduced to the meridian, as will be explained. In either case, let Z denote the observed or reduced meridional zenith distance of the star at its lower culmination, and r its refraction at that point; also let Z' and r' denote the zenith distance and refraction at its upper culmination. Then the correct zenith distance of the pole, or the co-latitude of the place, will be = (Z + Z') + } (r + r'). According to Baily, a difference of about half a second may result from using different tables of refraction. Method 2nd. By means of the meridional zenith distance (or co-altitude) of the sun, or a star whose declination is known. The altitude of the sun or star being determined at the moment of its superior transit, as before explained, and corrected for refraction, and also for parallax and semidiameter when necessary, the latitude required will be— Z+D, if the observation is to the south of the zenith. D-Z, if to the north above the pole. 180—(Z + D) to the north below the pole. Z being put to denote the meridional zenith distance, and D the declination (-when south). |