ON PARALLAX, REFRACTION, AND DIP OF THE HORIZON. PARALLAX (or diurnal parallax) is the difference between the true altitude of the sun, moon, or star, if it were observed at the centre of the earth, and the apparent altitude, observed, at the same instant, by a spectator, at any point on the surface of the earth. Thus, in Plate XII, figure 3, let ABC be the earth, C its centre, A the place of a spectator, ZAK a vertical plane, passing through the place D of the moon, or the place d of a planet; EDF, ed G, circular arcs drawn about C as a centre, and KZ part of the starry heavens. Then, if at any time the moon be at D, she will be referred to the point H, by a spectator supposed to be placed at the centre of the earth, and this is called the true place of the moon; but the spectator at A will refer the moon to the point b, and this is called the apparent place of the moon; the difference Hb (or the angle HDb=ADC) is called the moon's parallax in altitude, which is evidently greatest when the moon is in the horizon at E, being then equal to the arc KI, and it decreases from the horizon to the zenith, and is there nothing. The parallax is less as the objects are farther from the earth: thus the parallax of a planet at dis represented by a b, being less than that of the moon at D; and the horizontal parallax Kf of the planet is less than the horizontal parallax KI of the moon. As the parallax makes the objects appear lower than they really are, it is evident that the parallax must be added to the apparent altitude to obtain the true altitude. Having the horizontal parallax, the parallax in altitude is easily found by the following rule:-As radius is to the cosine of the apparent altitude, so is the horizontal parallax to the parallar in altitude. This rule may be easily proved; for in the triangle CAE we have CE: CA:: radius: sine CEA; and in the triangle CDA we have CD (or CE): CA :: sine CAD: sine CDA; hence we have radius: sine CEA:: sine CAD: sine CDA; but CEA horizontal parallax, CDA = parallax in altitude, and sine CAD cosine app. alt. Hence we have radius: cosine app. alt. :: sine hor. par. : sine par. in alt.; but the parallaxes of the heavenly bodies being very small, the sines are nearly proportional to the parallaxes; hence we may say, As radius: cosine app. alt. :: hor. par.: par. in alt. = The sun's mean parallax in altitude is given in Table XIV., for each 5° or 10° of altitude. The moon's horizontal parallax is given in the Nautical Almanac, for every noon and midnight at the meridian of Greenwich; also that of the sun for every ten days, and the parallaxes of Venus, Mars, Jupiter, and Saturn, for every five days, throughout the year. Refraction of the heavenly bodies. It is known, by various experiments, that the rays of light deviate from their rectilinear course, in passing obliquely out of one medium into another of a different density; and if the density of the latter medium continually increase, the rays of light, in passing through it, will deviate more and more from the right lines in which they were projected towards the perpendicular to the surface of the medium. This may be illustrated by the following experiment:-Make a mark at the bottom of any basin, or other vessel, and place yourself in such a situation that the hither edge of the basin may just hide the mark from your sight; then keep your eye steady, and let another person fill the basin gently with water; as the basin is filled, you will perceive the mark come into view, and appear to be elevated above its former situation. In a similar manner, the light, in passing from the heavenly bodies through the atmosphere of the earth, deviates from its rectilinear course. By this means the objects appear higher than they really are, except when in the zenith. This apparent elevation of the heavenly bodies above their true places, is called the refraction of those bodies. To illustrate this, let ABC (Plate XII., fig. 2) represent the atmosphere surrounding the earth DEF, and let an observer be at D, and a star at a; then, if there were no refraction, the observer would see the star according to the direction of the right line Da; but as the light is refracted, it will, when entering the atmosphere near A, be bent from its rectilinear course, and will describe a curve line from A to D, and, at entering the eye of the observer at D, will appear in the line D b, which is a tangent to the curve at the point D, and the arc ab will be the refraction in altitude, or, simply, the refraction, which must be subtracted from the observed altitude to obtain the true. At the zenith, the refraction is nothing; and the less the altitude, the more obliquely the rays will enter the atmosphere, and the greater will be the refraction: at the horizon, the refraction is greatest. In consequence of the refraction, any heavenly body may be actually below the horizon when appearing above it. Thus, when the sun is at T below the horizon, a ray of light TI, proceeding from T, comes in a right line to I, and is there, on entering the atmosphere, turned out of its rectilinear course, and is so bent down towards the eye of the observer at D, that the sun appears in the direction of the refracted ray above the horizon at S. The mean quantity of the refraction of the heavenly bodies is given in Table XII All observed altitudes of the sun, moon, planets, or other heavenly bodies, must be decreased by the numbers taken from that table corresponding to the observed altitude of the object. The refraction varies with the temperature and density of the air, increasing by cold or greater density, and decreasing by heat and rarity of the atmosphere. The corrections to be applied to the numbers taken from Table XII., for the different heights of Fahrenheit's Thermometer and the Barometer, are given in Table XXXVI. Thus, if the refraction be required for the apparent altitude 5o, when the thermometer is at 20°, and the barometer at 30.67 inches, we shall have the mean refraction by Table XII. equal to 9′ 53′′, and by Table XXXVI. the correction corresponding to the height of the thermometer 20° equal to +48", and for the barometer 30.67 equal to + 22′′; hence the true refraction will be 953"-48" 22′′ = 11′ 3′′. There is sometimes an irregular refraction near the horizon, caused by the vapors near the surface of the earth; the only method of avoiding the error arising from this source, which is sometimes very great, is to take the observations at a time when the object which is observed is more than 10° above the horizon. The refraction makes any terrestrial object appear more elevated than it really is. The quantity of this elevation varies, at different times, from to of the angle formed, at the centre of the earth, between the object and the observer; but, in general, this refraction is about 4 of that angle. Dip of the horizon. Dip of the horizon is the angle of depression of the visible horizon below the true or sensible horizon (touching the earth at the observer), arising from the elevation of the eye of the observer above the level of the sea. Thus, in Plate XII., figure 1, let ABC represent a vertical section of the earth, whose plane, being produced, passes through the observer and the object, and let AE be the height of the eye of the observer above the surface of the earth; then FEG, drawn parallel to the tangent to the surface at A, will represent the true horizon, and EIH, touching the earth at I, will represent the apparent horizon; therefore the angle FEH will be the dip of the horizon. Let M be an object whose altitude is to be observed by a fore observation by bringing the image in contact with the apparent horizon at H; then will the angle MEH be the observed altitude, which is greater than the angle MEF (the altitude independent of the dip) by the quantity of the angle FEH; so that, in taking a fore observation, the dip must be subtracted from the observed altitude to obtain the altitude corrected for the dip. In a back observation, the apparent horizon is in the direction EK; and, by continuing this line in the direction EL, we shall have the observed altitude MEL; and it is evident that to this the dip LÉF (= KEG) must be added to obtain the altitude corrected for the dip. In Table XIII. is given the dip, for every probable height of the observer, expressed in feet. In calculating this table, attention is paid to the terrestrial refraction, which decreases the dip a little, because IE becomes a curve line instead of a straight one, and EH is a tangent to that curve in the point E. *This table is to be entered with the height of the thermometer or barometer at the top, and the apparent altitude at the side; under the former, and opposite the latter, will be the correction corre sponding to the thermometer or barometer, which is to be applied to the mean refraction, by addition or What has been said concerning the dip of the horizon, supposes it free from all encumbrances of land or other objects; but, as it often happens, when ships are sailing along shore, or at anchor in a harbor, that an observation is wanted when the sun is over the land, and the shore nearer the ship than the visible horizon would be if it were unconfined, in this case, the dip of the horizon will be different from what it otherwise would have been; and greater the nearer the ship is to that part of the shore to which the sun is brought down. For this reason Table XVI. has been inserted, which contains the dip of the sea at different heights of the eye, and at different distances of the ship from the land. This table is to be entered at the top with the height of the eye of the observer above the level of the sea in feet; and in the left-hand side column, with the distance of the ship from the land in sea miles and parts. Under the former, and opposite the latter, stands the dip of the horizon, which is to be subtracted from the altitude observed by a fore observation, instead of the numbers in Table XIII. The distance of the land requisite in finding the dip from Table XVI., may be found nearly in the following manner:-Let two observers, one placed as high on the main-mast as he can conveniently be, and the other on the deck immediately beneath him, observe, at the same instant, the altitude of the sun or other object that may be wanted, and let the height of the eye of the upper observer above that of the lower be measured in feet, and multiplied by 0.56; then the product, being divided by the difference of the observed altitudes of the sun in minutes, will be the distance in sea miles, nearly. Thus, if the eye of the upper observer was 68 feet higher than that of the lower, and the two observed altitudes of the sun 20° 0′ and 20° 12′, the distance of the land, in sea miles, would be 3.2. For 68 × 0.56=38.08, and this, being divided by the difference of the two observed altitudes of the sun 12, gives 3.2, nearly. Now, if the lower observer be 25 feet above the level of the sea, the dip corresponding to this height and the distance 3.2 miles will be 6', which, being subtracted from 20° 0′, leaves 19° 54', the altitude corrected for the dip. The dip may be calculated, in this kind of observations, to a sufficient degree of accuracy, without using Table XVI., in the following manner;-Divide the difference of the heights of the two observers in feet, by the difference of the observed altitude in minutes, and reserve, the quotient. Divide the height of the lower observer in feet by this reserved number, and to the quotient add one quarter of the reserved number, and the sum will be the dip in minutes corresponding to the lower observer. Thus, in the above example, $5.6 is the reserved number, and 254.4; to this add one fourth of 5.6 or 1'.4, and the sum will be the dip 5'.8, or nearly 6', corresponding to the lower observer, being the same as was found by the table. TO FIND THE SUN'S DECLINATION. THE declination of the sun is given, to the nearest minute, in Table IV., for every noon, at Greenwich, from the year 1833 to 1848; and this table will answer for some years beyond that period, without any material error. If great accuracy is required, the declination may be taken from the Nautical Almanac.* This declination may be reduced to any other meridian, by means of Table V., in the following manner: To find the sun's declination, at noon, at any place. RULE. Take out the declination at noon, at Greenwich, from Table IV., or from the Nautical Almanac; then find the longitude from Greenwich in the top column of Table V., and the day of the month in the side column; under the former, and opposite to the latter, is a correction, in minutes and seconds, to be applied to the declination taken from Table IV.; to know whether this correction be additive or subtractive, you must look at the top of the column where you found the day of the month, and you will see it noted whether to add or subtract, according as the longitude is east or west. This correction being applied, you will have the declination at noon at the given place. EXAMPLE I. Required the declination of the sun, at the end of the sea day, October 10, 1836, in the longitude of 130° E. from Greenwich. Sun's declination, October 10, at Greenwich, at the end of the sea day, or True dec. noon, October 10, in long. 130° E..... EXAMPLE II. 6° 46′ S. .sub. 0 8 6 38 S. Required the sun's declination at noon ending the sea day of March 12, 1836, in the longitude of 65° W. from Greenwich. The preceding correction ought always to be applied to the declination used in working a meridian observation to determine the latitude, though many mariners are in the habit of neglecting it. In finding the declination, or any other quantity, in the Nautical Almanac, you must be careful to note the difference between the civil, nautical, and astronomical account of time. The civil day begins at midnight, and ends the following midnight, the interval being divided into 24 hours, and is reckoned in numeral succession from 1 to 12, then beginning again at 1 and ending at 12. The nautical or sea day begins at noon, 12 hours before the civil day, and ends the following noon; the first 12 hours are marked P. M., the latter A. M. The astronomical day begins at noon, 12 hours after the civil day, and 24 hours after the sea day, and is divided into 24 hours, numbered in numeral succession from 1 to 24, beginning at noon, and ending the following noon. All the calculations of the Nautical Almanac are adapted to astronomical time; the declination marked in the Nautical Almanac, or in Table IV., is adapted to the beginning of the astronomical day, or to the end of the sea day, it being at the end of the sea day when mariners want the declination to determine their latitude. It would be much better if seamen would adopt the astronomical day, and wholly neglect the old method of counting by the |