Hence it may be seen that lunars may be used not alone for finding the longitude, but as checks upon the chronometer, and should frequen ly be taken on a long voyage for the latter purpose; and by taking the mean of all the results a very reliable correction to the chronometer is ascertained when approaching the land; and a tolerable rate may be found by comparing the mean of a number of observations in the begini ing of a voyage with a similar mean in the latter part of it. The observation of the lunar distance is subject to unavoidable errors which are diminished by making a number of measurements. Errors of the instrument may be diminished by measuring distances on opposite sides of the moon, when possible, and combining the results. Art. 304. Before taking the observation, the Nautical Almanac must be examined to see from what objects the distances are computed. There are only nine fixed stars and four planets from which the angular distances are computed in the Nautical Almanac; and as it is of the greatest importance to be able to discover them easily, we shall here add a number of remarks which will be found useful for that purpose. The best way of discovering any star or p'anet is by means of a celestial globe; observing that, when a planet is used, we must estimate roughly, by inspecting the Nautical Almanac, the right ascension and declination of the planet, and make a mark on the corresponding point of the globe with a pencil, or by attaching a small piece of moist paper, and this must be considered as the place of the planet. If a globe cannot be obtained, the time of passing the meridian, and the meridian altitude of the object, may be calculated; and by observing at that time, the object may be easily discovered. The distances marked in the Nautical Almanac afford also to the observer an easy method of knowing the star or planet f.om which the moon's distance is to be observed; for he has nothing to do but to set the sextant or circle to the distance computed roughly for the apparent time, estimated nearly for the meridian of Greenwich, and direct his sight to the east or west of the moon, according as the object is marked E. or W. in the Nautical Almanac; and, having found the reflected image of the moon upon the horizon glass, sweep the instrument to the right or left, and the image will pass over the sought star or planet, if above the horizon, and the weather clear: the star or planet is always one of the brightest, and is situated nearly in the arc passing through the moon's centre, perpendicular to the line connecting the two horns. The computed distance made use of in sweeping for the star may be found in this manner: Reckon the apparent time at the ship, and to this time apply the longitude turned into time, by adding in west, or subtracting in east longitude; the sum or difference will be the apparent time at Greenwich nearly. Take the distances from the Nautical Almanac for the time immediately preceding and following this estimated time, and note the difference of these distances; then say, As 3, or 180m, is to the difference of the distances, so is the difference between the apparent time at Greenwich and the next preceding time, set down in the Nautical Almanac, to a proportional part to be added to the next preceding distance taken from the Nautical Almanac, if the distance be increasing, but subtracted if decreasing; the sum or difference will be the distance at which the quadrant or sextant is to be fixed. In sweeping for the stars by this method, it will often happen that two or more are swept upon at once; this might cause some difficulty to an inexperienced observer, who would be at a loss to know which to make use of. To remove this, the following description of these stars is added: This star bears about west, distant 23° from the Pleiades, or the Seven Stars; it is of the second magnitude, and may be known by means of the star, of the third magnitude, situated S. W. from a Arietis, at the distance of 31⁄2 degrees. South from the star, at the distance of 121⁄2°, is the star v, of the fourth magnitude. The northernmost of these stars is a Arietis. About 35° E. S. E. from a Arietis, and 14° S. E. from the Pleiades, or Seven Stars, is the bright star Aldebaran. Near this star, to the westward, are six or seven stars of the third or fourth magnitude, forming, with Aldebaran, a figure resembling the letter V, as is represented in the adjoined figure, where Aldebaran is marked a. At the distance of 23 from this star, in a S. E. direction, are three very bright stars, situated in a straight line, near to each other, forming the belt of Orion. At the distance of 45° from Aldebaran, in the direction of E. N. E., is the star Pollux, which is a bright star, though not of the first magnitude. N. W. from it, distant 5, is the star Castor, of nearly the same magnitude; and you will almost sweep both at once: the southernmost is the one used. * Regulus. +3 E. by S. 1⁄2 S. from Pollux, at the distance of 372°, is the star Regulus, of the first magnitude; to the northward of this star (at the distance of 8) is a star of the second magnitude; near to these are five stars of the third magnitude, the whole forming a cluster resembling a sickle, represented in the adjoined figure, Regulus being in the extremity of the handle. A line drawn from the northern polar star, through its pointers, passes about 12 to the eastward of Regulus. ха ++ E. S. E. from Regulus, at the distance of 54°, is the star Spica, of the first magnitude, with no very bright star near it; S. W. from this star, at the distance of about 16, are five stars of the third or fourth magnitude, situated as in the adjoined figure; the two northernmost of these stars, 7, v, form a straight line with Spici, and by this mark it may be easily discovered. A line drawn from the northern polar star, through the middle star of the tail of the Great Bear, will pass near to Spica. E. S. E. from Spica, at the distance of 46, is the star Antares, in 26 of south declination; it is a remarkable star, of a reddish color; on each side of it. to the W. N. W. and S. S. E., about 2 distant, is a star of the third or fourth magnitude, no very bright star being near. N. E. from Antares, at the distance of 60, is the very bright star a Aquile; N. N. W. from which, at 2 distance, is a star of the third magnitude, and S. S. E., at 3° distance, another star of a less magnitude. These three stars appear nearly in a straight line. The star a Aquila is nearly of the same color as Antares. S. E. from a Aquile, at the distance of 60, is the star Fomalhaut, which is a bright star of high southern declination, its altitude in northern latitudes Leng small, never exceeding 20 in the latitude of 40 N. This star bears nearly south from the star a Pegasi, distant 45. A line drawn from the pointers, through the northern polar star, and continued to the opposite meridian, will pass very near to a Pegasi and Fomalhaut. E. by N. from a Aquile, at the distance of 48, and westward from a Arietis, at the distance of 44, is the star a Pegasi, which may be known by means of four stars of different magnitudes, situated as in the adjoined figure; in which a represents a Pegasi, B a star of the second magnitude, bearing north of it, distant 13; the others are of less magnitudes, and two of them, 7, μ, form a straight line with the star a Pegasi; and by this mark it may easily be discovered. GENERAL REMARKS ON THE TAKING OF A LUNAR OBSERVATION. Art. 305. The accuracy of a lunar observation depends chiefly on the regulation of the chronometer, and on the exact measurement of the angular distance of the moon from the sun or star; a small error in the observed altitudes of those objects will not, in general, much affect the result of the calculation. The best method of regulating a chronometer at sea is by taking an altitude of the sun when rising or falling quickly, or when bearing nearly east or west, the altitude being sufficiently great to avoid the irregular refraction near the horizon, and noting the time by the chronometer. With this altitude, the latitude of the place, and the sun's declination, find the mean time of observation by either of the preceding methods; the difference between this time and that shown by the chronometer will show how much it is too fast or slow. A single observation, taken with care, will generally be exact enough, but if greater accuracy is required, the mean of a number of observations may be taken. If the distance of the sun and moon be observed when the sun is three or four points distant from the meridian, the mean time of observation may be deduced from the altitude of the sun taken at the precise time of measuring the distance; this will render the use of a chronometer unnecessary, and will prevent any irregularity in its going from affecting the result of the observation. If a night observation is to be taken, the chronometer should be regulated by an altitude of the sun taken the preceding evening, and its going examined by means of another observation taken the next morning; for the time found by an altitude of a star cannot be so well depended up n, except in the morning and evening twilight, as the horizon is generally ill defined; but the altitude may be sufficiently exact for finding the correction used in determining the angular distance. Ithough all the instruments used in these observations ought to be well adjusted, yet particular care should be tal.en of the sextant or circle used in measuring the angular distance of the moon from the sun or star, since an error of 1' in this distance will cause an error of nearly 30' in the longitude deduced therefrom. When a great angular distance is to be measured, it is absolutely necessary to use a telescope, and the parallelism of it, with respect to the plane of the instrument, must be carefully examined; but in measuring small distances the use of the telescope is not of such great importance, and a sight-tube may then be used, taking care, however, that the eye and point of contact of the objects on the horizon-glass be equally distant from the plane of the instrument. But it ought to be observed that it is always conducive to accuracy to use a telescope, and, after a little practice, it is easily done. While one person is observing the distance of the objects, two others ought to be observing the altitudes. The chronometer should be placed near one of the observers, or put into the hands of a fourth person appointed to note the time; the observer who takes the angular distance giving previous notice to the others to be ready with their altitudes by the time he has finished his observation; which being done, the time, altitudes, and distancef should be carefully noted, and other sets of observations taken, which must be done within the space of 15 minutes, and the mean of all these observations must be taken and worked as a single one. t When a ship is close-hauled to the wind, with a large sea, or when sailing before the wind, and rolling considerably, it is difficult to measure the stance of the objects; but when the wind is enough upon the quarter to keep the ship steady, there is no difficulty, especially in small distances, which are much more easily measured than large ones, and are not so liable to error from an ill adjustment of the telescope; an observer would therefore do well to choose those times for observation when the distance of the objects is less than 70 or 80°. An observation of the sun and moon is generally much easier to take when the altitude of the moon is less than that of the sun, because the instrument will be held in a more natural and easy manner. When the moon is near the zenith, the observation is generally difficult to take, and liable to be erroneous, because the observer is forced to place himself in a disagreeable posture. For the same reason an observation of the moon and a star or planet is generally much easier to take when the star or planet is lower than the moon. This situation of the objects may * It is not uncommon to find a difference in the regulation of a chronometer in the forenoon and afternoon; this difference generally arises from the uncertainty in the estimated latitude, or some slight error in the observation, and perhaps partly from the irregularity in the going of the chronometer. + If the distances are measured by a circular instrument, it will not be necessary to note the several distances measured, but only the times and altitudes, as the sum of all the distances measured by the circle will be given by the instrument at the end of the observations; and if the altitudes of the objects are also measured by circular instruments, it will not be necessary to note the several altitudes, but only the times of observation. in most cases be obtained by taking the observation at a proper time of the day. But it must be observed that neither of the objects, if possible, ought to be at a less altitude than 10, upon account of the uncertainty of the refraction near the horizon, for the horizontal refraction varies from 33′ to 36′ 40′′ only by an alteration of 40° in the thermometer. This alteration might cause an error of 2o in the longitude with an observer who uses the mean refraction. In measuring the distance of the moon from the sun we must bring the moon's round limb in contact with the nearest limb of the sun. In measuring the distance of the moon from a planet or fixed star her round limb must be brought in contact with the centre of the star or planet; observing that, the semi-diameter of the planet being only a few seconds, the centre of it can be estimated sufficiently near for all the purposes of this observation. In taking the altitude of the moon, the round limb, whether it be the upper or lower, must be brought to the horizon. In damp weather it is rather difficult to observe the altitude of the stars on account of their dimness, particularly a Pegasi and a Arietis. Sometimes they are so dim that they cannot be seen through the telescope of a sextant, particularly if the mirrors are not well silvered; in this case the telescope must be laid aside, and the altitude must be taken with a sight-tube. We have here supposed that there were observers enough to measure the altitudes when the distance was observed; but if that is not the case, the altitudes may be estimated by either of the methods which will be hereafter given. Art. 306. TO CORRECT THE LUNAR DISTANCE.-The method here given is that of Professor Chauvenet, and involves the use of Tables 29 to 36, inclusive. The object of these tables is to give the true correction of a lunar distance in all cases when, with the apparent distance of the moon from the sun, a planet, or star, the apparent altitudes of the two objects have also been obtained by observation. They enable us readily to take into account: Ist, the parallax of the moon in the latitude of the observer, allowing for the spheroidal figure of the earth; 2d, the parallax of the sun or a planet; 31, the true atmospheric refraction, allowing for the actual state of the air as shown by the barometer and thermometer; and 4th, that effect of refraction which gives the apparent discs of the moon and sun an oval or elliptical figure. The longitude deduced from a lunar observation, when no attention is paid to the spheroidal figure of the earth, to the barometer and thermometer, and the elliptical figure of the discs, may in certain cases be in error a whole degree. It is true these extreme cases are rare in practice; but cases are common in which from such neglect the error in the longitude is 10', 15', or 20'. Since lunars are now chiefly valuable as checks upon the chronometer, it is absolutely necessary to get rid of such errors, and to leave no other inaccuracy in the result than that which unavoidably follows from the observations. This is accomplished by means of these tables, with an amount of labor very little greater than that which is required by the inaccurate methods in common use. THE OBSERVATION. Art. 307. The record of a complete observation embraces 1. The latitude and approximate longitude of the place of observation. 2. The approximate local time. 3. The time of observation as shown by a chronometer, and the error of the chronometer, or its difference from mean Greenwich time. 4. The apparent distance of the moon's bright limb from a star or plane, or from the nearest limb of the sun. 5. The apparent altitude of the moon's upper or lower limb above the sea horizon. 6. The apparent altitude of the star, planet, or lower limb of the sun above the sea horizon. 7. The height of the barometer and thermometer. 8. The height of the eye above the level of the sea. 9. The index correction of the sextant, if a sextant is used. The index correction of the sextant may be supposed to be previously determined; but, since even in the best instruments it is not constant, its determination should be considered a necessary part of the observation; and when the greatest precision is sought, it should be found both before and after the measurement of the distance, and its mean value taken. The error of the chronometer above alluded to is that which is obtained by applying the daily rate (multiplied by the proper number of days) to the error found before leaving port. The agreement or disagreement of the error thus found with that found by the lunar observation will be the test of the good or bad going of the chronometer. PREPARATION OF THE DATA. Art. 308 Greenwich Date.-Correct the chronometer time for its error from Greenwich time and deduce the Greenwich date, i. e., the Greenwich day and hour (mean time), reckoning the hours in succession from o to 24, beginning at noon. Nautical Almanac.—With the Greenwich date enter the Almanac and take out the moon's semi-diameter and horizontal parallax; and if the sun is o' served, its semi-diameter and horizontal parallax;" but if a planet is observed, its horizontal parallax only. Apparent Altitude of the Moon.-To the altitude given by the sextant apply the index correction of the instru ment and subtract the dip of the horizon, Table 14. If the lower limb is observed, add the semi-diameter augmented by Table 18; if the upper limb is observed, subtract the augmented semi-diame er. The result is the apparent altitude of the moon's centre, denoted "'s App. Alt." Apparent Altitude of the Sun, Planet, or Star.-To the observed altitude apply the index correction of the sextant, and subtract the dip, Table 14; an if the sun is used, add its semi-diameter when the lower limb is bserved, or subtract it when the upper limb is observed. The result is the apparent altitude required, denoted by "O's or 's App. Alt." Apparent Distance.--Ist, when the sun is used, to the observed distance (corrected for index error when necessary) add the moon's augmented semi-diameter and the sun's semi diameter; 2, when a planet or star is used, add the moon's augmented semi-diameter if its nearest limb is observed, but subtract it if its farthest limb is observed. The result is "pp. Dist." Moon's Reduced Parallax and Refraction.-Enter Table 19 with the latitude of the place of observation and the moon's horizontal parallax, and take out the correction, which add to the horizontal parallax. all the result the moon's reduced parallax, or "('s Red. P." *The sun's horizontal pa'allax may be assumed as 8" 5. Enter Table 29 with the moon's apparent altitude, and take out the mean reduced refraction, and apply to this mean refraction the corrections given in Tables 21 and 22, adding or subtracting these corrections according to the directions in the Tables. The result is the moon's reduced refraction, or "'s Red. R." Subtract the "Q's Red. R." from the " ('s Red. P." and mark the result as "'s Red. P. and R.” Reduced Parallax and Refraction of Sun, Planet, or Star."-With the apparent altitude of the sun, planet, or star, take from Table 29 the mean reduced refraction, which correct by Tables 21 and 22, If the sun is observed, subtract its horizontal parallax (which may alw. ys be taken at 8.5) from its reduced refraction, and mark the result as "O's Red. P. and R." If a planet is observed subtract its horizontal parallax, and mark the result as "*'s Red. P. and R.” If a star is observed, its reduced refraction is at once the required “★'s Red. P. and R.” Art. 309. Computation of the True Distance. Take from Table 30 the four logarithms A, B, C, D, and place these logarithms each at the head of a column, marking the columns A, B, C, and D, respectively; then put the log of 's Red. P. and R. (Table 34) in columns A and B. log of O's Red. P. and R. (Table 34) in columns C and D. log cosec App. Dist. (Table 44) in columns B and D. The sum of the four logs in Col. A is the log (Table 34) of the First Part of 's Correction, which is to be marked when the app. dist. is less than 90°, but when the app. dist is greater than 90o. The sum of the four logs in Col. B is the log (Table 34) of the Second Part of 's Correction, which is always to be marked The sum of the four logs in Col. C is the log (Table 34) of the First Part of the ©'s or ★'s Correction, which is to be marked when the app. dist. is less than 90°, but when the app. dist. is greater than 90°. The sum of the four logs in Col. D is the log (Table 34) of the Second Part of the O's or 's Correction, which is always to be marked +. Combine the first and second parts of the 's correction according to the signs prefixed; that is, take their sum if they have the same sign, but their difference if they have different signs, and prefix the sign of the greater to the result, which call 's whole correction." In the same manner form the "O's or ✶'s whole correction.” First Correction of Distance.-Combine the 's whole corr and the O's or 's whole corr., according to their signs; the result is the First Correction of Distance, which is to be added to or subtracted from the apparent distance, according as its sign is + or -. Second Correction of Distance.-Enter Table 31 with the Apparent Distance and the First Correction of Distance, and take out the Second Correction of Distance, which is to be applied to the distance according to the directions in the side columns of the Table. Correction for the Elliptical Figure of the Moon's Disk, or Contraction of the Moon's Semi-diameter (Table 32).— Enter Table 32, A, with the 's App. Alt. and 's Red. P. and R., and take out the number. With this number and the 's whole correction enter Table 32, B, and take out the required contraction, which is to be added to the app. dist. when the farthest limb is observed, but subtracted when the nearest limb is observed. Correction for the Elliptical Figure of the Sun's Disk, or Contraction of the Sun's Semi-diameter (Table 33).— Enter Table 33, A, with the O's App. Alt. and O's Red. P. and R., and take out the number. With this number and the 's whole corr. enter Table 33, B, and take out the required contraction, which is always to be subtracted from the distance (the nearest limb of the sun being always observed). Correction for Compression, or for the Spheroidal Figure of the Earth.-Take from the Nautical Almanac for the Greenwich date the declinations of the bodies to the nearest whole degree. With the moon's declination and apparent distance take from Table 36, A, the first part of N, and mark it with the sign in the table if the declination is North; but if the declination is South, change the sign from to or from to. With the sun's or star's declination and the apparent distance take from Table 36, B, the second part of N, giving it the same sign as the declination. Take the sum, or difference, of the two parts, according as their signs are the same or different, and to the resulting number prefix the sign of the greater. The logarithm of this number of seconds, taken from Table 34, with its sign prefixed, is the required log N. To log N add the log sine of the latitude of the place of observation; the sum is the log (Table 34) of the required correction for compression. In north latitude add this correction if log N is +, or subtract it if log N is -; in south latitude subtract the correction when log N is +, and add it when log N is All these corrections being applied to the Apparent Distance, the result is the True Distance. Art. 310. To find the Greenwich Time. Find in the Nautical Almanac the two distances between which the true distance falls. Take cut the first of these, together with the Prop. Log following it, and the hours of Greenwich time over it. Find the difference between the distance taken from the Almanac and the true distance, and to the log of this difference (Table 34) add the Prop. Log from the Almanac; the sum is the log (Table 34) of an interval of time to be added to the hours of Greenwich time taken from the Almanac. The result is the approximate Greenwich time. To correct this Greenwich time, take the difference between the two Prop. Logs in the Almanac which stand against the two distances between which the true distance falls. With this difference and the interval of time just found, enter Table 35 and take out the seconds, which are to be added to the approximate Greenwich time when the Prop. Logs are decreasing, but subtracted when the Prop. Logs are increasing. The result is the true Greenwich time. By comparing with this the local mean time the longitude will be found; or, if testing the time shown by chronometer, the difference between the true Greenwich time and the time shown by the chronometer is the error of the chronometer as determined by the lunar observation. *The parallax of a star being zero, its "reduced parallax and refraction" become, of course, merely its "reduced refraction"; but as no mistake can arise from marking it as "'s Red. P. and R.," this designation has been retained in order to give simplicity and uniformity at once to the rules and the tables. † No interpolation is necessary in taking out these logarithms. DEGREE OF DEPENDENCE. Art. 311. If the error thus determined agrees with that deduced by means of the rate and original error, the chronometer has run well, and its rate is confirmed; if otherwise, more or less doubt is thrown upon the chronometer, according to the degree of accuracy of the lunar observation itself. An error of 10" in the measurement of the distance produces about 20$ error in the Greenwich time; and since, even with the best observers, a single set of distances is subject to a possible error of 10", it may be well to consider the chronometer as still to be trusted so long as it does not differ from the lunar by more than 20. Since, however, so much depends upon skill in measuring the distance, the observer can only form a correct judgment of the degree of dependence to be placed upon his own observations by repeated trials and a careful comparison of his several results. EXAMPLE. In Lat. 35° 30′ N., Long. 30 W., by account, at the local mean time, 1855, September 6, 18h 8m, the observed distance of O's and 's nearest limbs was 43° 52' 10"; observed alt. 4, 49 32' 50"; observed alt., 5° 27' 10"; barometer, 29in.1; thermometer, 75°; height of the eye above the sea, 20ft; I. C. o.; required the longitude. Preparation of the Data. |