General Remarks on the taking of a Lunar Observation. 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 Dean 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 upon, 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. * Although all the instruments used in these observations ought to be well adjusted, yet particular care should be taken 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. Whilst 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 distance, should be carefully noted, and other sets of Coservations 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. 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 distance 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 measier 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 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 sign. 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 ineasured, 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, is generally much easier to take when the star or planet is lower than the moon, This situation of the objects may 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 two degrees 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 semidiameter 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 & Pegasi and a Arietis. Sometimes they are so dim that they cannot be seen through the holes of the sight-vane of a quadrant, particularly if the mirrors are not well silvered; in this case, the vane must be turned aside, and the eye held in nearly the same place, or the altitude must be taken by a sextant furnished 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. Preparations necessary for working a Lunar Observation Find the mean time of observation by astronomical account, reckoning the hours from noon to noon in numerical succession from 1 to 24, and taking the day one less than the sea account; to this time apply the longitude turned into time by Table XXI.† oy adding if in west longitude, but subtracting if in east; the sum or difference ‡ will De the supposed time at Greenwich, or reduced time. In page III. of the month of the Nautical Almanac, find the moon's semidiameter and horizontal parallax, for the nearest noon and midnight before and after the reduced time, and find the difference of the parallaxes and the difference of the semidiameters; then enter Table XI. with these differences respectively in the side column, and the reduced time at the top; opposite the former, and under the latter, will stand the corrections § to be applied respectively to the semidiameter and horizontal parallax marked first in the Nautical Alman-c, additive if increasing, subtractive if decreasing; the sum or difference will be the horizontal semidiameter and the horizontal parallax, respectively, at the time of observation. To this horizontal semidiameter must be added the augmentation from Table XV. corresponding to the moon's altitude; the sum will be the true semidiameter of the moon. The sun's true semidiameter is to be found in page II. of the month of the Nautical Almanac. To the observed altitude of the sun's or moon's lower limb add 12'; but if the upper limbs were observed, subtract 20, and from the observed altitude of the star or planet subtract 4', and you will have nearly the apparent altitudes of those objects respectively.|| * If any one wishes to proceed with perfect accuracy, he may bring the round limb of the moon to the nearest limb of the planet, and then apply the planet's semidiameter, taken from the Nautical Almanac, in the same manner as in observations of the sun. Or by multiplying by 4 sexagesimally, in the manner directed in the note page 170. When the sum exceeds 24 hours, you must subtract 24 hours, and add one to the day of the month; and when the time to be subtracted is greater than the mean time, the latter must be increased by 21 Hours, and one day taken from the day of the month, conformably to the usual rules of addition and subtraction. If the chronometer used in taking the observation be regulated to Greenwich time, this part of the calculation will be unnecessary, because the reduced time at Greenwich will be given directly by the chronometer. These corrections may be found easily without the table, by saying, As 12 hours are to the reduced time, (rejecting 12 hours when it exceeds 12,) so is the difference of semidiameter or parallax for 12 hours to the corresponding correction. If the reduced time cannot be found accurately in the table, You must use the nearest numbers, which will, in general, be sufficiently accurate. These altitudes are supposed to be taken at sea by a fore observation; and the application of the above numbers will give the apparent altitudes corresponding to observations taken on the deck of a common-sized vessel (where the dip is about 4' or 5') to a sufficient degree of accuracy; if the observer was 40 or 50 feet above the water, 1 or 2 might be taken from these altitudes. The propriety of using these numbers wil appear by considering that every wave, by raising the ship above the level of the sea, will alter the dip, and that an error of 1 or 2' in the altitudes will in general cause but a To the observed distance of the moon from a star or planet add the moon's true semidiameter, if her nearest limb was observed, but subtract that semidiameter if her farthest limb was observed; the sum or difference will be the apparent distance. But to the observed distance of the sun and moon's nearest limbs, add their true semidiameters; the sum will be the apparent distance. These preparations are necessary in every method of working a lunar observation The most noted methods are those of Dunthorne, Borda, Maskelyne, Rios, Witchell, Lyons, &c., and improvements thereon by various authors. Dunthorne's and similar methods have one great advantage in not being liable to a variety of cases; but these methods are tedious, when tables of logarithms to minutes only are used, by reason of the great exactness required in proportioning the logarithms to seconds. This is obviated in the excellent methods published by Rios and Stansbury; but they require large and expensive tables, and on that account are not in very general use. Witchell's and Lyons's methods do not labor under the inconvenience of requiring large tables, nor do they require any particular notice of the seconds in finding the log. sines and log. tangents; but these methods, as they were originally published, are embarrassed with a variety of cases; sometimes the corrections are additive, sometimes subtractive; and learners find a difficulty in rightly applying them. To remedy this, a method was published in the first edition of this work, in which two corrections were constantly additive, two subtractive, and one small correction was additive when the distance was less than 90°, but subtractive when above 90°. This method was further improved in the Appendix to that edition, by means of four new tables, which are inserted in this edition, and numbered XVII. XVIII. XIX. and XX., by means of which the work is considerably shortened, and all the corrections rendered additive. This method will now be given, after making a few remarks on the manner of taking the corrections and logarithms from these new tables. Table XVII. contains a correction and logarithm to be used when the moon's distance from a star or planet is observed; and Table XVIII. is a similar one, to be used when the moon's distance from the sun is observed. Table XVII. contains six pages, corresponding to the horizontal parallax of the planet, supposing it to be either 0', 5", 10", 15", 20", 25", or 30", as at the top of the pages respectively; and tha page is to be used which agrees the nearest with the horizontal parallax of the plane at the time of observation.* These tables are so extended, that no proportional parts are necessary in taking out the corrections and logarithms, except the altitude of the sun or star be less than 7° 30', and at such altitudes an observation is liable to error on account of the uncertainty of the refraction; so that, in using these tables, it is sufficiently accurate to find the number nearest to the given altitude of the sun or star, and make use of the corresponding correction and logarithm. Thus, if the star's altitude be 12° 25', the nearest number in Table XVII. is 12° 24', corresponding to which are the correction 55′ 45′′, and the logarithm 1.3161. Table XIX. contains the corrections and logarithms corresponding to the moon's horizontal parallax and altitude, both being found at the same opening of the book. The corrections for seconds of parallax and minutes of altitude are easily taken out by means of Tables A, B, C, placed in the margin. The method of finding these corrections is given at the bottom of the table: they are always additive. Besides the two logarithms taken from Table XVII. (or XVIII.) and XIX., this new rule requires only four logarithms to be taken from Table XXVII. to four places of figures, and to the nearest minute, it being in general unnecessary to proportion for the seconds. We shall now give the rule for correcting the distance, and shall, for brevity, use the words sine, secant, and cosecant, instead of log. sine, log. secant, and log.cosecant, respectively, and the same practice will be observed in the second, third, and fourth methods of correcting the distance. small error in the result of the calculation of a lunar observation, so that for all practical purposes the above numbers may be esteemed as sufficiently exact. It may also be observed, that the error arising from this source will not generally be greater than that arising from neglecting the equations depending on the spheroidal form of the earth, and on the density and temperature of the air; equations which are almost always neglected. If any one wishes to obtain the apparent altitudes strictly, he must, from the observed altitudes subtract the dip of the horizon taken from Table XIII., and add or subtract the semidiameter of the object, according as the lower or upper limb is observed. In strictness, when the horizontal parallax differs from those in the table, we ought to take he numbers for the next greater and the next less number, and take a proportional part of the differences FIRST METHOD Of correcting the apparent distance of the moon from the sun,* in which there is no variety of cases, all the corrections being additive. Add the apparent distance of the moon from the sun to their apparent altitudes, and note the half-sum. The difference between the half-sum and the apparent distance call the first remainder; and the difference between the half-sum and the sun's apparent altitude call the second remainder. Take from Table XXVII. the following logarithms, which mark beneath each other m two columns, viz. the sine of the apparent distance, to be marked in both columns, the cosecant of the second remainder, to be marked also in both columns, the secant of the first remainder to be placed in the first column, and the secant of the half-sum in the second column.f Enter Table XVIII. (or Table XVII. if a star or planet be used), and take out the correction corresponding to the sun's altitude (or star or planet's); take also from the same table the corresponding logarithm, which place in column 1st. Enter Table XIX. with the moon's apparent altitude and horizontal parallax; find the corresponding correction, which place under the former correction, and the logarithm, which place in column 2d. The sum of the four logarithms of column first will be the proportional logarithm of the first correction, and the sum of the logarithms of column second will be the proportional logarithm of the second correction; these corrections being found in Table XXII. are to be placed under the former corrections. Enter Table XX., and find the numbers which most nearly agree with the observed distance and the observed altitudes of the objects, and take out the corresponding correction in seconds, which is to be placed under those already found. Then, by adding all these corrections to the apparent distance, decreased by 2°, we shall get the true distance nearly.‡ To determine the longitude from the true distance. if the true distance of the objects can be found in the Nautical Almanac, in either of the pages where the distances are marked, on the day of the observation, the time will be found at the top of the page. If the true distance cannot be found exactly, in the Nautical Almanac, you must find the two which are nearest to it, the one greater and the other less than the true distance; and take out that one which corresponds with the earliest or first of these times, with the corresponding proportional logarithm. Find the difference between this first distance and the true distance, and take out its proportional logarithm from Table XXII. The difference between these two proportional logarithms will be the proportional logarithm of a portion of time, to be added to the time standing over the first distance in the Nautical Almanac, and the sum will be the mean time of the observation at Greenwich. The difference between this time and the mean time at the ship, being turned into degrees and minutes by Table XXI., will be the true longitude of the ship from Greenwich, at the time of observation. This longitude will be east if the time at the ship be greater than that at Greenwich, otherwise west.§ To exemplify the preceding rules, we shall now give several examples of correcting the apparent distance, including also the preparation and the determination of the longitude from the true distance. * This rule is the same as that for correcting the distance of the moon from a star or a planet, except in reading star or planet for sun, and using Table XVII. instead of Table XVIII. Rejecting always the tens in the indices. The distance obtained by this rule is not perfectly correct, since several small corrections must be applied to obtain the true distance to the nearest second, viz. (1) The refraction taken from Table XII. which is made use of in constructing Tables XVII. XVIII. and XIX., ought to be corrected for the different heights of the barometer and thermometer, as directed in page 154. (2) A correction must be applied for the spheroidal figure of the earth. And (3) a very small correction ought to be made in the numbers of Table XX. when the D's horizontal parallax varies from 57′ 30′′. But to notice all these corrections would increase the calculation very much, and the result of a single observation, u. which all these things were noticed, would probably not be so accurate as the mean of two or three observations, taken at different times of the day, in which these corrections were neglected; and the time necessary to take and work the latter observations would not be much greater than to work a single observation, in which all the corrections were noticed. It may be necessary to observe that, if the times at the ship and Greenwich fall on different days the latest day is to be reckoned the greatest, though the hour of the day may be the least; thus, 17 day hour is to be esteemed greater than 16th day 22 hours EXAMPLE 1. Suppose that, on the 7th of January, 1836, sea account, at 11m 57 past midnight, mean time, in the longitude of 127° 30′ E., by account, the observed distance of the farthest limb of the moon from the star Aldebaran, was 68° 36' 00", the observed altitude of the star 32° 14', and the observed altitude of the moon's lower limb 34° 43' Required the true longitude. App. dist. 68° 21' 15 37 2 14 25 30 25 1 Rem. 0 38. Sec... 0.0000 Half-sum 67° 43'. Sec. 0.4212 Table XIX.*...... True distance..... 68° 03′ 00 This corr. = Corr. Tab. XIX. 15′ 05′′ + Corr. Tab. A. 29" + Corr. Tab. B. 3" 15′ 37. = |