in Geography and Navigation, is the distance of any place from another eastward or westward, counted in degrees upon the equator; but when the distance is reckoned by leagues or miles and not in degrees, or in degrees on the meridian, and not of the parallel of latitude, in which case it includes both latitude and longitude, it is called departure.

To find the longitude at sea, is a problem to which the attention of navigators and mathematicians has been drawn ever since navigation began to be improved.—The importance of this problem soon became so well known, that, in 1598, Philip III. of Spain offered a reward of 1000 crowns for the solution; and his example was soon followed by the States General, who offered 10,000 florins. In 1714 an act was passed in the British parliament, empowering certain commissioners to make out a bill for a sum not exceeding 2000l. for defraying the necessary expenses of experiments for ascertaining this point; and likewise granting a reward to the person who made any progress in the solution, proportionable to the degree of accuracy with which the solution was performed: 10,000l. was granted if the longitude should be determined to one degree of a great circle, or 60 geographical miles; 15,000l. if to two-thirds of that distance; and 20,000l. if to half the distance.

In consequence of these proffered rewards, innumerable attempts were made to discover this important secret. The first was that of John Morin professor of mathematics at Paris, who proposed it to Cardinal Richelieu; and though it was judged insufficient on account of the imperfection of the lunar tables, a pension of 2000 livres per annum was procured for him in 1645 by Cardinal Mazarine. Gemma Frisius had indeed, in 1530, projected a method of finding the longitude by means of watches, which at that time were newly invented: but the structure of these machines was then by far too imperfect to admit of any attempt; nor even Longitude in 1631, when Metius made an attempt to this purpose, were they advanced in any considerable degree. About the year 1664, Dr Hooke and Mr Huygens made a very great improvement in watchmaking, by the application of the pendulum spring. Dr Hooke having quarrelled with the ministry, no experiment was made with any of his machines; but many were made with those of Mr Huygens. One experiment particularly, made by Major Holmes, in a voyage from the coast of Guinea in 1665, answered so well, that Mr Huygens was encouraged to improve the structure of his watches; but it was found that the variations of heat and cold produced such alterations in the rate of going of the watch, that unless this could be remedied, the watches could be of little use in determining the longitude.

In 1714 Henry Sully, an Englishman, printed a small tract at Vienna upon the subject of watchmaking. Having afterwards removed to Paris, he applied himself to the improvement of time-keepers for the discovery of the longitude. He taught the famous Julian de Roy; and this gentleman, with his son, and M. Berthoud, are the only persons who, since the days of Sully, have turned their thoughts this way. But though experiments have been made at sea with some of their watches, it does not appear that they have been able to accomplish any thing of importance with regard to the main point. The first who succeeded in any considerable degree was Mr John Harrison; who, in 1726, produced a watch which went so exactly, that for ten years together it did not err above one second in a month. In 1736 it was tried in a voyage to Lisbon and back again, on board one of his majesty's ships; during which it corrected an error of a degree and a half in the computation of the ship's reckoning. In consequence of this he received public encouragement to go on; and by the year 1761 had finished three time-keepers, each of them more accurate than the former. The last turned out so much to his satisfaction, that he now applied to the commissioners of longitude for leave to make an experiment with his watch in a voyage to the West Indies. Permission being granted, his son Mr William Harrison set out in his majesty's ship the Deptford for Jamaica in the month of November 1761. This trial was attended with all imaginable success. The longitude of the island, as determined by the time-keeper, differed from that found by astronomical observations only one minute and a quarter of the equator; the longitudes of places seen by the way being also determined with great exactness. On the ship's return to England, it was found to have erred no more during the whole voyage than $1' 54''$ in time, which is little more than 28 miles in distance; which being within the limits prescribed by the act, the inventor claimed the whole 20,000l. offered by government. Objections to this, however, were soon started. Doubts were pretended about the real longitude of Jamaica, as well as the manner in which the time had been found both there and at Portsmouth. It was alleged also, that although the time-keeper happened to be right at Jamaica, and after its return to England, this was by no means a proof that it had always been so in the intermediate times; in consequence of which allegations, another trial was appointed in a voyage to Barbadoes. Precautions were now taken to obviate as many of these objections as possible. The commissioners sent out proper persons to make astronomical observations at that island; which, when compared with others in England, would ascertain beyond a doubt its true situation. In 1764 then, Mr Harrison junior set sail for Barbadoes; and the result of the experiment was, that the difference of longitude between Portsmouth and Barbadoes was shown by the time-keeper to be $3h. 55' 3''$, and by astronomical observations to be $3h. 54' 20''$, the error being now only $43''$ of time, or $10' 45''$ of longitude. In consequence of this and the former trials, Mr Harrison received one half of the reward promised, upon making a discovery of the principles upon which his time-keeper was constructed. He was likewise promised the other half of the reward as soon as time-keepers should be constructed by other artists which should answer the purpose as well as those of Mr Harrison himself. At this time he delivered up all his time-keepers, the last of which was sent to Greenwich to be tried by Mr Nevil Maskelyne, the astronomer-royal. On trial, however, it was found to go with much less regularity than had been expected; but Mr Harrison attributed this to his having made some experiments with it which he had not time to finish when he was ordered to deliver up the watch. Soon after this, an agreement was made by the commissioners with Mr Kendall to construct a watch upon Mr Harrison's principles; and this upon trial was found to answer the purpose even better than any that Harrison himself had constructed. This watch was sent out with Captain Cook in 1772; and during all the time of his voyage round the world in 1772, 1773, 1774, and 1775, never erred quite $1\frac{1}{4}$ seconds per day; in consequence of which, the house of commons, in 1774, ordered the other 10,000l. to be paid to Mr Harrison. Still greater accuracy, however, has been attained. A watch was lately constructed by Mr Arnold, which, during a trial of 13 months, from February 1779 to February 1780, varied no more than $6.69''$ during any two days; and the greatest difference between its rates of going on any day and the next to it was $4.11''$. The greatest error it would have committed therefore in the longitude during any single day would have been very little more than one minute of longitude; and thus might the longitude be determined with as great exactness as the latitude generally can.—This watch, however, has not yet been tried at sea.

Thus the method of constructing time-keepers for discovering the longitude seems to be brought to as great a degree of perfection as can well be expected. Still, however, as these watches are subject to accidents, and may thus alter the rate of their going without any possibility of a discovery, it is necessary that some other method should be fallen upon, in order to correct from time to time those errors which may arise either from the natural going of the watch, or from any accident which may happen to it. Methods of this kind are all founded upon celestial observations of some kind or other; and for these methods, or even for an improvement in time-keepers, rewards are still held out by government. After the discoveries made by Mr Harrison, the act concerning the longitude was repealed, excepting so much of it as related to the constructing, printing, publishing, &c. of nautical almanacks and other useful tables. It was enacted also, longitude, that any person who shall discover a method for finding the longitude by means of a time-keeper, the principles of which have not hitherto been made public, shall be entitled to a reward of £500. If, after certain trials made by the commissioners, the said method shall enable a ship to keep her longitude, during a voyage of six months, within 60 geographical miles, or a degree of a great circle. If the ship keeps her longitude within 40 geographical miles for that time, the inventor is entitled to a reward of £7500, and to £10,000 if the longitude is kept within half a degree. If the method is by improved astronomical tables, the author is entitled to £5000, when they show the distance of the moon from the sun and stars within 15 seconds of a degree, answering to about 7 minutes of longitude, after allowing half a degree for errors of observation and under certain restrictions, and after comparison with astronomical observations for a period of 18½ years, during which the lunar irregularities are supposed to be completed. The same rewards are offered to the person who shall with the like accuracy discover any other method of finding the longitude.

These methods require celestial observations; and any of the phenomena, such as the different apparent places of stars with regard to the moon, the beginning and ending of eclipses, &c., will answer the purpose: only it is absolutely necessary that some variation should be perceptible in the phenomenon in the space of two minutes; for even this short space of time will produce an error of 40 miles in longitude. The most proper phenomena therefore for determining the longitude in this manner are the eclipses of Jupiter's satellites. Tables of their motions have been constructed, and carefully corrected from time to time, as the mutual attractions of these bodies are found greatly to disturb the regularity of their motions. The difficulty here, however, is to observe these eclipses at sea; and this difficulty has been found so great, that no person seems able to surmount it. The difficulty arises from the violent agitation of a ship in the ocean, for which no adequate remedy has ever yet been found, nor probably will ever be found. Mr. Christopher Irwin indeed invented a machine which he called a marine chair, with a view to prevent the effects of this agitation; but on trying it in a voyage to Barbadoes, it was found to be totally useless.

A whimsical method of finding the longitude was proposed by Messrs Whiston and Ditton from the report and flash of great guns. The motion of sound is known to be nearly equable, from whatever body it proceeds or whatever be the medium. Supposing therefore a mortar to be fired at any place the longitude of which is known, the difference between the moment that the flash is seen and the report heard will give the distances between the two places; whence, if we know the latitudes of these places, their longitudes must also be known. If the exact time of the explosion be known at the place where it happens, the difference of time at the place where it is heard will likewise give the difference of longitude. Let us next suppose the mortar to be loaded with an iron shell filled with combustible matter, and fired perpendicularly upward into the air, the shell will be carried to the height of a mile, and will be seen at the distance of near 100; whence, supposing neither the flash of the Longitude, mortar should be seen nor the report heard, still the longitude might be determined by the altitude of the shell above the horizon.

According to this plan, mortars were to be fired at certain times and at proper stations along all frequented coasts for the direction of mariners. This indeed might be of use, and in stormy weather might be a kind of improvement in lighthouses, or a proper addition to them; but with regard to the determination of longitudes, is evidently ridiculous.

We shall now proceed to give some practical directions for finding the longitude at sea by proper celestial observations; exclusive of those from Jupiter's satellites, which, for reasons just mentioned, cannot be practised at sea. In the first place, however, it will be necessary to point out some of those difficulties which stand in the way, and which render even this method of finding the longitude precarious and uncertain. These lie principally in the reduction of the observations of the heavenly bodies made on the surface of the earth to similar observations supposed to be made at the centre; which is the only place where the celestial bodies appear in their proper situation. It is also very difficult to make proper allowances for the refraction of the atmosphere, by which all objects appear higher than they really are; and another difficulty arises from their parallaxes, which make them, particularly the moon, appear lower than they would otherwise do, excepting when they are in the very zenith. It is also well known, that the nearer the horizon any celestial body is, the greater its parallax will be; and as the parallax and refraction act in opposite ways to one another, the former depressing and the latter raising the object, it is plain, that great difficulties must arise from this circumstance. The sun, for instance, whose parallax is less than the refraction, must always appear higher than he really is; but the moon, whose parallax is greater than her refraction, must always appear lower.

To render observations of the celestial bodies more easy, the commissioners of longitude have caused an Ephemeris or Nautical Almanack to be published annually, containing every requisite for solving this important problem which can be put into any form of tables. But whatever may be done in this way, it will be necessary to make the necessary preparations concerning the dip of the horizon, the refraction, semidiameters, parallax, &c., in order to reduce the apparent to the true altitudes and distances; for which we shall subjoin two general rules.

The principal observation for finding the longitude at sea is that of the moon from the sun, or from some remarkable star near the zodiac. To do this, the operator must be furnished with a watch which can be depended upon for keeping time within a minute for six hours; and with a good Hadley's quadrant, or, which is preferable, a sextant; and this last instrument will still be more fit for the purpose if it be furnished with a screw for moving the index gradually; likewise an additional dark glass, but not so dark as the common kind, for taking off the glare of the moon's light in observing her distance from a star. A small telescope, which may magnify three or four times, is also necessary to render the contact of a star with the moon's limb more discernible. A magnifying glass of one Longitude, one and a half or two inches focus will likewise assist the operator in reading off his observations with the greater facility.

1. To make the observation. Having examined and adjusted his instrument as well as possible, the observer is next to proceed in the following manner: If the distance of the moon from the sun is to be observed, turn down one of the screens; look at the moon directly through the transparent part of the horizon-glass; and keeping her in view, gently move the index till the sun's image be brought into the silvered part of that glass. Bring the nearest limbs of both objects into contact; and let the quadrant librate a little on the lunar ray; by which means the sun will appear to rise and fall by the side of the moon; in which motion the nearest limbs must be made to touch one another exactly by moving the index. The observation is then made; and the division coinciding with that on the Vernier scale, will show the distance of the nearest limbs of the objects.

When the distance of the moon from a star is to be observed when the moon is very bright, turn down the lightest screen, or use a dark glass lighter than the screens, and designed for this particular purpose; look at the star directly through the transparent part of the horizon-glass; and keeping it there, move the index till the moon's image is brought into the silvered part of the same glass. Make the quadrant librate gently on the star's ray, and the moon will appear to rise and fall by the star: move the index between the librations, until the moon's enlightened limb is exactly touched by the star, and then the observation is made. In these operations, the plane of the quadrant must always pass through the two objects, the distance of which is to be observed; and for this purpose it must be placed in various positions according to the situation of the objects, which will soon be rendered easy by practice.

The observation being made, somebody at the very instant that the operator calls observe by the watch the exact hour, minute, and quarter minute, if there be no second hand, in order to find the apparent time; and at the same instant, or as quick as possible, two assistants must take the altitudes of those objects the distance of which is observed; after which the observations necessary for finding the longitude are completed.

The Ephemeris shows the moon's distance from the sun, and likewise from proper stars, to every three hours of apparent time for the meridian of Greenwich; and that the greater number of opportunities of observing this luminary may be given, her distance is generally set down from at least one object on each side of her. Her distance from the sun is set down while it is between 40 and 120 degrees; so that, by means of a sextant, it may be observed for two or three days after her first and before her last quarter. When the moon is between 40 and 90 degrees from the sun, her distance is set down both from the sun and from a star on the contrary side: and, lastly, when the distance is above 120 degrees, the distance is set down from two stars, one on each side of her. The distance of the moon from objects on the east side of her is found in the Ephemeris in the 8th and 9th pages of the month; and her distance from objects on the west is found in the 10th and 11th pages of the month.

When the Ephemeris is used, the distance of the moon must only be observed from these stars the distance of which is set down there; and these afford a ready means of knowing the star from which her distance ought to be observed. The observer has then nothing more to do than to set his index to the distance roughly computed at the apparent time, estimated nearly for the meridian at Greenwich; after which he is to look to the east or west of the moon, according as the distance of the star is found in the 8th or 9th, or in the 10th or 11th, pages of the month; and having found the moon upon the horizon-glass, the star will easily be found by sweeping with the quadrant to the right or left, provided the air be clear and the star be in the line of the moon's shortest axis produced. The time at Greenwich is estimated by turning into time the supposed longitude from that place, and adding it to the apparent time at the ship, or subtracting it from it as occasion requires. The distance of the moon from the sun, or a star, is roughly found at this time, by saying, As 180 minutes (the number contained in three hours) is to the difference in minutes between this nearly estimated time and the next preceding time set down in the Ephemeris; so is the difference in minutes between the distance in the Ephemeris for the next preceding and next following times, to a number of minutes: which being added to the next preceding distance, or subtracted from it, according as it is increasing or decreasing, will give the distance nearly at the time the observation is to be made, and to which the index must be set.

An easier method of finding the angular distance is by bringing the objects nearly into contact in the common way, and then fixing the index tight to a certain degree and minute; waiting until the objects are nearly in contact, giving notice to the assistants to get ready with the altitudes, and when the objects are exactly in contact to call for the altitudes and the exact time by the watch. The observer may then prepare for taking another distance, by setting his index three or four minutes backwards or forwards, as the objects happen to be receding from or approaching to each other; thus proceeding to take the distance, altitudes, and time by the watch, as before. Thus the observer may take as many distances as he thinks proper; but four at the distance of three minutes, or three at the distance of four minutes, will at all times be sufficient. Thus not only the eye of the observer will be less fatigued, but he will likewise be enabled to manage his instrument with much greater facility in every direction, a vertical one only excepted. If in taking the distances the middle one can be taken at any even division on the arch, such as a degree, or a degree and 20 or 40 minutes, that distance will be independent of the Nonius division, and consequently free of those errors which frequently arise from the inequality of that division in several parts of the graduated arch. The observation ought always to be made about two hours before or after noon; and the true time may be found by the altitude of the sun taken at the precise time of the distance. If three distances are taken, then... Longitude then find the time by the altitude corresponding with the middle distance; and thus the observation will be secured from any error arising from the irregularity of the going of the watch. As the time, however, found by the altitude of a star cannot be depended upon, because of the uncertainty of the horizon in the night, the best way of determining the time for a night observation will be by two altitudes of the sun; one taken on the preceding afternoon, before he is within six degrees of the horizon; and the other on the next morning, when he is more than six degrees high. It must be observed, however, that in order to follow these directions, it is necessary that the atmosphere should be pretty free from clouds; otherwise the observer must take the observations at such times as he can best obtain them.

2. To reduce the observed Distance of the Sun or a Star from the Moon to the true Distance. 1. Turn the longitude into time, and add it to the time at the ship if the longitude be west, but subtract it if it be east, which will give the supposed time at Greenwich; and this we may call reduced time. 2. Find the nearest noon or midnight both before and after the reduced time in the seventh page of the month in the Ephemeris. 3. Take out the moon's semidiameter and horizontal parallaxes corresponding to these noons and midnights, and find their differences. Then say, as 12 hours is to the moon's semidiameter in 12 hours, so is the reduced time to a number of seconds; which, either added to or subtracted from the moon's semidiameter at the noon or midnight just mentioned, according as it is increasing or decreasing, will give her apparent semidiameter; to which add the correction from Table VIII. of the Ephemeris, and the sum will be her true semidiameter at the reduced time. And as 12 hours is to the difference of the moon's horizontal parallax in 12 hours, so is the reduced time to a fourth number; which being added to or subtracted from the moon's horizontal parallax at the noon or midnight before the reduced time, according as it is increasing or decreasing, the sum or difference will be the moon's horizontal parallax at the reduced time. 4. If the reduced time be nearly any even part of 12 hours, viz., 6th, 3rd, &c., these parts of the difference may be taken, and either added or subtracted according to the directions already given, without being at the trouble of working by the rule of proportion. 5. To the observed altitude of the sun's lower limb add the difference betwixt the semidiameter and dip; and that sum will be his apparent altitude. 6. From the sun's refraction take his parallax in altitude, and the remainder will be the correction of the sun's altitude. 7. From the star's observed altitude take the dip of the horizon, and the remainder will be the apparent altitude. 8. The refraction of a star will be the correction of its altitude. 9. Take the difference between the moon's semidiameter and dip, and add it to the observed altitude if her lower limb was taken, or subtract it if her upper limb was taken; and the sum or difference will be the apparent altitude of her centre. 10. From the proportional logarithm of the moon's horizontal parallax, taken out of the nautical almanack (increasing its index by 10), taking the logarithmic cosine of the moon's apparent altitude, the remainder will be the proportional logarithm of her parallax in altitude; from which take her refraction, and the remainder will be the correction of the moon's altitude. 11. To the observed distance of the moon from a star add her semidiameter if the nearest limb be taken, but subtract it if the farthest limb was taken, and the sum or difference will be the apparent distance. 12. To the observed distance of the sun and moon add both their semidiameters, and the sum will be the apparent distance of their centres.

3. To find the true Distance of the Objects, having their apparent Altitudes and Distances. 1. To the proportional logarithm of the correction of the sun or star's altitude, add the logarithmic cosine of the sun or star's apparent altitude; the logarithmic sine of the apparent distance of the moon from the sun or star; and the logarithmic cosecant of the moon's apparent altitude. The sum of these, rejecting 30 from the index, will be the proportional logarithm of the first angle. 2. To the proportional logarithm of the correction of the sun or star's altitude, add the logarithmic cotangent of the sun or star's apparent altitude, and the logarithmic tangent of the apparent distance of the moon from the sun or star. The sum of these, rejecting 20 in the index, will be the proportional logarithm of the second angle. 3. Take the difference between the first and second angles, adding it to the apparent distance if it be less than 90°, and the first angle be greater than the second; but subtracting it if the second be greater than the first. If the distance be greater than 90°, the sum of the angles must be added to the apparent distance, which will give the distance corrected for the refraction of the sun or star. 4. To the proportional logarithm of the correction of the moon's altitude add the logarithmic cosine of her apparent altitude; the logarithmic sine of the distance corrected for the sun or star's refraction and the logarithmic cosecant of the sun's or star's apparent altitude. The sum, rejecting 30 in the index, will be the proportional logarithm of the third angle. 5. To the proportional logarithm of the correction of the moon's apparent altitude, add the logarithmic cotangent of her apparent altitude, and the tangent of the distance corrected for the sun or star's refraction; their sum, rejecting 20 in the index, will be the proportional logarithm of the fourth angle. 6. Take the difference between the third and fourth angles, and subtract it from the distance corrected for the sun or star's refraction if less than 90°, and the third angle be greater than the fourth; or add it to the distance if the fourth angle be greater than the third: but if the distance be more than 90°, the sum of the angles must be subtracted from it, to give the distance corrected for the sun or star's refraction, and the principal effects of the moon's parallax. 7. In Table XX. of the Ephemeris, look for the distance corrected for the sun and star's refraction, and the moon's parallax in the top column, and the correction of her altitude in the left-hand side column; take out the number of seconds that stand under the former, and opposite to the latter. Look again in the same table for the corrected distance in the top column, and the principal effects of the moon's parallax in the left-hand side column, and take out the number of seconds. The difference between these two numbers... Longitude numbers must be added to the corrected distance if less than 90, but subtracted from it if greater; and the sum or difference will be the true distance.

4. To determine the Longitude after having obtained the true Distance.—Look in the Ephemeris among the distances of the objects for the computed distance betwixt the moon and the other object observed on the given day. If it be found there, the time at Greenwich will be at the top of the column; but if it falls between two distances in the Ephemeris which stand immediately before and after it, and also the difference between the distance standing before and the computed distance; then take the proportional logarithms of the first and second differences, and the difference between these two logarithms will be the proportional logarithm of a number of hours, minutes, and seconds; which being added to the time standing over the first distance, will give the true time at Greenwich. Or it may be found by saying, As the first difference is to three hours, so is the second difference to a proportional part of time: which being added as above directed, will give the time at Greenwich. The difference between Greenwich time and that at the ship, turned into longitude, will be that at the time the observations were made; and will be east if the time at the ship is greatest, but west if it is least.

Having given these general directions, we shall next proceed to show some particular examples of finding the longitude at sea by all the different methods in which it is usually tried.

1. To find the longitude by Computation from the Ship's Course.—Were it possible to keep an accurate account of the distance the ship has run, and to measure it exactly by the log* or any other means, then both latitude and longitude would easily be found by setting the ship's account to that time. For the course and distance being known, the difference of latitude and departure is readily found by the Traverse Table: and the difference of longitude being known, the true longitude and latitude will also be known. A variety of causes, however, concur to render this computation inaccurate; particularly the ship's continual deflection from the course set by her playing to the right and left round her centre of gravity: the unequal care of those at the helm, and the distance supposed to be sailed being erroneous, on account of stormy seas, unsteady winds, currents, &c. for which it seems impossible to make any allowance. The place of the ship, however, is judged of by finding the latitude every day, if possible, by observations; and if the latitude found by observation agrees with that by the reckoning, it is presumed that the ship's place is properly determined; but if they disagree, it is concluded that the account of the longitude stands in need of correction, as the latitude by observation is always to be depended upon.

Currents very often occasion errors in the computation of a ship's place. The causes of these in the great depths of the ocean are not well known, though many of the motions near the shore can be accounted for. It is supposed that some of those in the great oceans are owing to the tide following the moon, and a certain libration of the waters arising from thence; likewise that the unsettled nature of these currents may be owing to the changes in the moon's declination. In the torrid zone, however, a considerable current is occasioned by the trade winds, the motion being constantly to the west, at the rate of eight or ten miles per day. At the extremities of the trade winds, or near the 30th degree of north or south latitude, the currents are probably compounded of this motion to the westward, and of one towards the equator; whence all ships sailing within these limits ought to allow a course each day for the current.

When the error is supposed to have been occasioned by a current, it ought if possible to be tried whether the case is so or not; or we must make a reasonable estimate of its drift and course. Then with the setting and drift, as a course and distance, find the difference of latitude and departure; with which the dead reckoning is to be increased or diminished; and if the latitude thus corrected agrees with that by observation, the departure thus corrected may be safely taken as true, and thus the ship's place with regard to the longitude determined.

Exam. Suppose a ship in 24 hours finds, by her dead reckoning, that she has made 96 miles of difference of latitude north and 38 miles of departure west; but by observation finds her difference of latitude 112, and on trial that there is a current which in 24 hours makes a difference of 16 miles latitude north, and 10 miles of departure east: Required the ship's departure.

| Miles. | Departure by account | Miles. | |--------|----------------------|--------| | Diff. lat. by account 96 N. | 38 W. | | Diff. lat. by current 16 N. | Departure by current 10 | | True diff. lat. 112 | 28 W. |

Here the dead reckoning corrected by the current gives the difference of latitude 112 miles, which is the same as that found by observation; whence the departure 28 is taken as the true one.

When the error is supposed to arise from the courses and distances, we must observe, that if the difference of latitude is much more than the departure, or the direct course has been within three points of the meridian, the error is most probably in the distance. But if the departure be much greater than the difference of latitude, or the direct course be within three points of the parallel, or more than five points from the meridian, the error is probably to be ascribed to the course. But if the courses in general are near the middle of the quadrant, the error may be either in the course, or in the distance, or both. This method admits of three cases.

1. When, by the dead reckoning, the difference of latitude is more than once and a half the departure; or when the course is less than three points: Find the course to the difference of latitude and departure. With this course and the meridional difference of latitude by observation, find the difference of longitude.

2. When the dead reckoning is more than once and a half the difference of latitude; or when the course is more than five points: Find the course and distance, with the difference of latitude by observation, and departure by account; then with the co-middle latitude by observation, and departure by account, find the difference of longitude.

3. When 3. When the difference of latitude and departure by account is nearly equal, or the direct course is between three and five points of the meridian: Find the course with the difference of latitude and departure by account since the last observation. With this course and the difference of latitude by observation find another departure. Take half the sum of these departures for the true one. With the true departure and difference of latitude by observation find the true course; then with the true course and meridional difference of latitude find the difference of longitude.

2. To find the Longitude at Sea by a Variation-chart.—Dr Halley having collected a great number of observations on the variation of the needle in many parts of the world; by that means was enabled to draw certain lines on Mercator's chart, shewing the variation in all the places over which they passed in the year 1700, at which time he first published the chart; whence the longitude of those places might be found by the chart, provided its latitude and variation were given. The rule is, Draw a parallel of latitude on the chart through the latitude found by observation; and the point where it cuts the curved line marked with the variation that was observed will be the ship's place.

Exam. A ship finds by observation the latitude to be $18^\circ 20'$ north, and the variation of the compass to be $4^\circ$ west. Required the ship's place.—Lay a ruler over $18^\circ 20'$ north parallel to the equator; and the point where its edge cuts the curve of $4^\circ$ west variation gives the ship's place, which will be found in about $27^\circ 10'$ west from London.

This method of finding the longitude, however, is attended with two inconveniences. 1. That when the variation lines run east or west, or nearly so, it cannot be applied; though as this happens only in certain parts of the world, a variation chart may be of great use for the rest. Even in those places indeed where the variation curves do run east or west, they may be of considerable use in correcting the latitude when meridian observations cannot be had; which frequently happens on the northern coasts of America, the Western ocean, and about Newfoundland; for if the variation can be found exactly, the east and west curve answering to it will show the latitude. But, 2. The variation itself is subject to continual change; whence a chart, though ever so perfect at first, must in time become totally useless; and hence the charts constructed by Dr Halley, though of great utility at their first publication, became at length almost entirely useless. A new one was published in 1746 by Messrs Mountain and Dodson, which was so well received, that in 1756 they again drew variation lines for that year, and published a third chart the year following. They also presented to the Royal Society a curious paper concerning the variation of the magnetic needle, with a set of tables annexed containing the result of more than 50,000 observations, in six periodical reviews from the year 1700 to 1756 inclusive, adapted to every five degrees of latitude and longitude in the more frequented oceans; all of which were published in the Philosophical Transactions for 1757.

3. To find the Longitude by the Sun's Declination.—Having made such observations on the sun as may enable us to find his declination at the place, take the difference between this computed declination and that shown at London by the Ephemeris; from which take also the daily difference of declination at that time; then say, as the daily difference of declination is to the above found difference, so is $360^\circ$ degrees to the difference of longitude. In this method, however, a small error in the declination will make a great one in the longitude.

4. To find the Longitude by the Moon's culminating.—Seek in the Ephemeris for the time of her coming to the meridian on the given day and on the day following, and take their difference; also take the difference betwixt the times of culminating on the same day as found in the ephemeris and as observed; then say, as the daily difference in the ephemeris is to the difference between the ephemeris and observation; so is $360^\circ$ degrees to the difference of longitude. In this method also a small difference in the culmination will occasion a great one in the longitude.

5. By Eclipses of the Moon.—This is done much in the same manner as by the eclipses of Jupiter's satellites: For if, in two or more distant places where an eclipse of the moon is visible, we carefully observe the times of the beginning and ending, the number of digits eclipsed, or the time when the shadow touches some remarkable spot, or when it leaves any particular spot on the moon, the difference of the times when the observations were made will give the difference of longitude. Phenomena of this kind, however, occur too seldom to be of much use.

6. In the 76th volume of the Philosophical Transactions, Mr Edward Pigot gives a very particular account of his method of determining the longitude and latitude of York; in which he also recommends the method of determining the longitude of places by observations of the moon's transit over the meridian. The instruments used in his observations were a gridiron pendulum clock, a two feet and a half reflector, an eighteen inch quadrant made by Mr Bird, and a transit instrument made by Mr Sisson.

By these instruments an observation was made, on the 10th of September 1783, of the occultation of a star of the ninth magnitude by the moon, during an eclipse of that planet, at York and Paris. Besides this, there were observations made of the immersions of $\phi$ Aquarii and $\delta$ Piscium; the result of all which was, that between Greenwich and York the difference of meridians was $4' 27''$.

In 1783, Mr Pigot informs us, that he thought of finding the difference of meridians by observing the meridian right ascensions of the moon's limb. This he thought had been quite original: but he found it afterwards in the Nautical Almanack for 1769, and in 1784 read a pamphlet on the same subject by the abbé Toaldo; but still found that the great exactness of this method was not suspected; though he is convinced that it must soon be universally adopted in preference to that from the first satellite of Jupiter.

After giving a number of observations on the satellites of Jupiter, he concludes, that the exactness expected from observations, even on the first satellite, is much overrated. "Among the various objections (says he), there is one I have often experienced, and, which proceeds solely from the disposition of the eye, that of seeing more distinctly at one time than another. It may not be improper also to mention, that the observation..." Longitude I should have relied on as the best, that of August 30, 1785, marked excellent, is one of those most distant from the truth."

After giving a number of observations on the eclipse of the moon September 10, 1783, our author concludes, that the eclipses of the moon's spots are in general too much neglected, and that it might be relied upon much more were the following circumstances attended to:

1. To be particular in specifying the clearness of the sky. 2. To choose such spots as are well defined, and leave no hesitation as to the part eclipsed. 3. That every observer should use, as far as possible, telescopes equally powerful, or at least let the magnifying powers be the same.

"A principal objection (says he) may still be urged, viz. the difficulty of distinguishing the true shadow from the penumbra. Was this obviated, I believe the results would be more exact than from Jupiter's first satellite: Undoubtedly the shadow appears better defined if magnified a little; but I am much inclined to think, that, with high magnifying powers, there is greater certainty of choosing the same part of the shadow, which perhaps is more than a sufficient compensation for the loss of distinctness."

The following rule for meridian observations of the moon's limb is next laid down: "The increase of the moon's right ascension in twelve hours (or any given time found by computation) is 12 hours, as the increase of the moon's right ascension between two places found by observation is to the difference of meridians.

Example.

November 30, 1782.

| h. | " | |----|---| | 13 | 12 57.62 Meridian transit of moon's second limb | By clock at Greenwich. | | 13 | 13 29.08 Ditto of a mR |

31.46 Difference of right ascension.

| h. | " | |----|---| | 13 | 14 8.05 Meridian transit of moon's second limb | By clock at York. | | 13 | 14 30.13 Ditto of a mR |

32.08 Difference at York

31.46 Difference at Greenwich,

9.38 Increase of the moon's apparent right ascension between Greenwich and York, by observation.

141" in seconds of a degree, ditto, ditto, ditto.

The increase of the moon's right ascension for 12 hours, by computation, is 23,349 seconds; and 12 hours reduced into seconds is 43,200. Therefore, according to the rule stated above,

\[ \frac{23,349}{43,200} : \text{diff. of merid.} = 261" \]

"These easy observations and short reduction (says Mr Pigot) are the whole of the business. Instead of computing the moon's right ascension for 12 hours, I have constantly taken it from the Nautical Almanacs, which give it sufficiently exact, provided some attention be paid to the increase or decrease of the moon's motion. Were the following circumstances attended to, the results would be undoubtedly much more exact.

1. Compare the observations with the same made in several other places. 2. Let several and the same stars be observed at these places. 3. Such stars as are nearest in right ascension and declination to the moon are infinitely preferable. 4. It cannot be too strongly urged, to get, as near as possible, an equal number of observations of each limb, to take a mean of each set, and then a mean of both means. This will in a great measure correct the error of telescopes and sight. 5. The adjustment of the telescopes to the eye of the observer before the observation is also very necessary, as the sight is subject to vary. 6. A principal error proceeds from the observation of the moon's limb, which may be considerably lessened, if certain little round spots near each limb were also observed in settled observatories; in which case the libration of the moon will perhaps be a consideration. 7. When the difference of meridians, or of the latitudes of places is very considerable, the change of the moon's diameter becomes an equation.

Though such are the requisites to use this method with advantage, only one or two of them have been employed in the observations that I have reduced. Two-thirds of these observations had not even the same stars observed at Greenwich and York; and yet none of the results, except a doubtful one, differ 15" from the mean; therefore I think we may expect a still greater exactness, perhaps within 10", if the above particulars be attended to.

When the same stars are not observed, it is necessary for the observers at both places to compute their right ascension from tables, in order to get the apparent right ascension of the moon's limb. Though this is not so satisfactory as by actual observation, still the difference will be trifling, provided the star's right ascensions are accurately settled. I am also of opinion, that the same method can be put in practice by travellers with little trouble, and a transit instrument, constructed so as to fix up with facility in any place. It is not necessary, perhaps, that the instrument should be perfectly in the meridian for a few seconds of time, provided stars, nearly in the same parallel of declination with the moon, are observed; nay, I am inclined to think, that if the instrument deviates even a quarter or half a degree, or more, sufficient exactness can be attained; as a table might be computed, showing the moon's parallax and motion for such deviation; which last may easily be found by the well-known method of observing stars whose difference of declination is considerable.

As travellers very seldom meet with situations to observe stars near the pole, or find a proper object for determining the error of the line of collimation, I shall recommend the following method as original.—Having computed the apparent right ascension of four, six, or more stars, which have nearly the same parallel of declination, observe half of them with the instrument inverted, and the other half when in its right position. If the difference of right ascensions between each set by observation agrees with the computation, there is no error; but if they disagree, half that disagreement is the error of the line of collimation. The same observations may also serve to deter- longitude, mine, whether the distances of the corresponding wires are equal. In case of necessity, each limb of the sun might be observed in the same manner, though probably with less precision. By a single trial I made above two years ago, the result was much more exact than I expected. Mayer's catalogue of stars will prove of great use to those that adopt the above method.—I am rather surprised that the immersions of known stars of the sixth and seventh magnitude, behind the dark limb of the moon, are not constantly observed in fixed observatories, as they would frequently be of great use."

The annexed rule for finding the ship's place, with the miscellaneous observations on different methods, were drawn up by Mr John McLean of Edinburgh.

1. With regard to determining the ship's place by the help of the course and distance sailed, the following rule may be applied.—It will be found as expeditious as any of the common methods by the middle latitude or meridional parts; and is in some respects preferable, as the common tables of sines and tangents only are requisite in applying it.—Let \(a\) and \(b\) be the distances of two places from the same pole in degrees, or their complete latitude; \(c\) the angle which a meridian makes with the rhumb line passing through the places; and \(L\) the angle formed by their meridians, or the difference of longitude in minutes: then \(A\) and \(B\) being the logarithmic tangents of \(\frac{a}{2}\) and \(\frac{b}{2}\), the sine of \(C\), and \(S\) the sine of \((C + L)\), we shall have the following equation:

\[ L = \frac{A \times B}{S' - S} \quad (A) \]

Also, from a well known property of the rhumb line, we have the following equation:

\[ S + E = R + D, \quad \text{where } S \text{ is the logarithmic cosine of } C, \quad E \text{ the logarithm of the length of the rhumb line, or distance, } D \text{ the logarithm of the minutes difference of latitude, and } R \text{ the logarithm of the radius.} \]

By the help of these two equations, we shall have an easy solution of the several cases to which the middle latitude, or meridional parts, are commonly applied.

Exam. A ship from a port in latitude \(56^\circ\) N. sails S.W. by W. till she arrives at the latitude of \(40^\circ\) N.: Required the difference of longitude?

Here \(a = 34^\circ\), \(b = 50^\circ\), \(c = 56^\circ 15'\), \(A = 9.48534\), \(B = 9.56107\), \(S = 9.9199308\), \(S' = 9.9198464\); therefore,

\[ L = \frac{A \times B}{S' - S} = \frac{757390}{844} = 897 \quad \text{the minutes difference of longitude.} \]

Also, \(S = 9.74474\), \(D = 2.98227\); therefore \(E = R + D - S = 3.23753\), to which the natural number is 1728, the miles in the rhumb line sailed over.

2. The common method of finding the difference of longitude made good upon several courses and distances, by means of the difference of latitude and departure made good upon the several courses, is not accurately true.

For example: If a ship should sail due south 600 miles, from a port in \(60^\circ\) north latitude, and then due west 600 miles, the difference of longitude found by the common methods of solution would be 1053: whereas the true difference of longitude is only 933, less than the former by 120 miles, which is more than one-eighth of the whole. Indeed every considerable alteration in the course will produce a very sensible error in the difference of longitude. Though, when the several rhumb lines sailed over are nearly in the same direction, the error in longitude will be but small.

The reason of this will easily appear from the annexed figure, in which the ship is supposed to sail from \(Z\) to \(A\), along the rhumb lines \(ZB, BA\); for if the meridians \(PZ, PkocBL\) be drawn; and very near the latter other two meridians \(PhD, Pmn\); and likewise the parallels of latitude \(Bn, De, mo, hk\); then it is plain that \(De\) is greater than \(hk\), (for \(De\) is to \(hk\) as the sine of \(DP\) to the sine of \(hP\)): and since this is the case everywhere, the departure corresponding to the distance \(BZ\) and course \(BZC\), will be greater than the departure to the distance \(oZ\) and course \(oZC\). And in the same manner, we prove that \(nB\) is greater than \(mo\); and consequently, the departure corresponding to the distance \(AB\), and course \(ABL\), is greater than the departure to the distance \(Ao\), and course \(AoL\). Wherefore, the sum of the two departures corresponding to the courses \(ABL\) and \(BZC\), and to the distances \(AB\) and \(BZ\), is greater than the departure corresponding to the distance \(AZ\) and course \(AZC\): therefore the course answering to this sum as a departure, and \(CZ\) as a difference of latitude, (\(AC\) being the parallel of latitudes passing through \(A\)), will be greater than the true course \(AZC\) made good upon the whole. And hence the difference of longitude found by the common rules will be greater than the true difference of longitudes; and the error will be greater or less according as \(BA\) deviates more or less from the direction of \(BZ\).

3. Of determining the ship's longitude by lunar observations.

Several rules for this purpose have been lately published, the principal object of which seems to have been

(A) \(A \propto B\) signifies the difference between \(A\) and \(B\). to abbreviate the computations requisite for determining the true distance of the sun or star from the moon's centre. This, however, should have certainly been less attended to than the investigation of a solution, in which considerable errors in the data may produce a small error in the required distance. When either of the luminaries has a small elevation, its altitude will be affected by the variability of the atmosphere; likewise the altitude, as given by the quadrant, will be affected by the inaccuracy of the instrument, and the uncertainty necessarily attending all observations made at sea. The sum of these errors, when they all tend the same way, may be supposed to amount to at least one minute in altitude; which, in many cases, according to the common rules for computing the true distance, will produce an error of about 30 minutes in the longitude. Thus, in the example given by Mons. Callet, in the Tables Portatives, if we suppose an error of one minute in the sun's altitude, or call it $6^\circ 26' 34''$, instead of $6^\circ 27' 34''$; we shall find the alteration in distance according to his rule to be $54''$, producing an error of about 27 minutes in the longitude; for the angle at the sun will be found, in the spherical triangle whose sides are the complement of the sun's altitude, complement of the moon's altitude, and observed distance, to be about $26^\circ$; and as radius is to the cosine of $26^\circ$, so is $16$ the supposed error in altitude, to $54''$ the alteration in distance. Perhaps the only method of determining the distance, so as not to be affected by the errors of altitude, is that by first finding the angles at the sun and moon, and by the help of them the corrections of distance for parallax and refraction. The rule is as follows:

Add together the complement of the moon's apparent altitude, the complement of the sun's apparent altitude, and the apparent distance of centres; from half the sum of these subtract the complement of the sun's altitude, and add together the logarithmic coscant of the complement of the moon's altitude, the logarithmic cosecant of the apparent distance of centres, the logarithmic sine of the half sum, and the logarithmic sine of the remainder; and half the sum of these four logarithms, after rejecting $20$ from the index, is the logarithmic cosine of half the angle at the longitude moon.

As radius is to the cosine of the angle at the moon; so is the difference between the moon's parallax and refraction in altitude to a correction of distance; which is to be added to the apparent distance of centres when the angle at the moon is obtuse; but to be subtracted when that angle is acute, in order to have the distance once corrected.

In the above formula, if the word sun be changed for moon, and vice versa, wherever these terms occur, we shall find a second correction of distance to be applied to the distance, once corrected by subtraction when the angle at the sun is obtuse, but by addition when that angle is acute, and the remainder or sum is the true distance nearly.

In applying this rule, it will be sufficient to use the complement, altitude, and apparent distances of centres, true to the nearest minute only, as a small error in the angles at the sun and moon will very little affect the corrections of distances.

If $D$ be the computed distance in seconds, $d$ the difference between the moon's parallax and refraction in altitude, $S$ the sine of the angle at the moon, and $R$ the radius; then $\frac{d^2 S^3}{2DR}$ will be the third correction of distance, to be added to the distance twice corrected: But it is plain from the nature of this correction, that it may be always rejected, except when the distance $D$ is very small, and the angle at the moon nearly equal to $90^\circ$.

This solution is likewise of use in finding the true distance of a star from the moon, by changing the word sun into star, and using the refraction of the star, instead of the difference between the refraction and parallax in the altitude of the sun, in finding the second correction of distance.

Ex. Given the observed distance of a star from the centre of the moon, $5^\circ 8' 41''$; the moon's altitude, $55^\circ 58' 5''$; the star's altitude, $19^\circ 18' 5''$; and the moon's horizontal parallax, $10^\circ 5' 5''$: Required the true distance.

| Cosec. | 0.02512 | *s co. alt. | 70° 42' | |--------|----------|-------------|---------| | D's co. alt. | 34 4 | Cossec. | 0.25169 | | Cossec. | 0.11479 | obs. dist. | 50 9 | | Cossec. | 0.11479 |

2) 154 55

Sine | 9.98950 | 77 27 | Sine | 9.98950 |

Rem. 6 45 | Sine | 9.07618 |

Sine | 9.83688 | Rem. 43 23 | 2) 19.42616 |

Cosec. | 9.71308 | 58° 54' |

Cosec. | 9.98314 | 15° 54' |

31 48 = *s angle.

Rad. : Cosec. 117° 48' : D's diff. parall & refract, 1980'' : 923'' = 1st correct. of distance.

Rad. : Cosec. 31° 48' : star's refract, 162'' : 138'' = 2nd correct. of distance.

Here Here the first correction of distance is additive, since the angle at the moon is obtuse; and the second correction is also additive, since the angle at the star is acute: therefore their sum $92\frac{1}{2}'' + 138\frac{1}{2}'' = 106\frac{1}{2}'' = 17' 41''$, being added to $50^\circ 8' 41''$, the apparent distance of the star from the moon's centre, gives $50^\circ 26' 21''$ for the true distance of centres nearly; and $2 \times L (d+S) - L (2LR + L_2 + LD) = L_8'$, which, being added to the distance twice corrected, gives $50^\circ 26' 29''$ for the true distance. By comparing this distance with the computed distances in the Ephemeris, the time at Greenwich corresponding to that of observing the distance will be known; and the difference of those times being converted into degrees and minutes, at the rate of 15 degrees to the hour, will give the longitude of the place of observation; which will be east if the time at the place be greater than that at Greenwich, but west if it be less.