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sun will be upon the meridians of places, or it will be apparent noon, or 12 o'clock at places situated 15°, 30°, 90°, and 180° west longitude from the meridian of Greenwich; while the several corresponding hours of apparent time at Greenwich will be 1 o'clock in the afternoon, 2 o'clock, 6 o'clock, and 12 o'clock at night, or midnight.
Beyond the 180° west longitude, east longitude commences. The only difference in the two cases is, that places to the west of Greenwich are said to have their noon later, and their reckoned time earlier: those to the east have their noon earlier and consequently their reckoned time later than at Greenwich. Hence, if, when it is the hour of apparent noon at any place situated either to the east or west of Greenwich, the corresponding hour of apparent time at Greenwich could be ascertained, the longitude of that place might be directly found by turning the difference of their times into degrees and parts of degrees, reckoning 15° for every hour of apparent time, and for proportionate parts of an hour taking proportionate parts of 15°.
But, as it has been already explained, the variation of the apparent solar day makes apparent time ill adapted as a standard to refer to for the purpose of ascertaining the difference of longitudes by the difference of the apparent times at two different meridians: it is necessary, therefore, to show how the difference of the mean time at two different meridians may be substituted in its stead. It has been stated that, at four times in the year, the equation of time is nothing, or that at some particular moment of four days in the year the hour of mean time exactly corresponds with the hour of apparent time. Thus, it appears, from the Nautical Almanack, that on the 24th of December of the present year, at the hour of apparent noon, when the sun will be on the meridian of Greenwich, the apparent time will be in advance of the mean time at Greenwich by 20.3, that is, when it is 12 o'clock in the day, by the sun, it will want 20".3 to 12 o'clock by the watch; so that it will be then necessary to subtract 20".3 from the apparent time deduced from observation in order to ascertain the corresponding mean time at Greenwich for that day. But on the 25th of December, or at the hour of apparent noon at Greenwich on the following day, the apparent time will be behind the mean
time by 9".8, which quantity therefore must then be added to the apparent time to get at the mean; and the watch will be 9".8 past 12, when it is noon by the sun. Hence, as in the space between these two successive passages of the sun over the meridian of Greenwich, the equation of time, or the difference between apparent and mean time, has, from being subtractive, become additive, it has, at some moment of that interval, been 0, or has passed through 0; or, in other words, the mean time at Greenwich having overtaken the apparent time at that place, the hour of apparent time and that of mean time will, for some one moment, between the two successive noons, be the same. Now, as the difference between mean and apparent time, or the equation of time, depends upon the variable velocity of the sun in his apparent annual motion in the ecliptic, and upon the obliquity of the ecliptic or the angle it makes with the equator; these circumstances being independent of place, the equation of time is for all parts of the earth the same that it is at Greenwich at any given moment. Hence, as at some particular moment between the noons at Greenwich of the 24th and 25th of December, the equation of time is nothing, at that moment it is also nothing at every other place upon the globe, or the apparent and mean times are then every where exactly the same. But we have already proved that the longitude might always be determined by turning the difference of the apparent times at Greenwich and any other place into degrees at the rate of 15° to every hour of apparent time. At the particular moment, however, when the equation of time is 0, the dif. ference of the apparent times is the same with the difference of the mean times at Greenwich and every other place upon a different meridian. Hence, at this moment the longitudes of all places may in like manner be determined by turning the difference of mean times at Greenwich and at all other places into degrees at the same rate of 15° for every hour of mean time. But what is true of mean time and of the difference of mean times at one particular moment, is true always, because mean time is not variable; so that the difference of mean times at Greenwich and all other places will always give the longitudes of places; and therefore by knowing on any day in the year the mean time at Greenwich, and also the corresponding mean time at
any other place, the longitude of that place will be found by converting the difference of their reckoned mean times into degrees, at the rate of 15° for every hour of mean time; it will be east longitude if the time at the place in question be later than the time at Greenwich, and west longitude if it be earlier.
All, therefore, that is required is to ascertain, 1st, the hour of mean time at the place, the longitude of which we wish to know; 2dly, the corresponding hour of mean time at Greenwich. Now the hour of mean time at any place may always be obtained by means of the corresponding apparent time, by adding to or subtracting from it the equation of time for the moment, which is given (or may be computed from what is given) in the Nautical Almanack. The hour of apparent time may always be found by means of an observed altitude of the sun, or, if the place be on land, by means of a sun-dial. * The corresponding mean time at Greenwich may then be ascertained by a chronometer or time-keeper, adjusted and regulated so as to show Greenwich mean time. If, therefore, a time-piece could be made so perfect as always to show the mean time at Greenwich without error; or if its error in going were always the same, that is, if it gained or lost the same quantity every day, the longitude of places might be correctly found by such a chronometer.
This desirable object has not hitherto been attained: the most ingenious and accomplished mechanics, although prompted by the liberal rewards held out by the legislature to encourage their exertions, have failed of complete success. Time-pieces have, however, been made, which from their near approach to an equable rate of going, might appear to justify even sanguine hopes that at some period or other a perfect machine may be constructed; but it is highly improbable that these hopes will ever be realized. The imperfection of the human mind seems to oppose even a moral obstacle to the attainment of absolute perfection in any of its productions. In other works of art an apparent perfection may be obtained, because their defects are not visible to our senses, and we have no other means of ascertaining their existence; but in a machine which is to measure time, the smallest errors accumulate so as to become at last apparent, and in the daily equable motion of the earth on her axis,
nature herself affords a perfect measure of time, by a comparison with which the errors and defects of the measure constructed by human art cannot in the long run escape detection.
At sea, where other methods cannot be resorted to with facility, chronometers are generally used for finding the longitude; but the mere circumstance that the best chronometer is liable to error, and to error which may escape notice, makes it dangerous to trust to the chronometer alone; nor ought it to be relied on but under circumstances excluding the adoption of some of the other methods of finding the longitudes.
These methods, therefore, form the next subject of consideration.
There are various appearances from time to time taking place among the heavenly bodies, that afford the means of finding the longitude nearly. These appearances are the following: 1st, Eclipses of the moon; 2d, Eclipses of Jupiter's satellites or moons; 3d, Occultations or concealments of fixed stars, by the moon's passing over them; 4th, Eclipses of the sun; 5th, The passage of the moon over the meridian of the place the longitude of which is required; 6th, The same passage compared with that of one or more (stars immediately preceding or following the moon, and having nearly the same declination; 7th, The distance of the moon from particular fixed stars or from the sun. There is also another method, of limited application, by means of artificial appearances upon the earth, as explosions of gunpowder made at one place and seen at another, the longitude of which is required.
The first and second and the last of these appearances are observed at all places where they happen to be visible at the same instant of absolute time. The difference, therefore, in the reckoned times, either mean or apparent, at two places where they are visible, is owing to the difference of their longitude. The time at Greenwich of eclipses of the moon and of Jupiter's satellites is previously computed and set down in the Nautical Almanack, and the corresponding time at the place whose longitude is wanted, being obtained at the moment of these appearances happening, the difference turned into degrees in the usual way is the longitude. By means of explosions of gunpowder or other signals made on the earth, the difference of the longitudes of any two
places not far distant from each other may be determined with very great exactness; the mean time for each place may be known by separate chronometers previously adjusted and regulated for the purpose; the difference of the times at the moment of the explosion or other signal, which is made at one place and seen instantaneously at the other, converted into degrees, will give the difference of longitudes. This method has of late become the more interesting from its having been adopted, in the course of the operations now in progress on the continent for measuring an arc of a parallel of latitude, as the best means of determining the longitude of the extremities of the arc. The space between the two extremities of this arc was divided into a great number of smaller arcs, all of such a length, that one of the extremities of each smaller arc might be made visible to an observer at the other extremity. At each point of division of the principal arc, were fixed stations, at which the requisite instantaneous signals were made and observed. The difference of times when these signals were made at one station and observed at another, gave the difference of longitudes of the extreme points of every smaller arc; and the sum of all the differences gave the difference of longitudes of the extremities of the prin cipal arc. It is scarcely necessary to remark, that any thing answering the purpose of an instantaneous signal, may be used instead of explosions of gunpowder-such as the discharge of a rocket, or the sudden display or extinction of a lamp: a contrivance called a Heliostat (which is from two Greek words, and signifies any thing the position of which has some reference to the sun) has been employed on the continent: it has a strongly reflecting surface, and is placed in such a manner that the rays of the sun are reflected by it towards the desired point of observation; the reflection is then made to disappear suddenly by interposing a screen between the Heliostat and the distant spectator, and thus conveys an instantaneous signal.
The third and fourth methods, by occultations of fixed stars by the moon, and by eclipses of the sun, likewise depend upon the difference of the times at which these appearances take place at Greenwich (and which times are computed by means of tables); and of the times at which they are actually observed
to take place at the spot the longitude of which is required; but with this qualification, that as these appearances are not observed at all places at the same point of absolute time, the difference in the absolute times of their happening must be allowed for: thus, if at Greenwich the occultation of a certain fixed star by the moon, happen at six o'clock in the morning; and at some other place to the west of Greenwich it be observed to happen at midnight, thus making a difference of six hours in the reckoned times of the appearance, it will not follow that this is all due to the longitude, and that the place in question is 90° west longitude, for the occultation does not happen at both places at the same moment of absolute time; but the star is seen at the place in question for some time after it is hidden at Greenwich. This time, which being caused by parallax may be computed, must be added to the Greenwich time, computed from the tables; and then the difference between the resulting time at Greenwich, and the time at the place at the moment of the occultation there, will give the true difference of corresponding reckoned times between that place and Greenwich; and from this difference the longitude may be deduced. The difference in the absolute time of these appearances occurring at different places, is owing to the sun and fixed stars shining by a light of their own, and to the moon's parallax.
The fifth method is by means of the moon's passage over the meridian.
If the sun and moon be upon the meridian of Greenwich together, on any particular day, on the following day when the sun is again on that meridian, the moon will be considerably to the east of it; and some time will consequently elapse before the moon reaches the meridian of Greenwich after the sun has left it. This easterly separation of the moon from the sun after they have been together, is caused by the moon's quicker motion in her orbit or course round the earth; and the time which elapses between the passage of the sun over the meridian of Greenwich, and that of the moon, is called the moon's retardation. The moon's motion in her orbit continuing, the distance between the sun and moon continually and gradually increases; so that if the moon's retardation be of a certain amount at the time of its passing the meridian of Greenwich, the retardation
at a place to the west of Greenwich will be of a greater amount, in proportion to the time that is required to bring the moon from the meridian of Greenwich, to the meridian of the place to the west of Greenwich. Hence, as the increase of the moon's retardation is for 24 hours proportional to the times in which it is produced, by knowing the retardation at two different meridians, and the time during which the retardation at one of the meridians has been produced, the time during which the greater retardation at the other meridian has been produced, may be found by the rule of three. Thus, suppose that the sun and moon having been upon the meridian of Greenwich together on one day, the retardation of the moon at Greenwich on the following day, or in 24 hours, is 52'; that at a place to the west of Greenwich the retardation of the moon is observed to be 57', or 5' more than it was at Greenwich; then we shall have this proportion; as 52: 57:: 24 hours: 24 hours + the additional time necessary to produce the additional retardation of 5'. This additional time is due to and expresses the difference of the longitudes, and 24 hours correspond with 360° of longitude. Hence, 52′ of time: 57' of time :: 360°: 360° + difference of the longitudes; and as in this case we have taken the meridian of Greenwich, the longitude of which is 0, we shall have 52' of time: 57' of time: 360° : 360+ longitude of the place; or the longitude of the place
is equal to 360 × of time, and expressing the time in parts of degrees at the rate of 15° to an hour, the longitude is obtained. Hence generally the longitude of a place is equal to 360°, multiplied by the difference between the retardation of Greenwich and the retardation of the place the longitude of which is required, divided by the increase of the retardation at Greenwich in the 24 hours preceding the time of observation. The increase of retardation at the place the longitude of which is required, is known from observation. The increase of retardation at Greenwich, for the 24 hours preceding, may be found by means of the Nautical Almanack.
The principle of this method is applicable to the fixed stars as well as to the sun; the only difference being, that the moon's retardation is greater with respect to the fixed stars, as they have
none of the daily easterly motion which the sun has in its apparent yearly path in the heavens. The application of this principle to the fixed stars for finding the difference of the longitudes of two places, was first successfully made by M. Nicolai, a distinguished astronomer, at Manheim, and is now very generally practised on the continent. Mr. Francis Baily, in his valuable paper on this subject, lately published in the Memoirs of the Astronomical Society of London (vol. ii.), observes, "That already at several observatories, the observers have been enabled to determine their difference of meridians in a few months with as much accuracy as they formerly could in as many years." The improvement introduced by M. Nicolai consists in the choice of those stars which have very nearly the same declination or distance from the equator as the moon, and which pass the meridian very soon after, or a little before the moon. The advantages of the method are to be found in avoiding a great number of errors and troublesome calculations, which in practice were found to detract from the value of other methods, and in the frequency with which observations may be made, being every night that the moon is visible. It was employed with very great success by Lieutenant Foster on Captain Parry's last voyage but one, in determining the longitude of Port Bowen in Prince Regent's Inlet. His observations have been calculated and compared with those made at the observatories of Greenwich and Dublin, and by the late Colonel Beaufoy at Bushey Heath; and the results, which will ap pear in a volume of the Astronomical Society's Memoirs, show, as far as one example can do so, the great value of this method of determining the longitude on land.
None of the previous methods, however, (except that which consists in the use of chonometers,) are adapted to the situation of a person on board a ship. The late Astronomer Royal, Dr. Maskelyne, in his Preface to the Nautical Almanack, observes, It was hoped that some means might be found of using proper telescopes on shipboard to observe these eclipses [the eclipses of Jupiter's Satellites]: and could this be effected, it would be of great service in ascertaining the longitude of a ship from time to time. In my voyage to Barbadoes, under the directions of the Commissioners of Longitude, in 1763,
I made a full trial of the late Mr. Irwin's marine chair proposed for the purpose, but could not derive any advantage from the use of it; and considering the great power requisite in a telescope for making these observations well, and the violence as well as irregularities of the motion of a ship, I am afraid the complete management of a telescope on shipboard will always remain among the desiderata."
The longitude may, however, be found at sea, when the moon is visible, by the observed distance of the moon either from the sun or from nine of the principal fixed stars mentioned in the Nautical Almanack. This distance is observed by means of a Hadley's Sextant. In consequence of the moon's quick motion in her orbit she is every moment changing her situation in the heavens with respect to the sun and stars. Her distance, therefore, from the sun, or a particular star, is at one moment of time different from what it was at the previous moment, and what it will be at the next; so that a particular or given distance is proper or due to a given moment, which moment will be expressed or reckoned differently at different meridians, according to the apparent time of day. This difference in the apparent times, being therefore due to the difference of meridians will, converted into degrees," give the longitude. The distance of the moon from the sun, and from nine principal fixed stars, is given in the Nautical Almanack, for every three hours of Greenwich time. This distance is such as it would appear at the centre of the earth; allowance having been made in computing the distance given in the Almanack as well for parallax as for refraction. The observed distance at the place the longitude of which is required, is in a similar manner to be reduced to the centre of the earth by correcting for the moon's and sun's parallax, and for refraction. The apparent time, at the place and moment of observation, is obtained in the usual manner, by taking the contemporary altitude of the sun or star. The difference between this apparent time and the apparent time at Greenwich, given in the tables as corresponding to the same distance, converted into degrees, will be the longitude of the ship. This method of finding the longitude is called the lunar method; it will generally give the longitude to
the earth's surface, Ea Q the equator, P the pole, A a the latitude of A, BQ that of B; a Q, which is the difference of their longitudes, is known, as the longitudes of both places are supposed to be known; then A B, being the arc of a great circle passing through A and B, is the shortest distance, and may be found as a side of the spherical triangle ABP by spherical trigonometry. With a trifling inaccuracy, the distance, A B, may also be determined mechanically, by means of a common terrestrial globe and a pair of compasses. The opening of the compasses given by applying the extremity of either leg to each place on the globe, will be the measure of that arc of a great circle which lies between the two places. The number of degrees contained in this arc may then be ascertained by applying the compasses, thus open, to any graduated great circle on the globe, or one which has the degrees marked, such as the equator or ecliptic. The number of degrees thus found, being turned into geographical miles, at the rate of 69.044 miles to