By the term geographical longitude, is meant an arc which measures the inclinations of two ter- restrial meridional planes, one of which passes through a known place, as a place of reference, the other through any place whatever. It is sometimes also defined as the distance E. or W. along the equator, of any place from a certain meridian. The selection of a station from which the longitudes of all other places are to be reckoned is en- tirely arbitrary; British astronomers and geographers have chosen the meridian of the Royal Observatory of Green- wich as their first meridian. The French and other con- tinental nations refer the longitudes of all places to the meridian of their principal observatory.

The longitude of a place may be expressed in hours, minutes, and seconds of time, or in degrees, minutes, and seconds of space; if it be given in either, it may be trans- lated into the other. The reason of this is, that the earth revolves on its axis from W. to E. in twenty-four mean solar hours, thereby causing the first meridian to describe during that time a space equal to 360°, and, therefore, in one hour 15°. Hence, if the plane of the first meridian pass at the present moment through the sun, then the meridian of a place 15° west of the former will pass through the sun exactly one hour after; if the place be 15° east of the first meridian, the plane of the former will pass through the sun one hour before the latter. The sun always passes the meridian of any place when highest in the heavens, i.e., at mid-day, or twelve o'clock mean solar time. Therefore, places lying to the E. of the first meridian will have every hour earlier, but places lying to the W. of that mer- idian will have every hour later than it; so that if, while the meridian of one place is passing through the sun, the time be known before the meridian of another place pass through the sun, then the longitude of that place from the former is determined, the time being turned into space, at the rate of 15° to the hour. Hence, therefore, places will have E. or W. longitude, according as they lie E. or W. of Greenwich Observatory, the longitude of the meridian of which is zero.

The problem of the longitude may be reduced to this,— Given the hour by calculation at the place of observation, to find the hour at Greenwich Observatory corresponding to the same time; the difference of times gives the longi- tude of the place from Greenwich. The solution of this problem was attempted in very early times, dating even from the time of the ancient Egyptians, but the results ob- tained were very inaccurate. These results were deduced from tables of celestial phenomena calculated for a certain meridian, and then the times were compared with the times at which the same phenomena appeared at a different place; actual admeasurement was also employed. But it was not till after the invention of watches that the problem was rendered solvable. Harrison, in the eighteenth cen- tury, was the first who gave a true solution by a watch; but the first accurate resolution of the problem may be said to date from the discovery by Galileo of Jupiter's satellites, and his tables of their motions. The result of the prob- lem at this period, as well as now, was, as Wolfius has expressed it, that means might be found whereby the art of navigation might be brought to its utmost pitch of perfection.

If the advantages of determining the longitude to a com- mercial and maritime people be considered, it will not ap- pear surprising that princes and others should have held out high rewards for a true solution of the problem. Philip III., king of Spain, saw its value, and in 1598 offered a reward of 1000 crowns to the person who would solve it. The States of Holland imitated his example by a prize of 10,000 florins. In the year 1714, the British government offered a premium of L20,000 for any method whereby the longi- tude might be determined at sea to within 30 miles; L15,000, if the proposed method would give it to within 40 geographical miles; L10,000, if it would determine the longitude to within 60 miles. It was also enacted, that a reward of L5000 would be given to the inventor of any time-keeper which should enable a ship, during a voyage of six months, to keep her longitude to within 60 miles; L7500 if within 40 miles; and L10,000 if within 30 miles. If the method were by improved astronomical tables, the reward was to be L5000, the tables being com- Longitude, pared with previous observations. France also, in 1716, under the regency of the Duke of Orleans, offered a prize of 100,000 livres. In consequence of these rewards, many and various methods were proposed, the best of which, at least as respects frequency of observation and shortness of calculation, is the method of Lunar Distances.

Jean Werner of Nuremberg appears to be the first who proposed, in his Ptolemy's Geography, 1514, a method of finding the longitude by the distance between the moon and a star. The lunar method was also recommended by Oronce Finé of Briançon, in his book De Invenienda Longitudine; by Gemma Frisius, in his treatise, Structura Radii Astronomici et Geometrici, 1545; by Kepler in his Rudolphine Tables; and by Christian Longomontanus in his Astronomia Danica, 1622. Gemma Frisius is, moreover, said to have attempted the longitude by a watch some time after 1530. Carpenter, in his Geography, 1685, says that the lunar method is to be ascribed to Pierre Appian, a German, born in 1495. John Baptiste Morin, in 1634, attempted to improve the lunar method, and received, in 1645, a pension of 2000 livres; but his improvements were useless, as Pascal declared, owing to the imperfect nature of the existing tables.

The tables of celestial observations previous to Flamsteed's time were imperfect and erroneous: those generally used were Tycho Brahe's or Kepler's, and to show that they were of little value in determining the longitude, although invaluable in other respects, it may be stated that Flamsteed's observed differed from Tycho's computed places by 3', 6", or more; and the tabulated distances of the latter differed from the observed distances of the former by 15' or 20', which would cause an error in the longitude of about 15°, or 300 leagues. Tycho's lunar theory, and the tables grounded on it, were in error 12' and more. The uncertainty, then, of these tables being known, as well as the paucity of astronomical observations generally, a Frenchman, named Le Sieur de St Pierre, contrived, in 1674, to get his pretensions to the discovery of the longitude brought under the notice of Charles II. of Britain and the court. Commissioners were appointed, and St Pierre's data necessary to work the problem were as follow:—(1.) The heights of two stars, and on which side of the meridian they were; (2.) The heights of the two limbs of the moon; (3.) The height of the pole; all to be given in degrees and minutes; and (4.) The year and day of observation. Flamsteed being in London at the time, was appointed, not only to act as a commissioner, but also to supply the necessary data. St Pierre, having received the data which he required, refused to work the problem, because he alleged the observations given him were feigned. Flamsteed on this wrote to the commissioners, assuring them that the observations were genuine, and at the same time stated, that the longitude could not be solved by the conditions proposed; but if the tables of celestial observations, especially those of the moon, could be rendered more accurate, then the longitude might be determined by them. On the letter being shown to Charles, his majesty was startled at the assertion of the computed places not agreeing with the observed, and said with some rehescence, he must have them observed, examined, and corrected anew for the use of his seamen. It was this simple incident which led to the formation of the Royal Observatory of Greenwich, the foundation of which was laid by Flamsteed on the 10th of August 1675; and it was in that building that Flamsteed laboured for forty-four years, under the most trying circumstances, to correct existing tables, and to commence the British Catalogue, one of the noblest monuments of British perseverance. So valuable were Flamsteed's observations to Newton, that they enabled him to form his lunar theory, which is now of such consequence in determining the longitude.

From the improvements made in watches by Huygens, Hooke, and others, previous to the year 1714, it was thought that the longitude would be solved by this machine. Hence, after 1714, the best artists applied themselves to the construction and improvement of watches. Henry Sully, an Englishman, but resident at Paris, tried in 1726 to determine the longitude by a marine watch, but without success. Julian Leroy, one of his pupils, would appear to lay claim to priority of invention; but it has never been disputed that the honour of solving the difficult problem of the longitude by means of a watch belongs wholly to Harrison. This ingenious workman began, at a very early period, to make experiments on pendulums made of different metals, in order to counteract the effects of heat and cold. In the year 1736 Harrison was brought into notice by a pendulum clock which he had made in 1726, and which, for ten successive years, kept remarkably exact. This clock was tried in a voyage to Lisbon during August 1736, when it corrected an error in the ship's reckoning of 1° 30'. At the special request of the commissioners of longitude, who advanced him money, he continued his experiments on watches from 1737 till 1761, when he produced three watches, or time-keepers,—the third the most accurate, and about 4 inches in diameter. This watch, or chronometer, was tried in a voyage to Jamaica as to its practicability in determining the longitude. The trial was eminently successful; the difference of time as shown by the chronometer indicating Greenwich or rather Portsmouth local time, and the local time of the place, being 4 seconds of time, which is equivalent to 1 nautical mile in the parallel of Jamaica. On the arrival of the vessel at Portsmouth, it was found that the error of the chronometer was only 1 min. 58.5 sec., or 28'37.5 for the entire voyage, which, in the parallel of Portsmouth, would be equivalent to 18 nautical miles. Since this error was within the limit prescribed by the act, Harrison claimed the full reward of L20,000; but the commissioners, considering the matter in all its details, came to the conclusion that the watch was not yet sufficiently tried. In order, however, to testify their appreciation of the invention, they gave Harrison a grant of L5,000, and requested him to improve the watch still further against a second voyage. This voyage was undertaken, in 1764, to Barbadoes; and that no misunderstanding might ensue, Maskelyne and Green were also sent out to make the necessary astronomical observations at that place. The difference of longitude, as shown by the chronometer and that by astronomical observation, was 43 seconds of time, which is equivalent to 10' 45" of space, or longitude. In consequence of the success attending this and the former trial, the House of Commons ordered one-half of the reward promised by the act of 1714, or L10,000, to be paid to Mr Harrison, the inventor of the longitude clock; the other half to be paid him when watches, constructed on principles stated by him, should determine by trial the longitude of any place to within 30 nautical miles. Another condition annexed to the payment of the other L10,000 was, that the inventor should give on oath a full explanation of the principles on which the watch was constructed. This was done most willingly, and Harrison delivered over all his watches to government. The first watch made on Harrison's principles was that by Mr Kendall; it was found to exceed the regularity of the best of its models. This instrument was committed to the care of Mr Wales, in his voyage round the world with Captain Cooke, during the years 1772, 1773, &c., and such was its success, that in 1774 an appeal was made to the House of Commons to order the remaining sum to be paid to Mr Harrison, which was accordingly done. Harrison realized by his invention alone upwards of L24,000.

Several other parties received rewards for their improvements in chronometers. Arnold and Son received L3000, and Mudge L500. Longitude. Since Harrison's time, remarkable improvements have been made in time-keepers, or chronometers as they are now termed; no one sustaining a good character that gains or loses more than a single second in one day.

But while watches were thus gradually being perfected, the tables of celestial motions were also attended to. Halley, on succeeding Flamsteed as astronomer-royal, continued improving what the latter had begun, so that for 1730, and consequently for the future, the Caroline Tables were presumed to give the true place of the moon, within the compass of 2' of her motion. But however perfect such tables may be made, they will be useless without a proper instrument with which to take angles accurately at sea. Dr Halley proposed to overcome this obstacle, by using on shipboard a telescope of 5 or 6 feet; but the error in such a case would nearly equal 2', or under the equator the longitude would be in excess or defect about 40 leagues.

But in 1761, Mr Halley communicated to the Royal Society the nature of the sextant which he had then invented. The sextant is an instrument for taking angles at sea with surprising accuracy; its principle depends on the law of the reflection of light. This instrument was tried in several voyages with wonderful success; but its results were most accurate when used with Professor Mayer's Tables of the Moon, computed for the meridian of Paris. These tables first appeared in the Memoirs of Gottingen for 1742, and a manuscript copy was sent in 1755 by Mayer to the Board of Longitude, setting forth, at the same time, his claim for some one of the rewards which he might be thought to merit. These tables were placed in the hands of Dr Bradley, astronomer-royal, who compared several hundred computed longitudes of the moon with his own observed longitudes, and never found a greater difference than 1.5'. Dr Bradley showed the commissioners the value of these tables. Mayer died in 1762; but having in the interval greatly improved his tables, his widow sent them in 1763 to the Board of Longitude. These are the tables which, in consideration of their value in finding the longitude at sea, were, by act of parliament, honoured with a reward of L.5000, which was paid, in 1765, to Mayer's widow. Dr Maskelyne, astronomer-royal, was at the same time requested to improve and correct them as far as possible, so that they might be compiled, and form the basis of a British Nautical Ephemeris or Almanac; and to print the same, in order to make the lunar tables of general utility. The first of the series of the Nautical Almanac and Astronomical Ephemeris was published in 1768, under the superintendence of Dr Maskelyne. It was published yearly by the Commissioners of the Board of Longitude. The Nautical Almanac has been greatly improved, corrected, and extended, under the able superintendence of Mr Airy, the present astronomer-royal; it is now published four or five years previous to the observations being made at Greenwich Observatory; hence in long voyages the set of tables may be taken out.

In consideration of Mayer having availed himself of Euler's lunar theory, the latter received from government L.300.

The several methods for finding the longitude are the following:

To find the Longitude by a Chronometer.—Suppose that a chronometer is warranted to measure equal portions of time uniformly, and always indicates Greenwich local time; it is evident that, were this instrument carried to any station on the surface of the earth where also the local time is known, the local times of Greenwich and that place can be compared with each other. If the chronometer be carried to any station on the meridian of Greenwich, the chronometer and local time of the place will always coincide; but if it be carried to any station W., or E., of the meridian of Greenwich, then the time as shown by the chronometer will be in excess in the former case, but in Longitude, the latter in defect of the local time of the place; the difference of local times gives the longitude of the place from Greenwich. The time may be converted into distance, at the rate of 15° to 1 hour. Chronometers can never be made perfect; they require, therefore, to be daily compared with the heavenly bodies, in order to ascertain if their motion has been uniform.

Such is the method of finding the longitude by means of a chronometer; but to show the care and trouble involved in the delicate operation of finding the difference of local times between any two places, it will suffice to state briefly the attempt of Mr Airy, astronomer-royal, to determine the longitude of Valentia on the W. coast of Ireland. Mr Airy considered it necessary to take into account the facility of land conveyance by railways, and of sea conveyance by steamboats. With respect to the former, the route by Bristol and Cork had the preference; but he found those advantages best combined by adopting the route by Liverpool and Kingston. It was at first proposed that the chronometers should be transmitted from Greenwich to Valentia without interruption, save for the purpose of winding them up and comparing them with a clock at Kingston. But Mr Sheepshanks, who acted in concert with Mr Airy, thought that, for the sake of accuracy, the arc of parallel from Greenwich to Valentia should be divided into two parts—that from Greenwich to Kingston, and that from Kingston to Valentia—which was the plan adopted. Thirty chronometers were employed, to be compared with the transit clocks which were set up at each station. The following were the courses of these chronometers:—The chronometers left Greenwich for the first time on the morning of 27th June, and reached Kingston for the last time on the morning of 27th July, having made nine journeys from Greenwich to Kingston, and eight from the latter to the former place. The chronometers then left Kingston for Liverpool on the evening of 27th July, and were returned to the former place the last time on the morning of 4th Aug., having made four journeys each way. They then left Kingston for Valentia on the evening of 5th August, and returned to Kingston the last time on the morning of 14th September, having made ten journeys each way. After a short delay, and transmission to Liverpool, the chronometers left that place for Greenwich, on the morning of 21st September, and arrived finally at the latter place on the evening of 28th Sept., having made four journeys from Liverpool to Greenwich, and three from Greenwich to Liverpool. It was by this transmission of chronometers from Greenwich to Valentia, that the comparison of the clocks at these two places was made. At each station, moreover, the transits of stars across the meridian were compared with the right ascensions of the Nautical Almanac list, using the mean places of the Greenwich Catalogue of 1439 stars. By this means, and with the requisite calculations, the inclination of the planes was obtained. The results on the whole are, in time,—

| Longitude of Liverpool W. from Greenwich | 12° 0' 0" | | Longitude of Kingston W. from Greenwich | 24° 31' 20" | | Longitude of Valentia W. from Greenwich | 41° 23' 23" |

The whole of the operations, including everything relating to the longitude of Liverpool, Kingston, and Valentia, occupied nearly from the end of June to the end of September 1844.

Mr Airy considered it proper, after all his labour with the transmission of chronometers, to compare the deduced results with that found by geodetic calculation. For this purpose, he made a survey-triangulation, which extended in an easterly and westerly direction from Greenwich to Valentia. From this survey the distance in yards was known between these two places, measuring along an arc of parallel. Knowing, therefore, this distance, as also the Longitude, inclination of the two meridian planes, the whole parallel passing through Greenwich was computed.

The inferences which Mr Airy deduces from the chronometrical and geodetic results are—That the verticals at Liverpool and Greenwich are less inclined to each other, or that the earth's surface in England is flatter than accords with geodetic calculation; that the inclination of the verticals at Liverpool and Kingston is sensibly precisely the same as that given by geodetic calculation; and that the inclination of the verticals at Kingston and Feagh Main is greater than that given by geodetic calculation, or that the earth's surface in Ireland is more curved than the elements of geodetic calculation imply: but that upon the whole are the difference of the chronometrical and geodetic results is very small; and, lastly, in latitude 51° 40' the complement of the logarithm of the number of feet in 1° for an arc perpendicular to the meridian is 7-9928932, or the length of 1° in an arc perpendicular to the meridian in latitude 51° 40' is 101,6499 feet. (See Transactions of the Royal Astronomical Society of London, 1847.)

To find the Longitude by Lunar Eclipses.—Since an eclipse of the moon is visible to one-half of the earth at the same time, this would seem to be an excellent method of finding the longitude. The different steps of the process are,—to compute the time at which an eclipse is to happen at the place of observation, and to compare this time with an accurate chronometer showing Greenwich time; or in the absence of this, the Greenwich time of the happening of the phenomenon must be looked for in the Nautical Almanac; or it may be computed by the observer from the lunar tables. But this method of determining the longitude is rarely used, owing to the difficulty of ascertaining the exact time of contact of the penumbra of the earth's shadow with the moon's limb at the beginning or ending of the eclipse. Sometimes, indeed, two observers of an eclipse at the same place may differ more than 2 minutes in noting the time of contact; and hence the error from this cause alone would be about 4 minutes of time, which would be equivalent to nearly 1° of longitude. It was proposed in the Philosophical Transactions of 1786 to diminish this source of error, by observing the contact of the earth's shadow with some remarkable spot on the moon's face. But although this method were more accurate, the unfrequency of lunar eclipses at sea renders the method of little use.

To find the Longitude by the Eclipses of Jupiter's Satellites.—Ever since the discovery by Galileo of Jupiter's satellites the observation of their eclipses by their primary has been used as a method of finding the longitude. Tables of these eclipses were constructed by Galileo; and it was the disagreement of these tables with actual observation that led Rosser to the discovery of the gradual propagation of light. (See Light.) The first astronomical solution of the great problem of the longitude really dates from the discovery of these secondaries, for the tables of their eclipses were framed on scientific principles. The three interior satellites of Jupiter pass through his shadow, and are eclipsed at every revolution; the fourth, or outer one, at times escapes eclipse, grazes the umbra, or is partially eclipsed. The computed times at which the eclipses are to happen at Greenwich Observatory are noted in the Nautical Almanac, published three or four years in advance; so that if these tables are in the hands of any one distant from Greenwich, he has but to observe the eclipse, and calculate the time at which it occurs, to find the difference of the local times between Greenwich and the place of observation, and thus ascertain the longitude. The times of immersion and emersion are noted with much greater accuracy than the contact of the moon's limb with the earth's shadow.

But before these eclipses can be observed with accuracy, a telescope of considerable power must be used; Longitude, and as it is extremely difficult to direct a telescope properly on shipboard, the method is practically useless at sea. But again, particular care is required in observing; for two observers at the same place, with telescopes of different magnifying powers and apertures, seldom agree within a second or two of each other; hence the mean of the results of immersion and emersion should be taken. But another source of error is, that no two or more observers will agree as to the instant of the total immersion, or of the complete emersion of the satellite; hence the only case in which this method is practically useful in determining terrestrial longitudes is that in which the instant of immersion and emersion are observed with the same telescope, and by the same observer, since in this manner he will find the precise instant of the satellite's opposition to the sun.

To find the Longitude by Signals.—If the difference of longitude between two places be small, it may be easily found by means of the bursting of a rocket, the oxy-hydrogen lime-ball light, or the explosion of gunpowder fired from the one place at a preconcerted time, and observed at the other place; the local times of these places being accurately ascertained, the longitude is known. These artificial signals, when fired from an elevated spot of country, may be seen, when the atmosphere is in a proper state, at distances varying from 30 to above 100 miles. An observer, therefore, distant from the spot at which the rocket or other signal is exposed, has only to observe the time when he sees it, and afterwards compare this time with the time when the rocket was set up, the difference of times giving the longitude of the one place from the other; if at one of the places the Greenwich time corresponding to that of the event is known, the longitudes of the places from that meridian are also known. It is here supposed that the gradual propagation of light leads to no appreciable error in the small distance between the two places.

If the distance between the two places be considerable, and if a rocket sent up at the one place cannot be seen at the other place the longitude of which is required, then a series of signals must be made and noted by observers placed at stations intermediate to the two extreme places.

Thus, let A and E be the two places, the longitude between which is required; B, C, and D, observers at intermediate stations; w, x, y, z, signal places, and let these places be arranged in the following manner:

A w B x C y D z E.

Before the signals are sent up at the previously arranged hours from w, x, y, z, the local times of the places along the whole line AE are supposed to be accurately known. Let then a signal be sent up at w, and noted at A and B, the difference of times of observation, as noted by the chronometers at those two places, will give the longitude AB. Let, again, another signal be sent up at x, and the time of appearance noted at B and C, then the difference of times, as shown by the chronometers, gives the longitude between B and C; and therefore between A and C. Similar results will be found when signals are sent up from the stations y and z, to be observed at C and D, D and E; and in this manner, the whole longitude AE between the extreme stations can be found. The longitude found on this principle, and the mode of deducing the most advantageous results from a combination of all the observations, is fully stated by Sir John Herschel in the Philosophical Transactions, 1826, on the Difference of Longitudes of Greenwich and Paris.

Natural signals might be adopted in place of artificial ones, especially if they occur in sufficient number. Thus, Halley is said to have first suggested the idea of employing Longitude. shooting stars, or meteoric stones, for determining the differences of longitude by simultaneous observation. Dr Mas- kelyne, in 1783, drew the attention of astronomers to these phenomena, and distinctly pointed out their application to this subject. The idea was revived in 1802 by Benzen- berg; but so long as these shooting stars were regarded as casual and irregular phenomena, it was not to be expected that they could be of much service in geodetic measurements. Since, however, these meteors are known to be regular in their appearance, especially on the 9th and 10th of August, and on November 12th and 13th; and since they are visible over extensive regions of the earth's surface, they may be advantageously applied, by previous concert and agreement between distant observers to watch and note them. Mr Cooper has thus employed the meteors of the 10th and 12th of August 1847 to determine the difference of longitudes of Markree and Mount Eagle, in Ireland. Those of the same epoch have also been used in Germany for ascertaining the longitudes of several stations, and with very satisfactory results.

To determine the Longitude by Moon-occulminating Stars.—This method consists in finding the increase of the moon's right ascension in the intervals between the passage of the moon over the meridian of Greenwich and over that place whose longitude is required. It is necessary to find the right ascension of the moon's bright limb, and of a star selected on, or as near as possible to, the moon's parallel of declination, and not differing much from her in right ascension at the two meridians; then, the moon's increase of right ascension being known, the difference of longitude is determined.

Let $T$, for example, be the time when the moon's enlightened limb transits the meridian of any place distant from Greenwich; $t$ the time of passage of a star over the meridian of the same place; let also $a$ be the error of the clock in the course of the day; then $24 + a$ will be the interval of time elapsing between two successive transits of the same star, and $24 + a; T - t = 360^\circ$; the difference of right ascension of the moon's bright limb and the star at the instant of the limb being on the meridian; and if to this the right ascension of the star be added, the right ascension, $= a'$, of the moon's bright limb when on the meridian is determined. Now the proper stars to be observed for this purpose, as well as the right ascension of the moon's bright limb when on the meridian of Greenwich, are given for every day of the year in the Nautical Almanac, from which the daily increment of right ascension may be determined.

Let $a$ be the right ascension of the moon's bright limb when on the meridian of Greenwich, $e$ the increment of right ascension in the time between two successive transits over the same meridian; then, whilst the moon, by her relative motion, separates from the meridian of Greenwich by an angle of $360^\circ$, its real motion in right ascension is $e$; and whilst it separates by an angle equal to the difference of longitude, the motion in right ascension is $a' - a$; and, therefore, supposing the change in right ascension uniform, the required longitude $= \frac{a' - a}{e} \cdot 360^\circ$. Where greater accuracy is required, the difference of longitude corresponding to the increase of right ascension $a' - a$, must be determined by interpolation. This method is considered one of the best which can be adopted for determining the longitude of distant places, when the observer, furnished with a transit instrument, can obtain a landing. (Hymers Astron. 1840.)

To find the Longitude by the passages of the Moon over the Meridian.—If the sun, moon, and a star be supposed to be on the meridian of Greenwich at the present moment, then in the next instant the three bodies will be separated from each other,—the star will be found most advanced to the W., the moon least advanced from the meridian, while the sun will occupy an intermediate situation. The meridian itself also leaves these bodies, but will approach them with different degrees of velocity, and reach each of them after certain intervals of time. It will pass the star after the lapse of a sidereal day, or after having described $360^\circ$; it will pass the sun at the end of a solar day, or after having described $360^\circ 59' 8''$; and it will pass the moon after a time $= \text{the sum of 24 hours and the moon's retardation}$ for that time, or after having described an angle $= \text{the sum of } 360^\circ$ and the moon's right ascension in 24 hours. This always takes place in the interval between two successive transits of the moon over the same meridian. So also a spectator on a different meridian will notice similar effects, but less in degree, and less proportional to the distance of his from the first meridian. The sun's right ascension will be increased (or the separation of the sun from the star), but less than $59' 8''$; the moon's right ascension (or the separation of the moon from the star) will also be increased to the spectator, but less than its increase between two successive transits; consequently there will be an excess of increase of the moon's right ascension above that of the sun's, but less than the excess that takes place between two successive transits of the moon over the meridian of Greenwich. Wherefore, since the spectator at the second meridian may compute the respective increments of right ascension of moon and sun that take place between two successive passages of the moon over the meridian of Greenwich; then, since he is also able to compute, by actual observation, the right ascensions of sun and moon at the times of their passage over his own meridian, he has determined the longitude. The spectator may choose the sun and a star, the moon and a star, or the moon and sun; the two former are preferable. (Woodhouse's Astron., 1821.)

To determine the Longitude by means of Eclipses of the Sun, or by Occultations of Stars by the Moon.—One of the most exact methods, and at the same time the simplest, for finding the longitude, is by means of solar eclipses and occultations. If the commencement and ending of an eclipse of the sun, or the immersion and emersion respectively of a star from the enlightened and dark limb of the moon or of a planet, be observed, it is only necessary to deduce the true time of conjunction for Greenwich and also for another place of observation; the difference of the times gives the difference of meridians, and therefore also of longitudes. Kepler employed this method, and it is one of the simplest. (Kepler, Astron. part opt.) The only inconvenience of this method is the large amount of calculation required.

To find the Longitude by Lunar Distances; that is, by the distance of the Moon from a Star or the Sun.—This method supposes that the face of the heavens is a dial-plate, the stars marks apparently irregularly distributed upon it, and the moon the hand moveable among them and round the earth as a variable centre. Three things require particular notice about this clock:—1. The intervals of space separating the principal and secondary marks from one another and from the moving hand—the moon. 2. The exact amount of the eccentricity of the earth, the centre of motion of the hand. 3. The proper motion of both moon and earth at any part of their respective paths. When these data are properly known, the time as shown by this clock may be read. The time as pointed out on this dial-plate is generally read at Greenwich Observatory, and tabulated in the Nautical Almanac, four or five years beforehand, for every three hours. But this clock is supposed to be accurately seen by a spectator at the centre of the earth, and consequently, since observers are on the surface, the moving hand being rather near, and the marks immensely distant from the earth, it is evident that this moveable hand will be displaced, or undergo a parallax with respect to the stars, which must be allowed for, ere the true place is known which she occupies in space, as seen from the centre of the earth. A reduction must also be made to the Longitude centre of the earth. The necessary steps for computing the longitude by this method are—(1) Find by a sextant the distance between a star and one of the moon's limbs; or, between the limbs of the sun and moon; add or subtract, in the former case, the semi-diameter of the moon, and in the latter, the sum of the semi-diameters of sun and moon, which gives the distance of the moon's centre from the star, or that between the centres of sun and moon. (2.) When two observers are making the observations, one should take the above distance, while at the same instant the other takes the altitude above the horizon of the moon and star, or of the moon and sun. In the case of one observer, he must take the altitudes immediately before and after the distance has been found, and allow for the changes of altitude which may have taken place in the intervals between their observations and that of the distance. (3.) The true altitudes are derived from the apparent and observed, by correcting the latter for refraction and parallax; the apparent altitude being the observed altitude corrected for the dip of the horizon and instrumental errors. (4.) The observed is also an apparent distance, and must, like the altitude, be corrected for parallax and refraction in order to find the true distance. (5.) Since the true distance is found, the hour, minute, &c., of Greenwich time corresponding to it will also be found by the tables of the Nautical Almanac. (6.) The local time of the place of observation is now to be computed from the true and corrected altitude of a star or the sun, the sun's or star's N. polar distance, and the latitude. (7.) The difference between this local time and Greenwich time gives the longitude.

To find the Longitude by the Electric Telegraph.—This beautiful and ingenious application of electricity for recording astronomical observations is the latest method of finding the longitude, and was proposed by Mr Bond of the Cambridge Observatory, United States. Mr Airy, of the Greenwich Observatory, has also carried it into effect with great improvements. During the summer of 1847 experiments were made on the electric telegraph connecting New York, Philadelphia, and Washington, for the purpose of determining the differences of longitude between these three cities. A competent observer was stationed at each observatory. A continuous wire connected the three cities, so that telegraphic signals might be exchanged between any two of them at pleasure. In some of the first experiments, signals were exchanged between Philadelphia and Washington, but it was found impossible to transmit signals from Jersey City to Washington, the power of the battery being inadequate to that distance. This, however, was remedied on the 29th of July, when twenty clock signals were given at Jersey City, and recorded both at Philadelphia and Washington; twenty signals were given at Philadelphia and recorded at Jersey City and Washington; and twenty signals were given at Washington and recorded at Jersey City and Philadelphia. Thus the comparison of the three clocks was decisively made in a remarkably short period of time. The success of these experiments amply repaid the first unsuccessful efforts. The difference of longitude between Jersey City and Philadelphia is 40° 3'; and between Jersey City and Washington, 12° 3'; omitting in each case the small fractional part of a second, which was ultimately allowed for. The distance between New York and Washington is 225 miles, and the time required to make a communication pass betwixt these two places was a fraction of a second which cannot be measured.

Soon after a system of telegraphic wires was erected on the principal English lines of railway, Mr Airy had them put in communication with Greenwich Observatory, his object being to give Greenwich time on a given day to the United Kingdom. It was at first proposed that a hall should be dropped from the upper part of Greenwich Observatory, so as to touch a spring communicating with all the telegraphic wires in the kingdom, and then, by the striking of a bell, give instantaneously true Greenwich time to Liverpool, Manchester, and all the northern towns. But this method was found impracticable, owing to the non-completion of all the lines with Greenwich. On the 1st of December 1847, true Greenwich time was communicated directly from the observatory to the several stations of the London and North-Western and Midland lines in connection with it; but to all other stations of these lines special messengers were sent with chronometers indicating true Greenwich time. Hence, since Greenwich time is used over the whole of the United Kingdom, if the local time of any place be known, its longitude from Greenwich is also determined.

Since submarine cables connect Greenwich with Brussels and Paris, and these again with the principal cities of Europe, Mr Airy was very lately enabled to correct the latitudes and longitudes of their observatories. Hence, also, when the submarine cables which are to connect India, Australia, and America, with Greenwich, shall have been completed, the true longitudes of the principal cities of the world will easily be determined.