of an intermediate length, depending on the latitude of the place; and fince the Moon's horizontal parallax is the angle under which the Earth's femidiameter appears at the Moon, it hence follows, that the horizontal parallax of the Moon will vary with the latitude, being greatest at the equator, and leaft at the poles; but the horizontal parallax of the Moon, as given in the Nautical Almanac, is that answering to the equatorial radius, and therefore, in order to reduce it to any given latitude, it must be diminished by a certain quantity. Table XXXII. contains this quantity, agreeable to Sir Ifaac Newton's hypothefis.

Summary of the Corrections.

When the altitude of the lower limb of any object is obferved. its femidiameter is to be added thereto, in order to obtain the central altitude; but if the upper limb be observed, the femidiameter is to be fubtracted. If the altitude be taken by the back obfervation, the contrary rule is to be applied.

The dip is to be fubtracted from, or added to, the observed altitude, according as the fore or back obfervation is used.

The refraction is always to be fubtracted from, and the parallax added to, the observed altitude.

If the distance between the nearest limbs of any two objects be observed, that distance is to be increased by the fum of their femidiameters; but if the remote limbs be obferved, the distance is to be diminished by the above fum. If the Moon's enlightened limb be compared with the nearest and remote limbs of the Sun alternately, half the fum of these distances will be the distance between the Sun's center and the Moon's enlightened limb. The distance between a ftar and the Moon's nearest limb is to be increased by the Moon's apparent femidiameter; and the distance between the Moon's remote limb and a star is to be diminished by the Moon's femidiameter. If, at the time of full Moon, both limbs of the Moon be compared with aftar, half the fum of these distances will be the apparent central distance between these objects. In like manner, if the distance between the Moon's enlightened limb, and the nearest and remote limbs of a planet, apparently full, be observed-then half the fum of these distances will be the apparent diftance, between the Moon's enlightened limb and the center of the planet. If the planet be not full, and the distance between the enlightened limbs obferved, allowance is to be made for the planet's femidiameter: this may, however, be avoided, by obferving the distance between the Moon's Himb and the planet's center,

BOOK

The Method of finding the Longitude at Sea or Land by Lunar Obfervations.

CHA P. I.

INTRODUCTION.

F the latitude and longitude of a fhip at fea were accurately known, the course and distance therefrom to any given port, might be easily deduced by charts or otherwife. The latitude is af certained from obfervation; but the common method of deducing, the fhip's longitude, from the courfe and diftance made good, is at beft only an approximation to the truth, and, at the end of the voyage, the accumulated error is fometimes very confiderable; having, in a run from Britain to the West Indies, been found to exceed eight or ten degrees. It has therefore been the wish of the practical navigator, and of every maritime nation, to have fome certain method of afcertaining the longitude of a fhip at fea. In order to obtain fo useful a discovery, feveral very confiderable rewards have been offered to any perfon, who would give a method, by which the ship's longitude might be determined within proper limits, as often as neceffary.

The firft who offered a reward for the discovery of the longitude at fea, was Philip III. of Spain in the pear 1598, and foon after, the States of Holland followed his example. The reward offered by Philip was 1000 crowns, and that by the States 10,000 florins. In the year 1714, the British Parliament offered a premium of 20,000l. and in 1716, a reward of 100,000 livres was promised by the Duke of Orleans, who was then Regent of France. In confequence of which, many and various methods have been proposed to folve this important problem; of these which have hitherto appeared, that by obferving the distance between the Moon and the Sun or a fixed star feems to be the most proper for this purpose, both on account of the frequency of the obfervations, and of the fhortnefs of the calculations.

John Werner of Nuremberg appears to be the first who propofed the method of finding the longitude, by obferving the distance between the Moon and a star, in his annotations on the first book of Ptolemy's Geography, printed 1514. He recommends the cross staff as

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a very proper inftrument for the purpose of observing the distance between these objects.

This method was again recommended by Gemma Frifius, in his treatife, intitled, Structura radii aftronomici et geometrici, printed at Antwerp in 1545, which ufually accompanies his edition of Apian's cofmography. At folio 39th, he expreffes himself as follows," Sed "quotidie fere, fi quis velit longitudinem loci alicujus perquirere, is dili "genter confideret luna diftantiam ab aliquo fidere, fydere firmamento "per radium noftrum. Ita tamen ut illa ftella fixa fecundum rectum "ecliptica ductum, lunam, præcedit aut fequatur. He then proceeds to show how the Moon's longitude is found from the above obfervation, and directs the apparent time of obfervation to be inferred from the altitude of the ftar obferved at the fame inftant; the Moon's longitude at this time is also to be computed from the best astronomical tables. Hence the difference between the obferved and computed longitudes of the Moon will be known; with which, and the Moon's horary motion, the difference of longitude, between the place of observation and that to which the tables are adapted, may be found.

The celebrated Kepler was fully perfuaded of the utility of this method of finding the longitude at fea. In his Rudolphine tables, he gave directions for obferving the distance between the Moon and a ftar, and for making the neceffary computations, nearly the fame as thofe given by Gemma Frifius. This method was alfo recommended by Longomontanus.

Mr Blundevil defcribes this method of afcertaining the longitude at fea, in his Exercises, printed at London, the invention of which he attributes to Apian. It is alfo taken notice of by Carpenter in his Geography, printed at Oxford in the year 1635, wherein he fays, This way was taught by Apian, illuftrated by Gemma Frifius and Blundevil. He then proceeds to defcribe the manner of obferving the distance, from whence the longitude of the place of obfervation is to be inferred as formerly. He concludes as follows: This way, though more difficult, may feeme better than all the reft, for as much as an eclipfe of the Moon feldom happens, and a watch, clocke, or hour glaffe cannot fo well be preferved, or at least fo well obferved in fo long a voyage, whereas every night may feem to give occafion to this experiment, if fo bee the ayre be freed from clouds, and the Moone fhew her face above the horizon. M. Jean Baptifte Morin, profeffor of mathematics at Paris, improved the above method; which, by order of Cardinal Richlieu, was examined in 1634. It was, however, judged to be incomplete, upon account of the imperfections of the lunar tables. Morin therefore wrote against those appointed to examine his method, particularly M. Pierre Herigone, who had propofed feveral methods of finding the longitude, in the fourth volume of his Courfe of Mathematics; which methods Morin endeavoured to confute. An an

fwer was given thereto by Herigone, at the end of the fifth volume of the above work, printed at Paris in 1637.- Although Morin did not obtain the reward he claimed for his improvement of the above method, yet, in 1645, Cardinal Mazarin procured him a penfion of 2000 livres.

A method for the above purpose was now much defired in England. The Royal Obfervatory at Greenwich was founded by Charles II. in th year 1675, and Mr Flamfteed appointed Astronomer Royal. The words of his commiffion were, To apply himself with the utmost care and diligence to the rectifying the tables of the mctions of the heavens, and the places of the fixed flars, in order to find out the fo much defired longitude at fea, for perfecting the art of navigation.

Among all the celestial objects, the Moon, upon account of the quicknefs of its motion, appeared to be the beft adapted for the above purpose. Any method, however, depending on obfervations of the Moon, which were to be compared with lunar tables, was very un. certain, as long as thofe tables remained fo inaccurate. The longitude, fays Mr Flamsteed*, might be alfo attained by obfervations of the Moon, if we had tables that would answer her motions exactly; but after 2000 years, we find the beft tables extant, erring fometimes 12 minutes or more in her appareut place, which would caufe a fault of half an hour, or 7% degrees in the longitude deduced, by comparing her place in the heavens with that given by the tables.

That juftly celebrated astronomer and navigator, Dr. Edmund Halley, recommended obfervations of the Moon, as the most certain method of afcertaining the longitude at fea, having, by his own experience, found the impractibility of all the other methods propofed for that purpofe. He gave an excellent paper on that fubject, in the Philofophical Transactions, No. 420, wherein he shews the defects of the lunar tables extant. By comparing the place of the Moon, as given by the tables, with that deduced from observation, he found the errors of the tables recur with great regularity, at the end of 18 years 11 days; fo that whatever error was found in a former period, the fame error was again repeated, under the like circumftances, of the fame diftance of the Moon from the Sun and apogeon. Being encouraged by this, he next examined what difference might arife, from the period of nine years wanting nine days; in which time there are performed very nearly, one hundred and eleven lunations; but the return of the Sun to the apogee in that time, differing above four times as much from an exact revolution, as in the period of eighteen years, a like agreement was not to be expected. Having, however, entered upon the 10th year of his obfervations of the Moon's tranfit, he compared his late obfervations of 1730 and 1731, with thofe he had made in 1721 and 1722, and very feldom

Phil. Tranfactions, Lowthorp's abridgement, vol. 1. page 537.

feldom found a difference of more than one fingle minute of motion, but most commonly this difference was wholly infenfible; fo that, by the help of what he obferved in 1722, he prefumed that he was able to compute the true place of the Moon with certainty within two minutes of motion, during the year 1731, and fo on for the future. He finishes this paper, by recommending Hadley's quadrant as a proper inftrument for taking the neceffary obfervations at fea. This inftrument had been defcribed in the preceding number of the Tranfactions of the Royal Society.

Dr Halley again treats of this fubject in his Aftronomical Tables, in which are given two complete examples of finding the longitude, from the observed distance between the Moon and the stars,, Leonis and Tauri, and concludes with obferving, that by a like method of computation, may the difference of the meridians be found from obfervations of the Moon's distance from the Sun, in her first and last quarter.

M. Emanuel Swedenborg takes notice of the above method of finding the longitude, in his Daedalus Hyperboreus, printed in Swedifh, at Stockholm, in the year 1716-and other tracts. It is alfo briefly explained in a very small pamphlet, intitled, An introduction to a true method for the discovery of longitude at fea, by Stephen Plank, printed at London in 1720.

The Abbé de la Caille recommends it as the only practical method at fea. He was, however, fenfible, from his own experience, of the errors to which it was then liable. In his edition of Bouguer's Navigation, printed at Paris in 1781, he says, "La grande incertitude à laquelle nous avons dit qu'étoit fujette la méthode d'employer les obfervations de la Lune faites fur mer, ne doit pas découra ger le marin, ni la lui rendre suspecte, puifque dans les voyages de long cours, où l'on a effuyé beaucoup de vents contraires, & de long coups de vents, il arrive fouvent qu'aux atterrages on fe trouve en erreur de fept ou huit degrés fur la longitude eftimée felon les regles du pilotage." In the fame work, he gave a new and eafy method of reducing the apparent to the true diftance, by means of fcales conftructed exprefsly for that purpofe.

In 1755, Profeffor Mayer of Gottingen fent a manuscript copy of his lunar tables to the British Admiralty, claiming at the fame time fome one of the rewards, promifed by Parliament, which he might be thought to merit. In thefe tables, the arguments are investigated on the Newtonian principle of universal gravitation, and the maxima of the equations are deduced from his own obfervations, and from those of Dr Bradley and others. The above tables were delivered to Dr Bradley, to be examined; who, having compared them with a great number of his own obfervations, was convinced of their excellence. He then fent an account of them to Mr Cleveland, fecretary

* Mayer's tables first appeared in the Memoirs of Gottingen for 1742.