which makes them, particularly the moon, appear lower than they would otherwife do, excepting when they are in the very zenith. It is also well known, that the nearer the horizon any celeftial body is, the greater its parallax will be; and as the parallax and refraction act in oppofite ways to one another, the former depreffing, and the latter raifing, the object, it is plain, that great difficulties muft arife from this circumftance. The fun, for inftance, whofe parallax is less than the refraction, muft always appear higher than he really is; but the moon, whofe parallax is greater than her refraction, must always appear lower. To render obfervations of the celestial bodies more eafy, the commiffioners of longitude have caufed an EPHEMERIS, or NAUTICAL ALMANACK, to be published annually, containing every requifite for folving this important problem, which can be put into the form of tables. But whatever may be done in this way, it will be proper to make the neceffary preparations concerning the dip of the horizon, the refraction, femidiameters, parallax, &c. in order to reduce the apparent to the true altitudes and diftances; for which we fhall here fubjoin two general rules.

The principal obfervation for finding the longitude at fea is that of the moon from the fun, or from fome 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 fix hours: and with a good Hadley's quadrant, or, which is preferable, a fextant: and this laft inftrument will still be more fit for the purpose, if it be furnished with a fcrew for moving the index gradually; likewife an additional dark glass, but not fo dark as the common kind, for taking off the glare of the moon's light, in obferving her distance from a ftar. A fmall telescope, which may magnify 3 or 4 times, is alfo neceffary to render the contact of a ftar with the moon's limb more difcernible. A magnifying glass of 1 or 2 inches focus will affift the operator in reading off his obfervations with the greater facility.

1. To MAKE the OBSERVATION. Having examined and adjusted his inftrument as well as poffible, the obferver is next to proceed in the follow ing manner: If the diftance of the moon from the fun is to be obferved, turn down one of the fcreens; look at the moon directly through the transparent part of the horizon-glafs; and keep ing her in view, gently move the index till the fun's image be brought into the filvered part of that glafs. Bring the neareft limbs of both objects into contact, and let the quadrant librate a little on the lunar ray; by which means the fun will appear to rife and fall by the fide of the moon; in which motion the neareft limbs muft be made to touch one another exactly, by moving the index. The obfervation is then made; and the divifion coinciding with that on the Vernier fcale, will show the distance of the nearest limbs of the objects.

When the diftance of the moon from a ftar is to be observed when the moon is very bright, turn down the lighteft fcreen, or ufe a dark glafs lighter than the fcreens, and defigned for this particular purpofe; look at the ftar directly through

the tranfparent part of the horizon-glaís; and keeping it there, move the index till the moon's image is brought into the filvered part of the fame glafs. Make the quadrant librate gently on the ftar's ray, and the moon will appear to rife and fall by the ftar: move the index between the librations, until the moon's enlightened limb is exactly touched by the ftar, and then the obfervation is made. In thefe operations, the plane of the quadrant muft always pafs through the two objects, the diftance of which is to be observed; and for this purpose it must be placed in various pofitions, according to the fituation of the objects, which will foon be rendered eafy by practice.

The obfervations being made, fomebody, at the very inftant when the operator calls, muft obferve, by the watch, the exact hour, minute, and quarter minute, if there be no fecond hand, in order to find the apparent time; and at the fame inftant, or as quick as poffible, two affiftants must take the altitudes of those objects, the distance of which is obferved; after which, the observations neceffary for finding the longitude are completed.

The ephemeris fhows the moon's distance from the fun, and likewife from proper ftars, to every 3 hours of apparent time for the meridian of Greenwich; and that the greater number of op portunities of obferving this luminary may be gi ven, her diftance is generally fet down from at least one object on each fide of her. Her diftance from the fun is fet down while it is between 40 and 120 degrees; fo that, by means of a fextant, it may be obferved for 2 or 3 days after her firft, and before her last quarter. When the moon is between 40 and 90 degrees from the fun, her diftance is fet down both from the fun, and from a ftar on the contrary fide; and, laftly, when the diftance is above 120 degrees, the diftance is fet down from two, ftars, one on each fide of her. The diftance of the moon from objects on the east fide of her, is found in the ephemeris, in the 8th and 9th pages of the month; and her diftance from objects on the weft, is found in the 10th and 11th pages of the month.

When the ephemeris is ufed, the distance of the moon muft only be obferved from thofe ftars, the distance of which is fet down there; and these afford a ready means of knowing the ftar from which her diftance ought to be obferved. The obferver has then nothing more to do, than to fet his index to the distance roughly computed at the apparent time, eftimated nearly for the meridian at Greenwich; after which he is to look to the E. or W. of the moon, according as the distance of the ftar is found in the 8th or 9th, or in the roth or 11th, pages of the month; and having found the moon upon the horizon-glafs, the ftar will eafily be found by fweeping with the quadrant to the right or left, provided the air be clear, and the ftar be in the line of the moon's fhorteft axis produced. The time at Greenwich is eftimated by turning into time the fuppofed longitude from that place, and adding it to the apparent time at the fhip, or fubtracting it from it, as occafion requires. The diftance of the moon from the fun, or à ftar, is roughly found at this time, by faying, As 180 minutes (the number contained in three hours) is to the difference in minutes between this,

nearly

fun or ftar's apparent altitude, and the logarithmic tangent of the apparent diftance of the moon from the fun or ftar. The fum of thefe, rejecting zo in the index, will be the proportional logarithm of the second angle. 3. Take the difference between the firft and fecond angles, adding it to the apparent diftance if it be lefs than 90, and the firft angle be greater than the fecond; but fubtracting it if the fecond be greater than the firft. If the distance be greater than go, the fum of the angles must be added to the apparent distance, which will give the diftance corrected for the refraction of the fun or ftar. 4. To the proportional logarithm of the correction of the moon's altitude add the logarithmic cofine of her apparent altitude; the logarithmic fine of the distance correct for the fun or atar's refraction, and the logarithmic co-fecant of the fun or ftar's apparent altitude. The fum, 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 co-tangent of her apparent altitude, and the tangent of the diftance corrected for the fun or ftar's refraction; their fum, 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 fubtract it from the diftance corrected for the fun or star's refraction if lefs 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 diftance be more than 90, the fum of the angles must be fubtracted from it, to give the distance corrected for the fun or ftar's refraction, and the principal effects of the moon's parallax. 7. In Table XX. of the ephemeris, look for the dif tance corrected for the fun and ftar's refraction, and the moon's parallax in the top column, and the correction of her altitude in the left-hand fide column; take out the number of seconds that ftand under the former, and oppofite to the latter. Look again in the fame table for the corrected distance in the top column, and the correction of the moon's altitude in the left-hand fide column; take out the number of feconds that ftand under the former, and oppofite to the latter. Look again in the fame table for the corrected diftance in the top column, and the correction of the moon's altitude in the left-hand fide column; take out the number of feconds that ftand under the former, and oppofite to the latter. Look again in the fame table for the corrected diftance in the top-column, and the principal effects of the moon's parallax in the left-hand fide column, and take out the number of feconds. The difference between these two numbers muft be added to the corrected diftance if lefs than 90, but fubtracted from it if greater; and the fum or difference will be the true distance.

4. To DETERMINE the LONGITUDE, after having obtained the TRUE DISTANCE. Look in the ephemeris among the diftances of the objects for the computed distance betwixt the moon and the other object obferved 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 diftances in the ephemeris, which ftand immediately before and after it, and also the difference between the distance standing before and the computed diftance; then take the proportional logarithms of the firft and fecond differences, and the difference between these two logarithms will be the proportional logarithm of a number of hours, minutes, and feconds; which being added to the time ftanding over the firft diftance, will give the true time at Greenwich. Or it may be found by faying. As the firft difference is to three hours, fo is the fecond 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 fhip, turned into longitude, will be that at the time the obfervations were made; and will be E. if the time at the ship is greatest, but W. if it is least. SECT. IV. EXAMPLES of FINDING the LONGITUDE at SEA, by all the DIFFERENT METHODS ufually tried.

I. TO FIND the LONGITUDE by COMPUTATION from the SHIP'S COURSE. Were it poflible to keep an accurate account of the distance the ship has run, and to measure it exactly by the log (fee Log, § 8.) or any other means, then both latitude and longitude would easily be found by fettling the ship's accounts to that time. For the courfe and diftance 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 alfo be known. A variety of causes, however, concur to render this computation inaccurate; particularly the fhip's continual deflection from the courfe fet by her playing to the right and left round her centre of gravity; the unequal care of thofe at the helm, and the distance supposed to be failed being erroneous, on account of ftormy fcas, unfteady winds, currents &c. for which it feems impoffible to make any allowance. The place of the fhip, however, is judged of by finding the latitude every day, if poffible, by obfervations; and if the latitude found by obfervation agrees with that by the reckoning, it is prefumed that the fhip's place is properly determined; but if they difagree, it is concluded that the account of the longitude ftands in need of correction, as the latitude by observation is always to be depended upon

Currents very often occafion errors in the computation of a fhip's place. The causes of these in the great depths of the ocean are not well known, though many of the motions near the fhore can be accounted for. It is fuppofed, that fome of thofe in the great ocean are owing to the tide following the moon, and a certain libration of the waters arifing thence; likewife that the unfettled nature of thefe currents may be owing to the changes in the moon's declination. In the torrid zone, however a confiderable current is occafioned by the trade winds, the motion being conftantly to the W. at the rate of 8 or 10 miles per day. At the extremities of the trade winds, or near the 30th degree of N. or S. Lat. the cur

rents

rents are probably compounded of this motion to the weftward, and of one towards the equator; whence all ships failing within these limits ought to allow a course each day for the current.

When the error is fuppofed to have been occafioned by a current, it ought, if poffible, to be tried whether the cafe is fo or not; or a reafonable estimate must be made 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 obfervation, the departure thus corrected may be fafely taken as true, and thus the fhip's place with regard to the longitude determined.

EXAM. Suppofe a fhip in 24 hours finds, by her dead reckoning, that she has made 96 miles of difference of lat. N. and 38 miles of departure W. 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 lat. N. and 10 miles of departure E. Required the fhip's departure?

Diff. lat. by acc. 96 N. Diff. lat. by curr, 16 N.

Miles 112

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

When the error is fuppofed to arife from the courses and diftances, we muft obferve, that if the difference of latitude is much more than the departure, or the direct courfe has been within 3 points of the meridian, the error is most probably in the diftance. But if the departure be much greater than the difference of latitude, or the direct course be within 3 points of the parallel, or more than 5 points from the meridian, the error is probably to be ascribed to the courfe. But if the courses in general are near the middle of the quadrant, the error may be either in the courfe, or in the distance, or both. This method admits of three cafes.

1. When, by the dead reckoning, the difference of latitude is more than once and an half the departure, or when the courfe is less than 3 points; find the course to the difference of latitude and departure. With this courfe and the meridional difference of latitude by obfervation, find the difference of longitude.

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

3. When the difference of latitude and departure by account is nearly equal, or the direct course is between 3 and 5 points of the meridian, find the course with the difference of latitude and departure by account, fince the last obfervation. Within this course and the difference of latitude

by obfervation find another departure. Take half the fum of thefe departures for the true one. With the true departure and difference of latitude by obfervation 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, fhowing the variation in all the places over which they paffed in the year 1700, at which time he first published the chart; whence the longitude of thofe places might be found by the chart, provided its latitude and variation was 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 fhip's place.

EXAM. A fhip finds by obfervation the latitude to be 18° 20' north; and the variation of the compass to be 4° weft. Required the fhip's place? Lay a ruler over 18° 20' N. parallel to the equator; and the point where its edges cut the curve of 4° W. variation gives the fhip's place, which will be found in about 27° 10′ W. from London.

This method of finding the longitude, however, is attended with two inconveniences. 1. That when the variation line runs E. or W. or nearly fo, 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 reft. Even in thofe places indeed where the variation curves do run E. or W. they may be of confiderable use in correcting the latitude when meridian obfervations cannot be had; which frequently happens on the northern coafts of America, the Weftern Ocean, and about Newfoundland; for if the variation can be found exactly, the eaft and weft curve answering to it will fhow the latitude. But, 2. The variation itself is fubject to continual change; whence a chart, though ever so perfect at first, muft in time become totally ufelefs; and hence the charts constructed by Dr Halley, though of great utility at their firft publication, became at length almost entirely useless. A new one was published in 1746 by Mefirs Mountaine and Dodfon, which was fo well received, that in 1756 they again drew variation lines for that year, and published a third chart the year following. They alfo prefented to the Royal Society a curious paper concerning the variation of the magnetic needle, with a fet of tables annexed, containing the refult of more than 50,000 obfervations, in fix periodical reviews from 1700 to 1756 irclufive, adapted to every 5° of lat. and lon. in the more frequented oceans; all of which were published in the Philof. Tranf. for 1757.

3. To FIND the LONGITUDE by the SUN'S DECLINATION.-Having made fuch obfervations on the fun 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 Eec 2 declination

declination at that time; then fay, as the daily difference of declination is to the above found difference, fo is 360 degrees to the difference of longitude. In this method, however, a small error in the declination will make a great one in 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; alfo take the difference betwixt the times of culminating on the fame day, as found in the ephemeris, and as obferved; then say, as the daily difference in the ephemeris is to the difference between the ephemeris and obfervation; fo is 360° to the difference of longitude. In this method alfo a finall difference in the culmination will occafion a great one in the longitude.

5. By ECLIPSES of the MOON.-This is done much in the fame manner as by the eclipfes of Jupiter's fatellites: For if, in two or more diftant places where an eclipfe of the moon is vifible, we carefully obferve the times of the beginning and ending the number of digits eclipfed, or the time when the shadow touches fome remarkable spot,' or when it leaves any particular fpot 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 feldom to be of much ufe.

6. In the 76th vol. of the Philof. Tranf. Mr EDWARD PIGOT gives a very particular account of his method of determining the fon. and lat. of York; in which he alfo recommends the method of determining the longitude of places by obferva tions of the moon's tranfit over the meridian. The inftruments ufed in his obfervations were agridiron, pendulum clock, a two feet and an half reflector, an eighteen inch quadrant made by Mr Bird, and a tranfit inftrument made by Mr Siffon. By thefe inftruments an observation was made, on the roth Sept. 178, of the occultation of a star of the 9th magnitude by the moon, during an eclipfe of that planet, at York and Paris. Befides this, there were obfervations made of the immersions of Aquarii and Pifcium ; the result of all which was, that between Greenwich and York the difference of meridians was 4' 27'.

In 1783, Mr PIGOT thought of finding the dif. ference of meridians by obferving the meridian right afcenfions 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 fame fubject. by the Abbé ToALDO; but ftill found that the great exactnefs of this method was not fufpected; though he is convinced that it must foon be univerfally adopted in preference to that from the firft fatellite of Jupiter. After giving a number of obfervations on the fatellites of Jupiter, he concludes, that the exactness expected from obferva. tions, even on the first fatellite, is much overrated. "Among the various objections (fays he), there is one I have often experienced, and which proceeds folely from the difpofition of the eye, that of feeing more diftinctly at one time than another. It may not be improper alfo to mention, that the obfervation I should have relied on as the

beft, that of Aug. 30, 1785, marked excellent, is one of those moft diftant from the truth."

After giving a number of observations on the eclipfe of the moon, Sept. 10, 1783, he concludes, that the eclipfes of the moon's fpots 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 fky. 2. To chufe fuch spots as are well defined, and leave no hesitation as to the part eclipfed. 3. That every obferver should ufe, as far as poffible, telefeepes equally powerful, or at leaft let the magnifying powers be the fame. "A principal objection (fays he may ftill be urged, viz. the difficulty of diftinguishing the true fhadow from the penumbra. Was this obviated, I believe the refults would be more exact than from Jupiter's first fatellite: Undoubtedly the fhadow appears better defined if mágnified little; but I am much inclined to think, thất, with high magnifying powers, there is greater certainty of chooling the fame part of the fhadow, whichperhaps is more than a fufficient compensation for the lofs of distinctness.”

The following rule for meridian obfervations of the moon's limb is next laid down: "The increafe of the moon's right afcenfion in 12 hours (or any given time found by computation), is to 12 hours as the increase of the moon's right afcenfion between two places found by obfervation is to the difference of meridians. EXAMPLE.-Nov. 30, 1782.

141" in feconds of a degree, ditto, ditto, ditto. The increafe of the moon's right afcenfion for 12 hours, by computation, is 23,340 feconds; and 12 hours reduced into feconds is 43,200. Therefore, according to the rule ftated above,

23,340": 43,200": diff. of merid. =261′′. "Thefe cafy obfervations and short reduction (fays Mr Pigot) are the whole of the bufinefs. Inftead of computing the moon's right afcenfion for 12 hours, I have conftantly taken it from the Nautical Almanacks, which give it fufficiently exact, provided fome attention be paid to the increafe or decrease of the moon's motion. Were the following circumstances attended to, the rcfults would be undoubtedly much more exact.

"1. Compare the obfervations with the fame made in feveral other places. 2. Let feveral and

the