LONGITUDE. 



brought the lunar tables to an unexpected degree of pre- 

 cifion, aftronomers of every nation began to conceive the 

 molt rational hopes, that, by gradual improvement, this 

 method would at laft be found to eqya! the moll fanguine 

 expeftation. 



Thofe who firll attempted to praflife it had to ftruggle 

 with great difficulties ; and the requifite calcuhitions were 

 fo formidable, that none but aftronomers, or at leaft very 

 able calculators, could poflibly attempt them. 

 ' The late aftror.omer royal, Dr. Mufl^elyne, praftifed this 



method with the gieatell fucccfs ; and it is to him this 

 country is indebted for fome of the grcateft improvements 

 that hav^ been made. It was he who firft propofed and 

 fuperintended the conftruftion of the Nautical Almanac, 

 which relieves the calculator from all the very laborious part 

 of the procefs ; and the remaining part of the computation 

 has been fo fimplified by fucceffive improvements, both in the 

 formula; and conftruftion of tables, that, at prefent, the necef- 

 fary obfervations may be both made and coniputed by any ma- 

 nner, who has received a tolerably good nautical education. 



As the praftical methods of making and computing a 

 lunar obfervation are given at great length in every nautical 

 book, we fliaU confine ourfelves to explaining the general 

 nature and objedl of the problem, and refer the reader to 

 profeffcd treatifes on navigation for farther information. In 

 Mackay's trcatife on the Longitude, the reader will find 

 fome excellent methods of folving both this and a variety of 

 other nautical problems, accompanied by very ufcful tables. 

 Mendoza's tables contain his own valuable method of coin- 

 pnting a lunar obfervation, belide general tables for every 

 nautical purpofe. The requifite tables are well known, and 

 are in the hands of every navigator. 



Explanation of the princ'iplcs of the method by tuhich the longi- 

 ' tuele is found at fea, by ohferving the dijlance of the moon from 

 the fun, or a given fixed Jlar. 



The requifite data for determining the jlongltude at fea, 

 by the lunar inethod, are the apparent dittance of the centre 

 of the moon from the centre of the fun or Itar, and the ap- 

 parent altitude of the centres of each at the moment of 

 obfervation. Hence three obfervers are ufnally employed : 

 one obicrves the diftance between the fun and moon, one 

 the altitude of the fun, and the other the altitude of the 

 moon. When this cannot be done, the place of the other 

 two may be fupplied by computation. 



By means of lunar tables, the exaft diftance of the moon 

 from the fun or ftar is computed for every three hours, for 

 the meridian of Greenwich. We are not, however, to fup- 

 pofe that thefe dilhuices are fuch as the moon and fun would 

 appear to have at Greenwich ; but fuch as they would ap- 

 pear to an obferver at the centre of the earth. It is for 

 Creentvich time only that they arc computed ; a circumftance 

 not fufficiently infiflled upon by elementary writers on this 

 fubjeft. From thefe tables (of the Nautical Almanac) it 

 is eafy to infer the diftance for any intermediate interval : a 

 fimple proportion will be fufScient for this purpofe. We 

 may therefore confider ourfelves as in poffeirion of an in- 

 llantaneous phenomenon, anfwering to every inftant of time 

 at Greenwich ; fince the diftance of the fun and the moon 

 are never the fame for two fucceffive inftants of time. Now 

 if we confider the converfe of this propofition, it is equally 

 evident, that if we have given the diftance of the moon from 

 the fun, as feen from the centre of the earth ; we can, by 

 the fame tables, infer the exaft time at Greenwich cor- 

 refponduig to this diftance. Now the objeft of a lunar ob- 

 fervation is to determine this diftance at a given moment of 

 actual time, lo afcert.iin the apparent time at this moment 

 for the meiidJan of the obferver, and to compare it with the 

 moment of Greenwich time, which is to be inferred from 

 the given diftance. Now the difficulty of the procefs arifes 



from this circumftance, that fince, to an obferver on the 

 furface of the earth, the moon appears always deprefted bv 

 the efteft of parallax, and the fun elevated by the effiidt of 

 refrailion, the angular diftance obferved with a fextant, or 

 any other inftrument, is not the Hune as the diftance feen 

 from the centre of the earth, and for which alone the 

 nautical tables are calculated. Hence a fphcrical computa- 

 tion becomes necefl^ary. Two cafes of oblique fpherical 

 triangles mull be computed, before the obferved diftance 

 can be corrected, and the true diftance afcertained. 



The general nature of the problem may be m'lre eafily un- 

 derftood by a reference to the figure ( Plate XVII. jiflronomy, 

 fig. 1.), which is a projeftion of the fphere on the plane of 

 the meridian : i) i -j is the obferved or apparent dijiance of the 

 fun and moon ; Z ]) is the zenith diftance ot the moon ; 

 Z that of the lun ; m is the true place of the moon, when 

 corrected for refi-aftion and parallax, which together tend to 

 apparently deprefs it ; j is the true place of the fun, when 

 corredted by refradtion and parallax, which together tend 

 apparently to elevate it : for the moon's parallax is always 

 greater than the refradtion, the fun's always Icfs. For a ilar, 

 the fimple corrtttion for retradtion is all that is required. 



We have now, therefore, given three fides in the triangle 

 "L Gj )) , and two fides (viz. Z m, Z .r) in the triangle Z m s. 



In the triangle Z Q J » the angle Z may be found from 

 the three given fides ; and then with Z m, "ZiS, and the in- 

 cluded angle Z found above, ms, or the true diftance, may 

 be obtained. 



To fiiorten the folution of this problem, and to reduce it 

 within the compals of a mariner's ordinary powers of com- 

 putation, has been an objctt with the firft geometricians in 

 Europe. It would lead us much beyond our limits to give 

 a hiftory of the numerous folutions that have been propofed. 

 The French mathematicians, probably not having a great 

 facility of conftrudting tables, have diredted their attention 

 chiefly to fuch methods as require only the common tables 

 of logarithms. In our own country, where the board of 

 longitude is always ready to publifti any ufeful tables that 

 may be approved, thofe methods and formulx have been in 

 general preferred, which admitted of the ffiorteft folution 

 by means of tables. In this refpedt, a progreffive feries of 

 improvement has taken place fince the firft introdudtion of 

 the method ; and a flcilful mariner will now compute the 

 true diftance from the apparent in five minutes, when 

 formerly as many hours were required. 



Befides the methods founded on a diredt trigonometrical 

 folution, there are many (fuch as Lyon's and Dr. Maf- 

 kelyne's) which are founded on rather a different principle. 

 The fmall triangle J) m m' is computed as if a plane one, 

 ]) m' being the effedt of the total depreffion of the moon in 

 changing the diftance : a fimilar triangle is formed for com- 

 puting the eftedt of refradtion for the fun or ftar. Various 

 formulae have been deduced from each of thefe principles, 

 for the inveftigation of which the reader may confult Cag- 

 nole's Trigonometry, and various volumes in the Connoif- 

 fance dcs Temps. A very clear and fcientific inveftigation 

 of all thefe methods was given by Mr. Mendoza, in the 

 Philofophical Tranfadtions for 1797. 



To enable the reader to judge of fome of the moft ap- 

 proved of thefe, we ftiall give a folution of the fame problem 

 by a variety of different ways. 



Given Apparent altitude - 32^34' 47" 



Apparent altitude B - 39 3 4 



Apparent diftance © ]) - 86 10 19 

 Horizontal parallax - O j8 28 



Required the true diltance. 



The firft example we fliall give is the method of Borda, 

 which is in general ufe among the more ikilful of the French 

 navigators. 



The 



