LUNAR OBSERVATIONS. 



r.-.i didanec. All celedial objefls, except the moon, are 

 'ore afifeiStcd by rcfraftion than by parallax, and therefore 

 , .ipear above tlieir true places ; but the moon is always fcen, 

 t .reptinfi^ in the zenith, below her true place, being more 

 p.iTccted by parallax ilian refratiion, on account of her 

 proximity to the cartli. 



Tliefe clTeCls of parallax and rcfraftion, though counter- 

 silincr each other, fcldom do it fo cqua ly as to render all 

 t-irreiSion unncceffary. Sometimes the apparent diftance is 

 r.earlr a whole degree more or lefs thia the true diftance ; 

 and jlie principal canfe of fo great a difference is the moon's 

 parallax : for this body, which is the chief guide to the 

 lj:ig;tude, is a'fo the great caufe of error in the diftances» 

 aiid is therefore the principal object of corredlion. 



In making a hmar obfervation, four perfons arc generally 

 en-ployed, one of whom takes the diftance, two the a!titudcs, 

 a :d the fourth'notes the time. Thefe things fliould be per- 

 f^.rmed at the fame inilant; and if the obfervation be re- 

 ;^C3ted feveral times, and a mean taken, the work is likely 

 '. . be the more correS i and great care is here neceffary, 

 i r an error in this part of ihe operation, particularly in 

 t.king the dillancc, will pervade tlie fubfequcnc parts of the 

 v.ork, and will of courfe produce a wrong foliition. The 

 ranner of adjufting the indruments, and cf making the ob- 

 ; ivations, is bi-'ft taught by practice. Thufe who with for 

 written inftruflions on the fubject are referred to the Britiih 

 Mariner's Guide by Dr. Mafkelyne, to Dr. Mackay's 

 book upon the longitude, or to profefibr Vince's Pradtical 

 Aftronomy. 



Of corred'mg thi Ahitudss of the obfrveJ Objefls. — When a 

 lun \r obfervation is made, the firll cbjeft is to clear the al- 

 ti'.udes from femidiameter, dip, refraClion, and parallax. 

 The moon's parallax in altitude mult be next calcu- 

 -laled; by faying, uls radius is to the fine of her zenith dif- 

 tance, fo is the fine of her horizontal parallax (as given in the 

 Nautical Almanac) to the fine of her parallax in altitude. 



In correfting the moon's altitude, an allowance iliould be 

 made for the sugmentation of her femidiameter, which gra- 

 dually takes place from the horizon to the zenith. I'his 

 inercafe is given, ii the IVth of the Requilite Tables, 

 for every five degrees of aliitude, which corrediion is to 

 be added to her horizontal femidiameter given in the Nauti- 

 cal Almarac. 



The augmentation of the moon's femidiameter is caufed 

 by her being nearer to the fpeftator in the .zenith than in 

 the horizon by a femidiameter of the earth — for the magni- 

 tude of a body is in the inverfe ratio of its di'lance from 

 the obferver ; and as the earth's femidiameter bears a very 

 fenfible proportion to the moon's diilance, (he is feen under 

 the greatefl angle in the zenith, which angle gradually dimi- 

 niflies to t!ic ho.izon. 



Thc.-e are other correftion: of the altitudes which may 

 be neceffary in cafes of particular nicety, but which are 

 feldom noticed at fea.. Thefe are — an allowance for the 

 contraction of the vertical femidiameters of the fun and 

 moon by rcfraftion ; a correftion of the moon's parallax, 

 fuppofing the earth an oblate fpheroid ; a correftion for 

 tlie refrattion, according to the aclual ftate of the atmo- 

 fphere, as (hewn by a thermometer and barometer, and not 

 according to the mean al^ronomical refraction which is com- 

 monly ufed. Thefe correftions, though perhaps neceffary 

 towards the perfedtion of this prrblem, being very fmall, and 

 frequently counterafting each other, are generally confidercd 

 of little confequence in nautical praClice, where greater errors 

 are unavoidable. 



From the or relied Altitudes to Jind the true D'flance. — It is 

 eafy to conceive, that by a lunar obfervation, the three iides 



of a fpheric triangle are meafured in the heavens, whicU 

 fides are the apparent co-altitudes of the obferved bodies, 

 and their apparent diftance afunder. 



The co-altitudcs or zenith diftances being ccrrefted, the 

 queftion is, to find the true diftance between the obferved 

 bodies ; but here there arc only tv.'o things given, and tlierc- 

 foic it cannot be performed unti' the angle at the zenith is 

 known, which is determined from the three given fides of the 

 triantjle, by the rules of fpheric trigonometry. 



As the efFefts of parallax, refraction, &c. operate only 

 in a veriical direction, it is evident that the corrcdlions of 

 the zenith dillanees or containing fides will not change the 

 included angle at the zenith ; and therefore three things are 

 now known, namely, the correcled zenith diftances and the 

 included angle, whence the other fide is determined by 

 Ipherics, and this fide is the true diftance fought. 



A General View cf the different Methods of •ivorling the 

 Lunar Olfervations, — Few problems have been ever more in- 

 veftigated or ftudied than that of clearing the lunar dif- 

 tance, and many ingenious methods lv>ve been devifed for 

 contrafting the operation. Thefe methods are founded on 

 fome of the following general principles. 



The firft; general principle is fpheric trigonometry, as be- 

 fore explained ; the fecond is the doftrine of proportional 

 errors, by which the effects that the errors in the alti- 

 tudes produce on the diftance are folved by fluxions, or by. 

 xXvi differential calculus : and a third principle has been lately 

 dilcovered, which is founded on the properties of a qnad- 

 i-angle infcribed in a circle, as explained and exemplified Ijy 

 the inventor, Dr. Andrew, in his Altronomical and Nautical' 

 Tables. ^ 



Various methods of working the lunar obfervations have 

 been devifed chiefly by Halley, Euler, Mayer, Maikelyne, . 

 Lyons, Witchcll, Burrow, Borda, Wales, Mackay, Kelly, 

 Gerrard, Andrew, and Mendoza. The methods of the 

 two laft authors appear the moft concife, but all are fuffi- 

 ciently correft, and leainen generally prefer fuch as they have 

 firft learnt. It may indeed be obferved, that operations which' 

 appear the moft concife are not always the moft cxpedi- 

 tioufly performed, as much depends on the number and. 

 variety of tables required, and the manner of applying 

 them. No method has hitherto obtained an exclufive pre- 

 ference over the reft, nor does it appear poffible to reduce 

 the calculation to a concifenefs to anfwer the general pur- 

 pofcs or wifhes of feamen ; and hence, other modes have 

 lieen devifed, of obtaining approximate folutions by pro- 

 jection or graphic operation. 



The firll graphic method for clearing the diftances was 

 that by La Cai'le, called the Chajfis de redudion, which has 

 fuice been copied by La Lande, Mackay, and others. It. 

 is an orthographic projeftiun, conliftmg of a great num- 

 ber of lines accurately drawn, and various fcaL-s for ob-^ 

 taining the different corrections. 



Another graphic operation, of a different defcription, was 

 executed by the late George Margetts, and p^bli(hed in 

 1790. It confifts of fevetity large jl.ttes, containing nume- 

 rous lines drawn from ttie folutions of lunar diftances in 

 Dr. Shepherd's Tables. By Margetts' Longitude Tables 

 the folution of a lunar obfervation may be obtained in 

 about one-fourth of the tirj-e required by calculatinn ; and 

 the anfwer, though not perfeftly accurate, is fulBciently. 

 correft for the general purpofes of navigation. 



An orthographic projection, founded on the fluxional 

 analogies of fpheric triangle?, has been devifed by Dr. 

 Kelly, and publifhed in his Introduction to Spherics and. 

 Nautical Ailronomy, where an inveftigation of its prin- 

 ciples is given (p. 195, edit. 2 and 3.) with a demon-- 



ftraticDj, 



