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1、CommentariiacademiaescientiarumPetropolitanae4(1729)11735,pp.98-101.TranslatedandannotatedbyIanBruce.TheSolutionofanAstronomicalProblem:FromthreegivenAltitudesofaFixedStartofindtheElevationofthePoleStarandtheDeclinationoftheStar.LEMMAAFig.1cαInanysphericaltriangleABC(Fig.1)
2、,theβbcosineoftheangleAisgivenby:BγcosBC−cosABcosACaCcosA=sinABsinAC,withtheradiusorthewholesineplacedas1.[Thisisastandardresultforsphericaltrianglesdefinedinthe'Euler'sense,sothatalltheanglesarelessthanπ;seeanystandardworkonsphericaltriangles,e.g.TheVNRConciseEncyclopediao
3、fMathematics(1975),p.262onwards,whichhassomenice3Deffectdiagrams,thoughitisratherdated;perhapsthereisaneweredition.Wehaveaddedα,β,and,andγastheangles;aswellasa,b,andcforthesidesinFig.1forconvenience.Forreference,wegivethestandardcosinerulesforsidesandangles:Forsides:cosa=co
4、sbcosc+sinbsinccosα;andlikewisebypermutationfortheothersides.Forangles:cosα=−cosβcosγ+sinβsinγcosa;andlikewisefortheotheranglesbypermutation.andthesinerule:sinbsincsinα=sincsinasinβ=sinasinbsinγ.]ThisisprovenfromthesetheoremsthatthemostdistinguishedProfessorMATERhassetoutin
5、hisTrigonometry.COROLLARYFromtheseitfollowsthatcosBC=cosABcosAC+cosAsinABsinAC.THEOREMInanysphericaltriangleABC:cos(AB+AC)+cos(AB−AC)cosAcos(AB−AC)−cosAcos(AB+AC)cosBC=+.22Withthewholesinetakenas1.Theproductoftwocosinesisequaltohalfthesumofthecosinesandthedifferenceofthecos
6、inesofthearcsorangles.Andtheproductoftwosinesisequaltohalfthecosineofthedifference,withhalfthecosineofthesumofthearcsoranglestakenaway.Ascanbeeasilygatheredorbecomeapparentfromthesecited[Forangles,theusualrulesofplanetrigonometryapply].Hence:cos(AB+AC)+cos(AB−AC)cosBCcosAC=
7、,and2cos(AB−AC)−cos(AB+AC)sinBCsinAC=.2CommentariiacademiaescientiarumPetropolitanae4(1729)21735,pp.98-101.TranslatedandannotatedbyIanBruce.Withthesesubstitutedintothecorollarytothelemma,thisgivestheequation:cos(AB+AC)+cos(AB−AC)cosAcos(AB−AC)−cosAcos(AB+AC)cosBC=+.22Q.E.D.
8、PROBLEMForagivenfixedstarsuccessivelyFig.2Zpobservedinthreeplaces(Fig.2)ABC,withth
