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1、Comm:Ac.Scient.Petr.Tom.IIIp.110;Nov.1728.L.Euler.1Concerningtheshortestlineonanysurfacebywhichanytwopointscanbejoinedtogether.E009:Translated&AnnotatedbyIanBruce.Concerningtheshortestlineonanysurfacebywhichanytwopointscanbejoinedtogether.AuthorLeonardEuler.1.Itiswe
2、ll-knownthattheshortestlineorpathfromagivepointtosomeotherpointisastraightline,whichisconsideredasanaxionbymanywriters.Itiseasilyunderstoodfromthisthatwhenthesurfaceisaplane,theshortestdistancejoininganytwopoints[intheplane]isthestraightlinedrawnfromonetotheother.On
3、asphericalsurface,onwhichitisnotpossibletodrawstraightlines,ithasbeenestablishedbythegeometersthattheshortestpathbetweentwogivenpointsisthe[shorterarcofthe]greatcirclejoiningthem.2.However,foranysurfaceeitherconvexorconcave,withoutbeingamixtureofthetwo,whattheshorte
4、stpathshallbe,drawnfromonegivenpointtoanyother,hasnotyetgenerallybeendetermined.ThemostcelebratedJohanBernoullihasproposedthisquestiontome,indicatingthathehimselfhasfoundthegeneralequation,inorderthattheshortestlinetobeappliedtoagivensurfacebetweenanytwogivenpointsc
5、anbefound.Itoohavesolvedthisproblem,andIwanttosetoutthesolutioninthisdissertation.3.Mechanicallythisproblemiseasilysolvedwiththehelpofathreadwhichisstretchedbetweenthetwogivenpoints:thelengthitbecomeswilldesignatetheshortestpathontheproposedsurface.Moreoveritisneces
6、sarythatthesurfaceisconvex,inorderthatthisthreadtouchesthesurfaceeverywhere,forwithconcavesurfacestheshortestlengthisnotrepresentedbythearcofacurvebutindeedbythechord[joiningthepoints;thoughthischorddoesnotlieonthesurface].Thereforeinthiscasethethreadoughttobeapplie
7、dthus,ortobesoconsideredinthisapplication,thatitalwaystouchesthesurfaceinaconvexpart.4.Truly,anyonewhowishestoexaminethenatureoftheinnermostsecretsofthisline,andwhoisaccustomedtohavinganequationsetup,cannotbesatisfiedwiththisgeometricalconstruction.Moreover,thelines
8、oughtthathasbeenseenfromamechanicalconstructionishardlyonethatissetout[inamathematicalsense],andneithercanthenatureofthelinebeexamined.Ona
