Euclidean Nature Research Articles

Comments on Miller's "The Myth of Gauss' Experiment on the Euclidean Nature of Physical Space"

Comments on Miller's "The Myth of Gauss' Experiment on the Euclidean Nature of Physical Space"

The Myth of Gauss' Experiment on the Euclidean Nature of Physical Space

Previous articleNext article No AccessThe Myth of Gauss' Experiment on the Euclidean Nature of Physical SpaceArthur I. MillerArthur I. Miller Search for more articles by this author PDFPDF PLUS Add to favoritesDownload CitationTrack CitationsPermissionsReprints Share onFacebookTwitterLinkedInRedditEmail SectionsMoreDetailsFiguresReferencesCited by Isis Volume 63, Number 3Sep., 1972 Publication of the History of Science Society Article DOIhttps://doi.org/10.1086/350941 Views: 15Total views on this site Citations: 27Citations are reported from Crossref Copyright 1972 History of Science Society, Inc.PDF download Crossref reports the following articles citing this article:Julien Bernard Riemann's and Helmholtz-Lie's problems of space from Weyl's relativistic perspective, Studies in History and Philosophy of Science Part B: Studies in History and Philosophy of Modern Physics 61 (Feb 2018): 41–56.https://doi.org/10.1016/j.shpsb.2017.05.010David E. Rowe On Gauss and Gaussian Legends: A Quiz, (Feb 2018): 19–28.https://doi.org/10.1007/978-3-319-67819-1_2David E. Rowe Looking Back on Gauss and Gaussian Legends: Answers to the Quiz from 37(4), The Mathematical Intelligencer 38, no.44 (Oct 2016): 39–45.https://doi.org/10.1007/s00283-016-9662-1Erik W. Grafarend, Rey-Jer You, Rainer Syffus “Sphere to Cylinder”: Oblique Aspect, (Jun 2014): 331–335.https://doi.org/10.1007/978-3-642-36494-5_12Erik W. Grafarend, Rey-Jer You, Rainer Syffus “Ellipsoid-of-Revolution to Cylinder”: Transverse Aspect, (Jun 2014): 361–413.https://doi.org/10.1007/978-3-642-36494-5_15Erik W. Grafarend, Rey-Jer You, Rainer Syffus Map Projections of Alternative Structures: Torus, Hyperboloid, Paraboloid, Onion Shape and Others, (Jun 2014): 609–671.https://doi.org/10.1007/978-3-642-36494-5_23Erik W. Grafarend, Rey-Jer You, Rainer Syffus “Sphere to Cylinder”: Transverse Aspect, (Jun 2014): 325–329.https://doi.org/10.1007/978-3-642-36494-5_11Erik W. Grafarend, Rey-Jer You, Rainer Syffus “Sphere to Tangential Plane”: Oblique Aspect, (Jun 2014): 247–253.https://doi.org/10.1007/978-3-642-36494-5_7Erik W. Grafarend, Rey-Jer You, Rainer Syffus “Sphere to Cylinder”: Polar Aspect, (Jun 2014): 311–323.https://doi.org/10.1007/978-3-642-36494-5_10Erik W. Grafarend, Rey-Jer You, Rainer Syffus “Ellipsoid-of-Revolution to Cylinder”: Oblique Aspect, (Jun 2014): 415–435.https://doi.org/10.1007/978-3-642-36494-5_16Erik W. Grafarend, Rey-Jer You, Rainer Syffus “Sphere to Cylinder”: Pseudo-Cylindrical Projections, (Jun 2014): 337–345.https://doi.org/10.1007/978-3-642-36494-5_13Erik W. Grafarend, Rey-Jer You, Rainer Syffus “Ellipsoid-of-Revolution to Cylinder”: Polar Aspect, (Jun 2014): 347–360.https://doi.org/10.1007/978-3-642-36494-5_14Erik W. Grafarend, Rey-Jer You, Rainer Syffus Ellipsoid-of-Revolution to Sphere and from Sphere to Plane, (Jun 2014): 293–310.https://doi.org/10.1007/978-3-642-36494-5_9Erik W. Grafarend, Rey-Jer You, Rainer Syffus Ellipsoid-of-Revolution to Tangential Plane, (Jun 2014): 255–291.https://doi.org/10.1007/978-3-642-36494-5_8Erik W. Grafarend, Rey-Jer You, Rainer Syffus C 10(3): The Ten Parameter Conformal Group as a Datum Transformation in Three-Dimensional Euclidean Space, (Jun 2014): 673–683.https://doi.org/10.1007/978-3-642-36494-5_24Erik W. Grafarend, Rey-Jer You, Rainer Syffus Optimal Map Projections by Variational Calculus: Harmonic Maps, (Jun 2014): 571–607.https://doi.org/10.1007/978-3-642-36494-5_22Florin Diacu The Curved N-Body Problem: Risks and Rewards, The Mathematical Intelligencer 35, no.33 (Jul 2013): 24–33.https://doi.org/10.1007/s00283-013-9397-1Florin Diacu, Ernesto Pérez-Chavela, Manuele Santoprete The n-Body Problem in Spaces of Constant Curvature. Part I: Relative Equilibria, Journal of Nonlinear Science 22, no.22 (Jan 2012): 247–266.https://doi.org/10.1007/s00332-011-9116-zFlorin Diacu, Ernesto Pérez-Chavela, J. Guadalupe Reyes Victoria An intrinsic approach in the curved n-body problem: The negative curvature case, Journal of Differential Equations 252, no.88 (Apr 2012): 4529–4562.https://doi.org/10.1016/j.jde.2012.01.002 Bibliography, (Mar 2011): 443–462.https://doi.org/10.1002/9781118032787.biblioFlorin Diacu, Ernesto Pérez-Chavela Homographic solutions of the curved 3-body problem, Journal of Differential Equations 250, no.11 (Jan 2011): 340–366.https://doi.org/10.1016/j.jde.2010.08.011Leo Corry The Origin of Hilbert's Axiomatic Method1, (Jan 2007): 1680–1777.https://doi.org/10.1007/978-1-4020-4000-9_40Colin Pask Answering junior ant's ‘why’ for Pythagoras' Theorem, International Journal of Mathematical Education in Science and Technology 33, no.55 (Sep 2002): 661–669.https://doi.org/10.1080/00207390210162476Robert Osserman Imaginäre Welten, (Jan 1997): 54–66.https://doi.org/10.1007/978-3-322-85025-6_5Christian Thiel Comments on Hans Hahn’s philosophical writings, (Jan 1997): 385–400.https://doi.org/10.1007/978-3-7091-6548-5_20Erhard Scholz Gauß und die Begründung der „höheren“ Geodäsie, (Jan 1992): 631–647.https://doi.org/10.1007/978-3-0348-8599-7_29Jeremy Gray Non-euclidean geometry — A re-interpretation, Historia Mathematica 6, no.33 (Aug 1979): 236–258.https://doi.org/10.1016/0315-0860(79)90124-1

Dirichlet's Principle and Generalized Boundary Values

Throughout this paper, we shall be dealing with differential forms of any degree in an n-dimensional manifold D. Although many of our results remain valid for an analytic manifold with an analytic metric, we shall deal with manifolds D of a Euclidean nature. The precise definition of these manifolds will shortly be given. A knowledge of differential forms and the exterior calculus for these forms will be assumed. They are set forth, say, in de Rham and Kodaira [1]. In our case, D will have Euclidean coordinates xl, * *, xn. The dual operator * will then be defined on a differential as follows: