Breaking in the 4-Vectors: The Four-Dimensional Movement in Gravitation, 1905–1910

  • Scott Walter
Chapter
Part of the Boston Studies in the Philosophy of Science book series (BSPS, volume 250)

In July, 1905, Henri Poincaré (1854–1912) proposed two laws of gravitational attraction compatible with the principle of relativity and all astronomical observations explained by Newton’s law. Two years later, in the fall of 1907, Albert Einstein (1879–1955) began to investigate the consequences of the principle of equivalence for the behavior of light rays in a gravitational field. The following year, Hermann Minkowski (1864–1909), Einstein’s former mathematics instructor, borrowed Poincaré’s notion of a four-dimensional vector space for his new matrix calculus, in which he expressed a novel theory of the electrodynamics of moving media, a spacetime mechanics, and two laws of gravitational attraction. Following another two-year hiatus, Arnold Sommerfeld (1868–1951) characterized the relationship between the laws proposed by Poincaré and Minkowski, calling for this purpose both on spacetime diagrams and a new 4-vector formalism.

Keywords

Mercury Manifold Assimilation Arena Ghost 

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References

  1. Abraham, Max. 1905. Elektromagnetische Theorie der Strahlung. (Theorie der Elektrizität, vol. 2.) Leipzig: Teubner.Google Scholar
  2. ——. 1909. “Zur elektromagnetischen Mechanik.” Physikalische Zeitschrift 10: 737–741.Google Scholar
  3. ——. 1910. “Sull' elettrodinamica di Minkowski.” Rendiconti del Circolo Matematico di Palermo 30: 33–46.CrossRefGoogle Scholar
  4. ——. 1912a. “Das Elementargesetz der Gravitation.” Physikalische Zeitschrift 13: 4–5.Google Scholar
  5. ——. 1912b. “Zur Theorie der Gravitation.” Physikalische Zeitschrift 13: 1–4. (English translation in this volume.)Google Scholar
  6. ——. 1914. “Die neue Mechanik.” Scientia (Rivista di Scienza) 15: 8–27.Google Scholar
  7. Andrade Martins, Roberto de. 1999. “The search for gravitational absorption in the early 20th century.” In (Goenner et al. 1999), 3–44.Google Scholar
  8. Arzeliès, Henri, and J. Henry. 1959. Milieux conducteurs ou polarisables en mouvement. Paris: Gauthier-Villars.Google Scholar
  9. Barrow-Green, June E. 1997. Poincaré and the Three Body Problem. (History of Mathematics, vol. 11.) Providence: AMS and LMS.Google Scholar
  10. Bork, Alfred M. 1966. “‘Vectors versus quaternions’—the letters in Nature.” American Journal of Physics 34: 202–211.CrossRefGoogle Scholar
  11. Born, Max. 1909. “Die Theorie des starren Elektrons in der Kinematik des Relativitätsprinzips.” Annalen der Physik 30: 1–56.CrossRefGoogle Scholar
  12. ——. 1912. “Besprechung von Max Laue, Das Relativitätsprinzip.” Physikalische Zeitschrift 13: 175–176.Google Scholar
  13. ——. 1914. “Besprechung von Max Weinstein, Die Physik der bewegten Materie und die Relativitäts- theorie.” Physikalische Zeitschrift 15: 676.Google Scholar
  14. Buchwald, Jed Z. 1985. From Maxwell to Microphysics. Chicago: University of Chicago Press.Google Scholar
  15. Buchwald, Jed Z., and Andrew Warwick. 2001. Histories of the Electron: The Birth of Microphysics. Dibner Institute Studies in the History of Science and Technology. Cambridge MA: MIT Press.Google Scholar
  16. Burali-Forti, Cesare, and Roberto Marcolongo. 1910. Éléments de calcul vectoriel. Paris: Hermann.Google Scholar
  17. Cantor, Geoffrey N., and Michael J. S. Hodge. 1981. Conceptions of Ether: Studies in the History of Ether Theories 1740–1900. Cambridge: Cambridge University Press.Google Scholar
  18. Carazza, Bruno, and Helge Kragh. 1990. “Augusto Righi's magnetic rays: a failed research program in early 20th-century physics.” Historical Studies in the Physical and Biological Sciences 21: 1–28.Google Scholar
  19. Cayley, Arthur. 1869. “On the six coordinates of a line.” Transactions of the Cambridge Philosophical Society 11: 290–323.Google Scholar
  20. Conway, Arthur W. 1911. “On the application of quaternions to some recent developments of electrical theory.” Proceedings of the Royal Irish Academy 29: 1–9.Google Scholar
  21. Corry, Leo. 1997. “Hermann Minkowski and the postulate of relativity.” Archive for History of Exact Sciences 51: 273–314.CrossRefGoogle Scholar
  22. ——. 2004. David Hilbert and the Axiomatization of Physics, 1898–1918: From “Grundlagen der Geometrie” to “Grundlagen der Physik”. Dordrecht: Kluwer.Google Scholar
  23. Coster, H. G. L. and J. R. Shepanski. 1969. “Gravito-inertial fields and relativity.” Journal of Physics A 2: 22–27.Google Scholar
  24. CPAE 2. 1989. John Stachel, David C. Cassidy, Jürgen Renn, and Robert Schulmann (eds.), The Collected Papers of Albert Einstein. Vol. 2. The Swiss Years: Writings, 1900–1909. Princeton: Princeton University Press.Google Scholar
  25. CPAE 5. 1993. Martin J. Klein, A. J. Kox, and Robert Schulmann (eds.), The Collected Papers of Albert Einstein. Vol. 5. The Swiss Years: Correspondence, 1902–1914. Princeton: Princeton University Press.Google Scholar
  26. Cunningham, Ebenezer. 1914. The Principle of Relativity. Cambridge: Cambridge University Press.Google Scholar
  27. Cuvaj, Camillo. 1968. “Henri Poincaré's mathematical contributions to relativity and the Poincaré stresses.” American Journal of Physics 36: 1102–1113.CrossRefGoogle Scholar
  28. ——. 1970. A History of Relativity: The Role of Henri Poincaré and Paul Langevin. Ph.D. dissertation, Yeshiva University.Google Scholar
  29. Darrigol, Olivier. 1993. “The electrodynamic revolution in Germany as documented by early German expositions of ‘Maxwell's theory’.” Archive for History of Exact Sciences 45: 189–280.CrossRefGoogle Scholar
  30. ——. 1994. “The electron theories of Larmor and Lorentz: a comparative study.” Historical Studies in the Physical and Biological Sciences 24: 265–336.Google Scholar
  31. ——. 2000. Electrodynamics from Ampère to Einstein. Oxford: Oxford University Press.Google Scholar
  32. Dugac, Pierre. 1986. “La correspondance d'Henri Poincaré avec des mathématiciens de A à H.” Cahiers du séminaire d'histoire des mathématiques 7: 59–219.Google Scholar
  33. ——. 1989. “La correspondance d'Henri Poincaré avec des mathématiciens de J à Z.” Cahiers du séminaire d'histoire des mathématiques 10: 83–229.Google Scholar
  34. Dziobek, Otto F. 1888. Die mathematischen Theorien der Planeten-Bewegungen. Leipzig: Barth.Google Scholar
  35. Einstein, Albert. 1905. “Zur Elektrodynamik bewegter Körper.” Annalen der Physik 17: 891–921, (CPAE 2, Doc. 23).CrossRefGoogle Scholar
  36. ——. 1907. “Relativitätsprinzip und die aus demselben gezogenen Folgerungen.” Jahrbuch der Radio-aktivität und Elektronik 4: 411–462, (CPAE 2, Doc. 47).Google Scholar
  37. Einstein, Albert, and Jakob J. Laub. 1908. “Über die elektromagnetischen Grundgleichungen für bewegte Körper.” Annalen der Physik 26: 532–540, (CPAE 2, Doc. 51).CrossRefGoogle Scholar
  38. Frank, Philipp. 1911. “Das Verhalten der elektromagnetischen Feldgleichungen gegenüber linearen Transformationen.” Annalen der Physik 35: 599–607.CrossRefGoogle Scholar
  39. Föppl, August O. 1894. Einführung in die Maxwell'sche Theorie der Elektricität. Leipzig: Teubner.Google Scholar
  40. Galison, Peter. 1979. “Minkowski's spacetime: from visual thinking to the absolute world.” Historical Studies in the Physical Sciences 10: 85–121.Google Scholar
  41. Gans, Richard. 1905. “Gravitation und Elektromagnetismus.” Jahresbericht der deutschen Mathematiker-Vereinigung 14: 578–581.Google Scholar
  42. Gibbs, Josiah W. and Edwin B. Wilson. 1901. Vector Analysis. New York: Charles Scribner's Sons.Google Scholar
  43. Gispert, Hélène. 2001. “The German and French editions of the Klein-Molk Encyclopedia: contrasted images.” In Umberto Bottazzini and Amy Dahan Dalmedico (eds.), Changing Images in Mathematics: From the French Revolution to the New Millennium, 93–112. Studies in the History of Science, Technology and Medicine 13. London: Routledge.Google Scholar
  44. Goenner, Hubert, Jürgen Renn, Tilman Sauer, and Jim Ritter (eds.). 1999. The Expanding Worlds of General Relativity. (Einstein Studies vol. 7.) Boston/Basel/Berlin: Birkhäuser.Google Scholar
  45. Gray, Jeremy. 1992. “Poincaré and the solar system.” In Peter M. Harman and Alan E. Shapiro (eds.), The Investigation of Difficult Things, 503–524. Cambridge: Cambridge University Press.Google Scholar
  46. Hargreaves, Richard. 1908. “Integral forms and their connexion with physical equations.” Transactions of the Cambridge Philosophical Society 21: 107–122.Google Scholar
  47. Harvey, A. L. 1965. “A brief review of Lorentz-covariant theories of gravitation.” American Journal of Physics 33: 449–460.CrossRefGoogle Scholar
  48. Havas, Peter 1979. “Equations of motion and radiation reaction in the special and general theory of relativity.” In Jürgen Ehlers (ed.), Isolated Gravitating Systems in General Relativity, 74–155. Proceedings of the International School of Physics “Enrico Fermi” 67. Amsterdam: North-Holland.Google Scholar
  49. Heaviside, Oliver. 1893. “A gravitational and electromagnetic analogy.” In Electromagnetic Theory, 3 vols., 1: 455–464. London: The Electrician.Google Scholar
  50. Hon, Giora. 1995. “The case of Kaufmann's experiment and its varied reception.” In Jed Z. Buchwald (ed.), Scientific Practice: Theories and Stories of Doing Physics, 170–223. Chicago: University of Chicago Press.Google Scholar
  51. Ishiwara, Jun. 1912. “Bericht über die Relativitätstheorie.” Jahrbuch der Radioaktivität und Elektronik 9: 560–648.Google Scholar
  52. Jackson, J. David. 1975. Classical Electrodynamics. New York: Wiley, 2nd edition.Google Scholar
  53. Jungnickel, Christa and Russell McCormmach. 1986. Intellectual Mastery of Nature: Theoretical Physics from Ohm to Einstein. Chicago: University of Chicago Press.Google Scholar
  54. Klein, Felix. 1907. Vorträge über den mathematischen Unterricht an den höheren Schulen, Rudolf Schim-mack (ed.). (Mathematische Vorlesungen an der Universität Göttingen, vol. 1.) Leipzig: Teubner.Google Scholar
  55. Klein, Felix, and Arnold Sommerfeld. 1897–1910. Über die Theorie des Kreisels. 4 vols. Leipzig: Teubner.Google Scholar
  56. Kottler, Felix. 1922. “Gravitation und Relativitätstheorie.” In Karl Schwarzschild, Samuel Oppenheim, and Walther von Dyck (eds.), Astronomie, 2 vols, 2: 159–237. Encyklopädie der mathematischen Wissen-schaften mit Einschluss ihrer Anwendungen 6. Leipzig: Teubner.Google Scholar
  57. Kretschmann, Erich. 1914. Eine Theorie der Schwerkraft im Rahmen der ursprünglichen Einsteinschen Relativitätstheorie. Ph.D. dissertation, University of Berlin.Google Scholar
  58. Krätzel, Ekkehard. 1989. “Kommentierender Anhang: zur Geometrie der Zahlen.” In Ekkehard Krätzel and Bernulf Weissbach (eds.), Ausgewählte Arbeiten zur Zahlentheorie und zur Geometrie, 233–246. Teubner-Archiv zur Mathematik 12. Leipzig: Teubner.Google Scholar
  59. Kuhn, Thomas S. 1970. The Structure of Scientific Revolutions. Chicago: University of Chicago Press, 2nd edition.Google Scholar
  60. Laue, Max von. 1911. Das Relativitätsprinzip. (Die Wissenschaft, vol. 38.) Braunschweig: Vieweg.Google Scholar
  61. ——. 1913. Das Relativitätsprinzip. (Die Wissenschaft, vol. 38.) Braunschweig: Vieweg, 2nd edition.Google Scholar
  62. ——. 1951. “Sommerfelds Lebenswerk.” Naturwissenschaften 38: 513–518.CrossRefGoogle Scholar
  63. Lewis, Gilbert N. 1910a. “On four-dimensional vector analysis, and its application in electrical theory.” Proceedings of the American Academy of Arts and Science 46: 165–181.Google Scholar
  64. ——. 1910b. “Über vierdimensionale Vektoranalysis und deren Anwendung auf die Elektrizitätstheorie.” Jahrbuch der Radioaktivität und Elektronik 7: 329–347.Google Scholar
  65. Liu, Chuang. 1991. Relativistic Thermodynamics: Its History and Foundations. Ph.D. dissertation, University of Pittsburgh.Google Scholar
  66. Lorentz, Hendrik A. 1900. “Considerations on Gravitation.” Proceedings of the Section of Sciences. Verslag Koninklijke Akademie van Wetenschapen 2: 559–574. (Printed in this volume.)Google Scholar
  67. ——. 1904a. “Electromagnetic phenomena in a system moving with any velocity less than that of light.” Proceedings of the Section of Sciences. Verslag Koninklijke Akademie van Wetenschapen 6: 809–831.Google Scholar
  68. ——. 1904b. “Maxwells elektromagnetische Theorie.” In (Sommerfeld 1903–1926), 2: 63–144.Google Scholar
  69. ——. 1904c. “Weiterbildung der Maxwellschen Theorie; Elektronentheorie.” In (Sommerfeld 1903–1926), 2: 145–280.Google Scholar
  70. ——. 1910. “Alte und neue Fragen der Physik.” Physikalische Zeitschrift 11: 1234–1257. (English translation in this volume.)Google Scholar
  71. ——. 1914. “La gravitation.” Scientia (Rivista di Scienza) 16: 28–59.Google Scholar
  72. Lützen, Jesper. 1999. “Geometrising configurations: Heinrich Hertz and his mathematical precursors.” In Jeremy Gray (ed.), The Symbolic Universe: Geometry and Physics, 1890–1930, 25–46. Oxford: Oxford University Press.Google Scholar
  73. Maltese, Giulio. 2000. “The late entrance of relativity into Italian scientific community (1906–1930).” Historical Studies in the Physical and Biological Sciences 31: 125–173.Google Scholar
  74. Manegold, Karl-Heinz. 1970. Universität, Technische Hochschule und Industrie. Berlin: Duncker & Humblot.Google Scholar
  75. Marcolongo, Roberto. 1906. “Sugli integrali delle equazioni dell'elettrodinamica.” Rendiconti della Reale Accademia dei Lincei 15: 344–349.Google Scholar
  76. McCormmach, Russell. 1976. “Editor's foreword.” Historical Studies in the Physical Sciences 7: xi–xxxv.Google Scholar
  77. Miller, Arthur I. 1973. “A Study of Henri Poincaré's ‘Sur la dynamique de l'électron’.” Archive for History of Exact Sciences 10: 207–328.CrossRefGoogle Scholar
  78. ——. 1981. Albert Einstein's Special Theory of Relativity: Emergence (1905) and Early Interpretation. Reading, MA: Addison-Wesley.Google Scholar
  79. Minkowski, Hermann. 1888. “Über die Bewegung eines festen Körpers in einer Flüssigkeit.” Sitzungsbe-richte der königliche preuβischen Akademie der Wissenschaften 40: 1095–1110.Google Scholar
  80. ——. 1890–1893. “H. Poincaré, Sur le problème des trois corps et les équations de la dynamique.” Jahrbuch über die Fortschritte der Mathematik 22: 907–914.Google Scholar
  81. ——. 1896. Geometrie der Zahlen. Leipzig: Teubner.Google Scholar
  82. ——. 1907. “Kapillarität.” In (Sommerfeld 1903–1926), 1: 558–613.Google Scholar
  83. ——. 1908. “Die Grundgleichungen für die electromagnetischen Vorgänge in bewegten Körpern.” Nachrichten von der Königlichen Gesellschaft der Wissenschaften zu Göttingen, 53–111. (English translation of the appendix “Mechanics and the Relativity Postulate” in this volume.)Google Scholar
  84. ——. 1909. “Raum und Zeit.” Jahresbericht der deutschen Mathematiker-Vereinigung 18: 75–88.Google Scholar
  85. ——. 1915. “Das Relativitätsprinzip.” Jahresbericht der deutschen Mathematiker-Vereinigung 24: 372–382.Google Scholar
  86. ——. 1973. Briefe an David Hilbert. Lily Rüdenberg and Hans Zassenhaus (eds.). Berlin: Springer-Verlag.Google Scholar
  87. Minkowski, Hermann, and Max Born. 1910. “Eine Ableitung der Grundgleichungen für die elektromagnetischen Vorgänge in bewegten Körpern.” Mathematische Annalen 68: 526–550.CrossRefGoogle Scholar
  88. Møller, Christian. 1972. The Theory of Relativity. Oxford: Oxford University Press, 2nd edition.Google Scholar
  89. North, John D. 1965. The Measure of the Universe: A History of Modern Cosmology. Oxford: Oxford University Press.Google Scholar
  90. Norton, John D. 1992. “Einstein, Nordström and the early demise of Lorentz-covariant theories of gravitation.” Archive for History of Exact Sciences 45: 17–94.CrossRefGoogle Scholar
  91. Pauli, Wolfgang. 1921. “Relativitätstheorie.” In (Sommerfeld 1903–1926), 2: 539–775.Google Scholar
  92. Planck, Max. 1906. “Das Prinzip der Relativität und die Grundgleichungen der Mechanik.” Verhandlungen der Deutschen Physikalischen Gesellschaft 8: 136–141.Google Scholar
  93. ——. 1907. “Zur Dynamik bewegter Systeme.” Sitzungsberichte der königliche preuβischen Akad. der Wiss.: 542–570.Google Scholar
  94. Poincaré, Henri. 1885. “Sur l'équilibre d'une masse fluide animée d'un mouvement de rotation.” Acta mathematica 7: 259–380.CrossRefGoogle Scholar
  95. ——. 1895. Capillarité. J. Blondin (ed.). Paris: Georges Carré.Google Scholar
  96. ——. 1898–1905. “Préface.” In Charles Hermite, Henri Poincaré, and Eugène Rouché (eds.), Œuvres de Laguerre 1: v–xv. Paris: Gauthier-Villars.Google Scholar
  97. ——. 1901. Électricité et optique: la lumière et les théories électrodynamiques. Jules Blondin and Eugène Néculcéa (eds.). Paris: Carré et Naud.Google Scholar
  98. ——. 1902a. Figures d'équilibre d'une masse fluide. Léon Dreyfus (ed.). Paris: C. Naud.Google Scholar
  99. ——. 1902b. “Sur la stabilité de l'équilibre des figures piriformes affectées par une masse fluide en rotation.” Philosophical Transactions of the Royal Society A 198: 333–373.CrossRefGoogle Scholar
  100. ——. 1904. “L'état actuel et l'avenir de la physique mathématique.” Bulletin des sciences mathématiques 28: 302–324.Google Scholar
  101. ——. 1906. “Sur la dynamique de l'électron.” Rendiconti del Circolo Matematico di Palermo 21: 129–176. (English translation of excerpt in this volume.)CrossRefGoogle Scholar
  102. ——. 1907. “La relativité de l'espace”. Année psychologique 13: 1–17.Google Scholar
  103. ——. 1908. “La dynamique de l'électron.” Revue générale des sciences pures et appliquées 19: 386–402.Google Scholar
  104. ——. 1909. “La mécanique nouvelle.” Revue scientifique 12: 170–177.Google Scholar
  105. ——. 1910. Sechs Vorträge über ausgewählte Gegenstände aus der reinen Mathematik und mathemati-schen Physik. (Mathematische Vorlesungen an der Universität Göttingen, vol. 4.) Leipzig/Berlin: Teubner.Google Scholar
  106. ——. 1912. “L'espace et le temps.” Scientia (Rivista di Scienza) 12: 159–170.Google Scholar
  107. ——. 1953. “Les limites de la loi de Newton.” Bulletin astronomique 17: 121–269.Google Scholar
  108. Pomey, Jean-Baptiste. 1914–1931. Cours d'électricité théorique, 3 vols. Bibliothèque des Annales des Postes, Télégraphes et Téléphones. Paris: Gauthier-Villars.Google Scholar
  109. Pyenson, Lewis. 1973. The Goettingen Reception of Einstein's General Theory of Relativity. Ph.D. dissertation, Johns Hopkins University.Google Scholar
  110. ——. 1985. The Young Einstein: The Advent of Relativity. Bristol: Hilger.Google Scholar
  111. Reich, Karin. 1994. Die Entwicklung des Tensorkalküls: vom absoluten Differentialkalkül zur Relativitäts-theorie. (Science Networks Historical Studies, vol. 11.) Basel/Boston: Birkhäuser.Google Scholar
  112. ——. 1996. “The emergence of vector calculus in physics: the early decades.” In Gert Schubring (ed.), Hermann Günther Graβmann (1809–1877): Visionary Mathematician, Scientist and Neohumanist Scholar, 197–210. Dordrecht: Kluwer.Google Scholar
  113. Reiff, Richard. and Arnold Sommerfeld. 1904. “Standpunkt der Fernwirkung: Die Elementargesetze.” In (Sommerfeld 1903–1926), 2: 3–62.Google Scholar
  114. Ritz, Walter. 1908. “Recherches critiques sur l'électrodynamique générale.” Annales de chimie et de physique 13: 145–275.Google Scholar
  115. Roseveare, N. T. 1982. Mercury's Perihelion: From Le Verrier to Einstein. Oxford: Oxford University Press.Google Scholar
  116. Rowe, David E. 1989. “Klein, Hilbert, and the Göttingen Mathematical Tradition.” In Katherina M. Olesko (ed.), Science in Germany, 186–213. Osiris 5. Philadelphia: History of Science Society.Google Scholar
  117. ——. 1992. Felix Klein, David Hilbert, and the Göttingen Mathematical Tradition. Ph.D. dissertation, City University of New York.Google Scholar
  118. ——. 1997. “In Search of Steiner's ghosts: imaginary elements in 19th-century geometry.” In Dominique Flament (ed.), Le nombre, un hydre à n visages: entre nombres complexes et vecteurs, 193–208.Paris: Éditions de la Maison des Sciences de l'Homme.Google Scholar
  119. Rüdenberg, Lily. 1973. “Einleitung: Erinnerungen an H. Minkowski.” In (Minkowski 1973), 9–16.Google Scholar
  120. Schwartz, Hermann M. 1972. “Poincaré's Rendiconti paper on relativity, III.” American Journal of Physics 40: 1282–1287.CrossRefGoogle Scholar
  121. Schwermer, Joachim. 1991. “Räumliche Anschauung und Minima positiv definiter quadratischer Formen.” Jahresbericht der deutschen Mathematiker-Vereinigung 93: 49–105.Google Scholar
  122. Serre, Jean-Pierre. 1993. “Smith, Minkowski et l'Académie des Sciences.” Gazette des mathématiciens 56: 3–9.Google Scholar
  123. Silberstein, Ludwik. 1914. The Theory of Relativity. London: Macmillan.Google Scholar
  124. de Sitter, Willem. 1911. “On the bearing of the principle of relativity on gravitational astronomy.” Monthly Notices of the Royal Astronomical Society 71: 388–415.Google Scholar
  125. Sommerfeld, Arnold. (ed). 1903–1926. Physik, 3 vols. (Encyklopädie der mathematischen Wissenschaften mit Einschluss ihrer Anwendungen, vol. 5.) Leipzig: Teubner.Google Scholar
  126. Sommerfeld, Arnold. 1904. “Bezeichnung und Benennung der elektromagnetischen Grössen in der Enzy-klopädie der mathematischen Wissenschaften V.” Physikalische Zeitschrift 5: 467–470.Google Scholar
  127. ——. 1910a. “Zur Relativitätstheorie, I: Vierdimensionale Vektoralgebra.” Annalen der Physik 32: 749–776.Google Scholar
  128. ——. 1910b. “Zur Relativitätstheorie, II: Vierdimensionale Vektoranalysis.” Annalen der Physik 33: 649–689.Google Scholar
  129. ——. 1913. “Anmerkungen zu Minkowski, Raum und Zeit.” In Otto Blumenthal (ed.), Das Relativitätsprinzip; Eine Sammlung von Abhandlungen, 69–73. (Fortschritte der mathematischen Wissenschaften in Monographien, vol. 2.) Leipzig: Teubner.Google Scholar
  130. ——. 2001–2004. Wissenschaftlicher Briefwechsel. Michael Eckert and Karl Märker (eds.). Diepholz: GNT-Verlag.Google Scholar
  131. Staley, Richard. 1998. “On the histories of relativity: propagation and elaboration of relativity theory in participant histories in Germany, 1905–1911.” Isis 89: 263–299.CrossRefGoogle Scholar
  132. Stein, Howard. 1987. “After the Baltimore Lectures: some philosophical reflections on the subsequent development of physics.” In Robert Kargon and Peter Achinstein (eds.), Kelvin's Baltimore Lectures and Modern Theoretical Physics: Historical and Philosophical Perspectives, 375–398. Cambridge, MA: MIT Press.Google Scholar
  133. Strobl, Walter. 1985. “Aus den wissenschaftlichen Anfängen Hermann Minkowskis.” Historia Mathematica 12: 142–156.CrossRefGoogle Scholar
  134. Tait, Peter G. 1882–1884. Traité élémentaire des quaternions. Paris: Gauthier-Villars.Google Scholar
  135. Tisserand, Francois-Félix. 1889–1896. Traité de mécanique céleste. Paris: Gauthier-Villars.Google Scholar
  136. Torretti, Roberto. 1996. Relativity and Geometry. New York: Dover Publications, 2nd edition.Google Scholar
  137. Varičak, Vladimir. 1910. “Anwendung der Lobatschefskijschen Geometrie in der Relativtheorie.” Physika-lische Zeitschrift 11: 93–96.Google Scholar
  138. Vizgin, Vladimir P. 1994. Unified Field Theories in the First Third of the 20th Century. (Science Networks Historical Studies, vol. 3.) Basel: Birkhäuser.Google Scholar
  139. Wacker, Fritz. 1906. “Über Gravitation und Elektromagnetismus.” Physikalische Zeitschrift 7: 300–302.Google Scholar
  140. Walter, Scott. 1997. “La vérité en géométrie: sur le rejet mathématique de la doctrine conventionnaliste.” Philosophia Scientiæ 2: 103–135.Google Scholar
  141. ——. 1999a. “Minkowski, Mathematicians, and the Mathematical Theory of Relativity.” In (Goenner et al. 1999), 45–86.Google Scholar
  142. ——. 1999b. “The Non-Euclidean Style of Minkowskian Relativity.” In J. Gray (ed.), The Symbolic Universe: Geometry and Physics, 1890–1930, 91–127. Oxford: Oxford University Press.Google Scholar
  143. Weinstein, Max B. 1913. Die Physik der bewegten Materie und die Relativitätstheorie. Leipzig: Barth.Google Scholar
  144. ——. 1914. Kräfte und Spannungen: Das Gravitations- und Strahlenfeld. (Sammlung Vieweg, vol. 8.) Braunschweig: Vieweg.Google Scholar
  145. Whitrow, Gerald J., and George E. Morduch. 1965. “Relativistic theories of gravitation: a comparative analysis with particular reference to astronomical tests.” Vistas in Astronomy 6: 1–67.CrossRefGoogle Scholar
  146. Whittaker, Edmund T. 1951–1953. A History of the Theories of Aether and Electricity. London: T. Nelson.Google Scholar
  147. de Wisniewski, Felix J. 1913a. “Zur Minkowskischen Mechanik I.” Annalen der Physik 40: 387–390.CrossRefGoogle Scholar
  148. ——. 1913b. “Zur Minkowskischen Mechanik II.” Annalen der Physik 40: 668–676.CrossRefGoogle Scholar
  149. Yaghjian, Arthur D. 1992. Relativistic Dynamics of a Charged Sphere: Updating the Lorentz-Abraham Model. (Lecture Notes in Physics: Monographs, vol. 11.) Berlin: Springer.Google Scholar
  150. Zahar, Elie. 1989. Einstein's Revolution: A Study in Heuristic. La Salle, Ill.: Open Court.Google Scholar
  151. Zassenhaus, Hans J. 1975. “On the Minkowski-Hilbert dialogue on mathematization.” Canadian Mathematical Bulletin 18: 443–461.Google Scholar
  152. Zenneck, Jonathan. 1903. “Gravitation.” In (Sommerfeld 1903–1926), 1: 25–67. (Printed in this volume.)Google Scholar

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