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Helmholtz, Riemann, and the Sirens: Sound, Color, and the “Problem of Space”

Abstract

Emerging from music and the visual arts, questions about hearing and seeing deeply affected Hermann Helmholtz’s and Bernhard Riemann’s contributions to what became called the “problem of space [Raumproblem],” which in turn influenced Albert Einstein’s approach to general relativity. Helmholtz’s physiological investigations measured the time dependence of nerve conduction and mapped the three-dimensional manifold of color sensation. His concurrent studies on hearing illuminated musical evidence through experiments with mechanical sirens that connect audible with visible phenomena, especially how the concept of frequency unifies motion, velocity, and pitch. Riemann’s critique of Helmholtz’s work on hearing led Helmholtz to respond and study Riemann’s then-unpublished lecture on the foundations of geometry. During 1862–1870, Helmholtz applied his findings on the manifolds of hearing and seeing to the Raumproblem by supporting the quadratic distance relation Riemann had assumed as his fundamental hypothesis about geometrical space. Helmholtz also drew a “close analogy … in all essential relations between the musical scale and space.” These intersecting studies of hearing and seeing thus led to reconsideration and generalization of the very concept of “space,” which Einstein shaped into the general manifold of relativistic space-time.

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Notes

  1. In present terminology, hue is the degree to which an area appears similar to the perceived colors red, yellow, green, blue or a combination of them; saturation its colorfulness relative to its own brightness; lightness (or value) its brightness relative to a similarly illuminated white.

  2. For instance, the famous Prelude to Wagner’s Tristan und Isolde (first performed in 1865) begins with an augmented sixth chord.

  3. In 1854, James Clerk Maxwell made the first color photograph, showing the wide range of contemporary work on the consequences of the Young-Helmholtz three-color theory of color perception, referred to in the first edition of Helmholtz, Handbuch der physiologischen Optik (ref. 10), pp. 288–297.

  4. Specifically, Riemann generalizes the quadratic Euclidean line element ds 2 = dx 1 2 + dx 2 2 + dx 3 2 (in terms of three spatial coordinates called, for convenience, x 1, x 2, x 3) to a general quadratic form ds 2 = g 11 dx 1 2 + g 12 dx 1 dx 2 +g 22 dx 2 2 + ··· = Σg μν dx μ dx ν (summed over all n dimensions, μ, ν = 1, …, n). In this modern notation (due to Einstein, following Levi-Civita), g μν is the “metric tensor.” Note also that Riemann did not include these calculations in his 1854 lecture, though they did appear in his paper.

  5. Helmholtz’s detailed description of the ear was superseded by later anatomical findings, particularly because the larger context of the processing of hearing became understood as involving the auditory system of the brain as well. Rather than being a kind of nerve-piano, its separate cilia sympathetically responding to incoming pitches, the cochlea currently is considered to comprise a series of chambers of variable resonant frequency, in which the cilia respond to the local amplitude of vibration, rather than its frequency. As Riemann surmised, the overall functioning of hearing may be described in terms of inputs and outputs of a complex electrical network. See, for example, Jonathan Sterne, The Audible Past: Cultural Origins of Sound Reproduction (Durham and London: Duke University Press, 2003), pp. 62–67.

  6. This is the form for ds given in footnote **, under the section "Riemann's Work on Space and Hearing."

  7. The influence of Grassmann should also be considered, though that by itself does not seem to have been sufficient for Helmholtz to speak of manifolds in his 1867 Handbuch, which does mention Grassmann. Riemann does not seem to have known Grassmann’s work, which still is “not a general theory of manifolds,” as argued by Torretti, Philosophy of Geometry (ref. 66), p. 109. See also Erhard Scholz, Geschichte des Mannigfaltigkeitsbegriffs von Riemann bis Poincaré (Boston, Basel, Stuttgart: Birkhäuser, 1980), esp. pp. 24–94, 113–123.

  8. A pseudosphere is a surface of constant negative curvature, roughly saddle-shaped, used by Beltrami in 1868 as a model for Lobachevky’s hyperbolic geometry.

  9. Helmholtz writes his four-dimensional line element as ds 2 = dx 2 + dy 2 + dz 2 + dt 2, in which he then allows t to become imaginary (t = ), so that the four-dimensional manifold is now pseudospherical and hence ds 2 = dx 2 + dy 2 + dz 2 −  2, exactly the form of the Lorentzian line-element used by Einstein and Minkowski, if τ = ct, where c is the speed of light.

References

  1. Einstein to Marić [early August 1899], in John Stachel, ed., The Collected Papers of Albert Einstein. Vol. 1. The Early Years 1879–1902 (Princeton: Princeton University Press, 1987), pp. 220–221, on p. 220; translated somewhat differently by Anna Beck (Princeton: Princeton University Press, 1987), p. 129.

  2. Albert Einstein, Lettres à Maurice Solovine (Paris: Gauthier-Villars, 1956), p. viii; Letters to Solovine (New York: Citadel Press, 1993), pp. 8–9; Gerald Holton, Einstein, History, and Other Passions (Woodbury, N.Y.: AIP Press, 1995), p. 84; David Cahan, “The Young Einstein’s Physics Education: H.F. Weber, Hermann von Helmholtz, and the Zurich Polytechnic Physics Institute,” in Don Howard and John Stachel, ed., Einstein: The Formative Years, 1879–1909 (Boston, Basel, Berlin: Birkhäuser, 2000), pp. 43–82, esp. pp. 59–74.

  3. Hermann Helmholtz, Über die Erhaltung der Kraft, eine physikalische Abhandlung (Berlin: G. Reimer, 1847): reprinted in Wissenschaftliche Abhandlungen. Erster Band (Leipzig: Johann Ambrosius Barth, 1882), pp. 12–75. Fabio Bevilacqua, “Helmholtz’s Ueber die Erhaltung der Kraft: The Emergence of a Theoretical Physicist,” in David Cahan, ed., Hermann von Helmholtz and the Foundations of Nineteenth-Century Science (Berkeley, Los Angeles, London: University of California Press), pp. 291–333.

  4. Hermann Helmholtz, “Messungen über den zeitlichen Verlauf der Zuckung animalischler Muskeln und die Fortpflanzungsgeschwindigkeit der Reizung in den Nerven” [1850], reprinted in Wissenschaftliche Abhandlungen, Zweiter Band (Leipzig: Johann Ambrosius Barth, 1883), pp. 764–843.

  5. Hermann Helmholtz, “Ueber die Methoden, kleinste Zeittheile zu messen, und ihre Anwendung für physiologische Zwecke” [1850], in ibid., pp. 862–880. Claude Debru, “Helmholtz and the Psychophysiology of Time,” Science in Context 14 (2001), 471–492. For the question of “lost time,” see Michel Meulders, Helmholtz: From Enlightenment to Neuroscience. Translated and edited by Laurence Garey (Cambridge, Mass. and London: The MIT Press, 2010), pp. 89–106.

  6. Hermann Helmholtz, “Beschreibung eines Augenspiegels zur Untersuchung der Netzhaut im lebende Auge” [1851], in ibid., pp. 229–260.

  7. Hermann von Helmholtz, “Die neureren Fortschritten in der Theorie des Sehens” [1868], in Vorträge und Reden. Fünfte Auflage. Erster Band (Braunschweig: Friedrich Vieweg und Sohn, 1903), pp. 265–365, on pp. 352–365; translated by Philip H. Pye-Smith as “The Recent Progress of the Theory of Vision,” in Popular Scientific Lectures (New York: Dover Publications, 1962), pp. 93–185, on pp. 181–185; and in Russell Kahl, ed., Selected Writings of Hermann von Helmholtz (Middletown, Conn.: Wesleyan University Press, 1971), pp. 144–222, on pp. 213–222. Gary Hatfield, The Natural and the Normative: Theories of Spatial Perception from Kant to Helmholtz (Cambridge, Mass. and London: The MIT Press, 1990); Timothy Lenoir, “The Eye as Mathematician: Clinical Practice, Instrumentation, and Helmholtz’s Construction of an Empiricist Theory of Vision,” in Cahan, Helmholtz and the Foundations (ref. 3), pp. 109–153, on pp. 124–126; Alexandra Hui, The Psychophysical Ear: Musical Experiments, Experimental Sounds, 1840–1910 (Cambridge, Mass. and London: The MIT Press, 2013), pp. 84–87.

  8. For an excellent treatment of the relation between Helmholtz’s physiological, mathematical, and philosophical work, see Joan L. Richards, “The Evolution of Empiricism: Hermann von Helmholtz and the Foundations of Geometry,” The British Journal for the Philosophy of Science 28 (1977), 235–253.

  9. Hermann Helmholtz, “Ueber die Zusammensetzung von Spectralfarben” [1855], in Wissenschaftliche Abhandlungen. Zweiter Band (ref. 4), pp. 45–70.

  10. H. Helmholtz, Handbuch der physiologischen Optik (Leipzig: Leopold Voss, 1867), pp. 282–288, 293; Professor Grassmann, “On the Theory of Compound Colours,” Philosophical Magazine 7 (1854), 254–264; reprinted as Hermann Günter Grassmann, “Theory of Compound Colors” [1853, translation 1854], in David L. MacAdam, ed., Sources of Color Science (Cambridge, Mass. and London: The MIT Press, 1970), pp. 53–60.

  11. H. von Helmholtz, Handbuch der Physiologischen Optik. Dritte Auflage ergänzt und herausgegeben in Gemeinschaft mit A. Gullstrand und J. von Kriess von W. Nagel. Zweiter Band (Hamburg und Leipzig: Leopold Voss, 1911), p. 54; translated as Helmholtz’s Treatise on Physiological Optics. Vol. II. Edited by James P.C. Southall (New York: Dover Publications, 1962), p. 64. See also Lenoir, “The Eye as Mathematician” (ref. 7).

  12. Helmholtz, Handbuch der Physiologischen Optik (ref. 11), p. 63; Helmholtz’s Treatise on Physiological Optics (ref. 11), p. 76; Isaac Newton, Opticks or A Treatise of the Reflections, Refractions, Inflections & Colours of Light (New York: McGraw-Hill Book Company, 1931; reprinted Dover Publications, 1979), pp. 126–128, 212. Peter Pesic, “Isaac Newton and the mystery of the major sixth: a transcription of his manuscript ‘Of Musick’ with commentary,” Interdisciplinary Science Reviews 31 (2006), 291–306.

  13. Helmholtz, Handbuch der Physiologischen Optik (ref. 11), p. 55; Helmholtz’s Treatise on Physiological Optics (ref. 11), p. 66.

  14. Ibid., p. 64; 77. Helmholtz originally published this observation as an appendix to a paper, “Ueber die Messung der Wellenlänge des ultravioletten Lichtes, von E. Esselbach” [1855], in Wissenschaftliche Abhandlungen. Zweiter Band (ref. 4), pp. 78–82.

  15. Helmholtz, Handbuch der Physiologischen Optik (ref. 11), p. 98; Helmholtz’s Treatise on Physiological Optics (ref. 11), p. 117.

  16. Ibid., p. 64; 77.

  17. For excellent discussions of Helmholtz within his larger musical and cultural milieux, see Erwin Hiebert and Elfrieda Hiebert, “Musical Thought and Practice: Links to Helmholtz’s Tonempfindungen,” in Lorenz Krüger, ed., Universalgenie Helmholtz: Rückblick nach 100 Jahren (Berlin: Akademie Verlag, 1994), pp. 295–311. Myles W. Jackson, Harmonious Triads: Physicists, Musicians, and Instrument Makers in Nineteenth-Century Germany (Cambridge, Mass. and London: The MIT Press, 2006). Hui, Psychophysical Ear (ref. 7), pp. 55–87.

  18. Ferdinand to Hermann Helmholtz, November 2, 1838, quoted in Leo Koenigsberger, Hermann von Helmholtz (Oxford: At the Clarendon Press, 1906; reprinted New York: Dover Publications, 1965), p. 14; quoted in German in David Cahan, ed, Letters of Hermann von Helmholtz to His Parents: The Medical Education of a German Scientist 1837–1846 (Stuttgart: Franz Steiner Verlag, 1993), p. 43, n. 2; see also p. 68.

  19. Hermann von Helmholtz, “Optisches über Malerei” [1871 to 1873], in Vorträge und Reden. Fünfte Auflage. Zweiter Band (Braunschweig: Friedrich Vieweg und Sohn, 1903), pp. 95–135; translated by E. Atkinson as “On the Relation of Optics to Painting,” in Popular Scientific Lectures (ref. 7), pp. 250–286. Regarding the relation between painting and visual science in Helmholtz, see Gary Hatfield, “Helmholtz and Classicism: The Science of Aesthetics and the Aesthetics of Science,” in Cahan, Helmholtz and the Foundations (ref. 3), pp. 522–558, on pp. 535–540. On the significance of the Kulturträger for science, see Gerhard Sonnert, Einstein and Culture (Amherst, N.Y.: Humanity Books, 2005), pp. 127–184.

  20. Hermann Helmholtz, “Ueber Combinationstöne” [1856], “Ueber musikalische Temperatur” [1860], and “Ueber die arabisch-persische Tonleiter” [1862], in Wissenschaftliche Abhandlungen. Erster Band (ref. 3), pp. 256–302, 420–423, 424–426.

  21. For helpful overviews of his project, see Stephan Vogel, “Sensations of Tone, Perception of Sound, and Empiricism: Helmholtz’s Physiological Acoustics,” in Cahen, Helmholtz and the Foundations (ref. 3), pp. 259–287; Meulders, Helmholtz (ref. 5), pp. 153–199.

  22. For Helmholtz’s use of “instruments as agents of change,” see David Pantalony, Altered Sensations: Rudolph Koenig’s Acoustical Workshop in Nineteenth-Century Paris (Dordrecht: Springer, 2009), pp. 20–36.

  23. Peter Pesic, “Thomas Young’s Musical Optics: Translating Sound into Light,” Osiris 28 (2013), 15–39; idem, “Euler's Musical Mathematics,” The Mathematical Intelligencer 35(2) (2013).

  24. For the larger context of this instrument, see Alexander Rehding, “Of Sirens Old and New,” in Sumanath S. Gopinath and Jason Stanyek, ed., The Oxford Handbook of Mobile Music Studies. Vol. 2 (Oxford: Oxford University Press, 2013). See also Julia Kursell, “Experiments on Tone Color in Music and Acoustics: Helmholtz, Schoenberg, and Klangfarbenmelodie,” Osiris 28 (2013), 191–211.

  25. Hermann L.F. Helmholtz, On the Sensations of Tone as a Physiological Basis for the Theory of Music. Translated … conformable to the Fourth (and last) German Edition of 1877 … by Alexander J. Ellis (London and New York: Longmans, Green, and Co.; reprinted New York: Dover Publications, 1954), pp. 8, 11, 13.

  26. Ibid., pp. 155–159, on p. 157. He provides the mathematical details in Appendixes XII and XVI, pp. 411–413, 418–420.

  27. Ibid., p. 158.

  28. Ibid., pp. 170, 173.

  29. Hermann von Helmholtz, “Die neureren Fortschritten in der Theorie des Sehens” [1868], in Vorträge und Reden. Fünfte Auflage. Erster Band (ref. 7), pp. 265–365, on p. 309; “Recent Progress of the Theory of Vision” (ref. 7), pp. 93–185, on pp. 130–131; Kahl, Selected Writings of Hermann von Helmholtz (ref. 7), pp. 144–222, on p. 177.

  30. Hermann von Helmholtz, “Ueber die physiologischen Ursachen der musikalishen Harmonie” [1857], in Vorträge und Reden. Erster Band (ref. 7), pp. 119–155, on p. 122; translated by A.J. Ellis as “On the Physiological Causes of Harmony in Music,” in Popular Scientific Lectures (ref. 7), pp. 22–58, on p. 23.

  31. Helmholtz, On the Sensations of Tone (ref. 25), pp. 369–370. In the second edition, H. Helmholtz, Die Lehre von den Tonempfindungen als physiologische Grundlage für die Theorie der Musik. Zweite Ausgabe (Braunschweig: Friedrich Vieweg und Sohn, 1865), following this paragraph the text then cuts to the final paragraph on p. 371. Regarding the importance of invariance in his thinking, see Hatfield, “Helmholtz and Classicism,” (ref. 19), pp. 552–553.

  32. Quoted in Meulders, Helmholtz (ref. 5), p. 71.

  33. According to Koenigsberger, Helmholtz (ref. 18), p. 207, in 1862 “his thoughts were already turning to the investigations of the axioms of geometry.”

  34. Carl Friedrich Gauss, General Investigations of Curved Surfaces, ed. Peter Pesic (Mineola, N.Y.: Dover Publications, 2005), pp. 3–44.

  35. Bernhard Riemann, “Ueber die Hypothesen, welche der Geometrie zu Grunde liegen” [1854], in Gesammelte Mathematische Werke, Wissenschaftlicher Nachlass und Nachträge: Collected Papers. Nach der Ausgabe von Heinrich Weber und Richard Dedekind neu hereausgegeben von Raghavan Narasimhan (Berlin, Heidelberg, New York, London, Paris, Tokyo: Springer-Verlag, 1990), pp. 304–319; translated as “On the Hypotheses That Lie at the Foundations of Geometry,” in Peter Pesic, ed., Beyond Geometry: Classic Papers from Riemann to Einstein (Mineola, N.Y.: Dover Publications, 2007), pp. 23–40.

  36. Pesic, Beyond Geometry (ref. 35), p. 2. See also E. Scholz, “Riemanns frühe Notizen zum Mannigfaltigkeitsbegriff und zu den Grundlagen der Geometrie,” Archive for History of Exact Sciences 27 (1982), 213–232, and Gregory Nowak, “Riemann’s Habilitationsvortrag and the Synthetic A Priori Status of Geometry,” in David E. Rowe and John McCleary, ed., The History of Modern Mathematics. Vol. I. Ideas and Their Reception (Boston: Academic Press, 1989), pp. 17–46.

  37. Riemann, “Ueber die Hypothesen” (ref. 35), p. 318; “On the Hypotheses” (ref. 35), p. 33. See also Jeremy Gray, “Bernhard Riemann, Posthumous Thesis ‘On the Hypotheses which lie at the Foundation of Geometry’ (1867),” in I. Grattan-Guinness, ed., Landmark Writings in Western Mathematics 1640–1940 (Amsterdam: Elsevier, 2005), pp. 506–520.

  38. Riemann, “Ueber dei Hypothesen” (ref. 35), pp. 304, 306; “On the Hypotheses” (ref. 35), pp. 23, 24.

  39. Ibid., pp. 309–310; 26–27.

  40. Bernhard Riemann, “Ein Beitrag zur Electrodynamik” [1858, published 1867], in Gesammelte Mathematische Werke (ref. 35), pp. 320–325. Concerning Riemann’s work in electrodynamics, see Nicholas Ionescu-Pallas and Liviu Sofonea, “Bernhard Riemann: A Forerunner of Classical Electrodynamics,” Organon 22 (1987), 259–272; Thomas Archibald, “Riemann and the Theory of Electrical Phenomena: Nobili’s Rings,” Centaurus 34 (1991), 247–271; Detlef Laugwitz, Bernhard Riemann 1826–1866: Turning Points in the Conception of Mathematics. Translated by Abe Shenitzer (Boston, Basel, Berlin: Birkhäuser, 1999), pp. 254–263, 269–272. For the context of natural philosophy, see Renato Pettoello, “Dietro la superficie dei fenomeni: frammenti di filosofia in Bernhard Riemann,” Rivista di storia della filosofia 4 (1988), 697–728; Rossana Tazzioli, “Fisica e ‘filosofia naturale’ in Riemann,” Nuncius 8 (1993), 105–120; Luciano Boi, “Die Beziehungen zwischen Raum, Kontinuum und Materie im Denken Riemanns; die Äthervorstelung und die Einheit der Physik: Das Enstehen einer neuen Naturphilosophie,” Philosophia Naturalis 31 (1994), 171–216; Umberto Bottazzini and Rossana Tazzioli, “Naturphilosophie and Its Role in Riemann’s Mathematics,” Revue d’histoire des mathématiques 1 (1995), 3–38; Umberto Bottazzini, “Riemann ‘filosofo naturale’,” Rivista di filosofia 87 (1996), 129–141.

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  41. J. Clerk Maxwell, “On a Method of making a Direct Comparison of Electrostatic with Electromagnetic Force; with a Note on the Electromagnetic Theory of Light,” Philosophical Transactions of the Royal Society of London 158 (1868), 643–657; reprinted in W.D. Niven, ed., The Scientific Papers of James Clerk Maxwell, M.A., LL.D. Edin., D.C.L., F.R.S. Vol. 2 (Cambridge: Cambridge University Press, 1890; reprinted New York: Dover Publications, 1965), pp. 125–143.

  42. Bernhard Riemann, “Ueber die Anzahl der Primzahlen unter einer gegebenen Grösse” [1859], in Gesammelte Mathematische Werke (ref. 35), pp. 177–185.

  43. For a selection of such passages, see Pesic, Beyond Geometry (ref. 35), pp. 41–45.

  44. Bernhard Riemann, “Mechanik des Ohres” [1866], in Gesammelte Mathematische Werke (ref. 35), pp. 370–382; translated by David Cherry, Robert Gallagher, and John Siegerson as “The Mechanism of the Ear,” Fusion 6 no. 3 (1984), 31–36. See also Laugwitz, Bernhard Riemann (ref. 40), pp. 281–287.

  45. The quotations are from “Bernhard Riemanns Lebenslauf,” in Gesammelte Mathematische Werke (ref. 35), pp. 571–590, on pp. 587–589; see also Laugwitz, Bernhard Riemann (ref. 40), pp. 2–3.

  46. Riemann, “Mechanik des Ohres” (ref. 44), p. 373; “Mechanism of the Ear” (ref. 44), pp. 32–33.

  47. Ibid., p. 375; 35.

  48. Tom Ritchey, “On Scientific Method—Based on a Study by Bernhard Riemann,” Systems Research 8, no. 4 (1991), 21–41. For another perspective, see Tito M. Tonietti, “Music between Hearing and Counting (A Historical Case Chosen within Continuous Long-Lasting Conflicts),” in Carlos Agon, Moreno Andreatta, Gérard Assayag, Emmanuel Amiot, Jean Bresson, and John Mandereau, ed., Mathematics and Computation in Music [Lecture Notes in Artificial Intelligence, 6726] (Berlin, Heidelberg: Springer-Verlag, 2011), pp. 285–296.

  49. Riemann, “Mechanik des Ohres” (ref. 44), p. 376; “Mechanism of the Ear” (ref. 44), p. 35.

  50. Hermann von Helmholtz, “Die Thatsachen in der Wahrnehmung ”[1878], in Vorträge und Reden. Fünfte Auflage. Zweiter Band (ref. 19), pp. 213–247, 387–406, on pp. 219–229; reprinted in Paul Hertz and Moritz Schlick, ed. Hermann v. Helmholtz: Schriften zur Erkenntnistheorie (Berlin: Julius Springer, 1921), pp. 109–152, on pp. 111–121; translated as “The Facts in Perception,” in Kahl, Selected Writings of Hermann von Helmholtz (ref. 7), pp. 366–408, on pp. 368–378; newly translated by Malcolm F. Lowe in Robert S. Cohen and Yehuda Elkana, ed., Hermann von Helmholtz: Epistemological Writings [Boston Studies in the Philosophy of Science, Vol. 37] (Dordrecht-Holland and Boston: D. Reidel Publishing Co., 1977), pp. 115–163, on pp. 117–128. Hui, Psychophysical Ear (ref. 7), pp. 84–87. Ernst Mach, in the earlier phase of his research on accommodation in musical hearing, held just such a view of the state of the ear muscles as signs (and physically observable agents) of the act of perception; see Alexandra Hui, “Changeable Ears: Ernst Mach’s and Max Planck’s Studies of Accommodation in Hearing,” Osiris 28 (2013), 119–145.

  51. Riemann, “Mechanik des Ohres” (ref. 44), p. 380; Riemann, “Mechanism of the Ear” (ref. 44), p. 37. For helpful treatments of the developing concept of attention, see Jonathan Crary, Suspensions of Perception: Attention, Spectacle, and Modern Culture (Cambridge, Mass. and London: The MIT Press, 1999), pp. 30, 64, 104–105; Benjamin Steege, Helmholtz and the Modern Listener (Cambridge: Cambridge University Press, 2012), pp. 80–122.

  52. Riemann, “Mechanik des Ohres” (ref. 44), p. 374; “Mechanism of the Ear” (ref. 44), p. 34.

  53. Jacob Henle, the friend and editor who published Riemann’s paper posthumously in the Zeitschrift für rationelle Medicin, noted that “Riemann thought that the mathematical problem to be solved was in fact hydraulic [hydraulisches]”; quoted in Riemann, “Mechanik des Ohres” (ref. 44), p. 370, note. See also Peter D. Lax, “On Riemann’s paper: Ueber die Fortpflanzung ebener Luftwellen von endlicher Schwingungsweite,” in Riemann, Gesammelte Mathematische Werke (ref. 35), pp. 807–810. See also the tendentious but still informative paper by Robert Gallagher, “Riemann and the Göttingen School of Physiology,” Fusion 6, no. 3 (1984), 24–30.

  54. Heremann Helmholtz, “Ueber die Mechanik der Gehörknöchelchen” [1867] and “Die Mechanik der Gehörknöchelchen und des Trommelfelles” [1869], in Wissenschaftliche Abhandlungen. Zweiter Band (ref. 4), pp. 503–514, 515–581. These papers were collected in Hermann von Helmholtz, The Mechanism of the Ossicles of the Ear and Membrana Tympani, translated by Albert H. Buck and Normand Smith (New York: William Wood & Co., 1873). See also Veit Erlmann, Reason and Resonance: A History of Modern Aurality (Brooklyn, New York: Zone Books, 2010), pp. 240–241.

  55. Helmholtz to Lipschitz, March 2, 1881: “The individual, even if he be a Riemann, will always be regarded as a crank who is discussing unfamiliar matters as an amateur”; quoted in Koenigsberger, Helmholtz (ref. 18), p. 267. Regarding the accuracy of Koenigsberger’s reporting of Helmholtz’s correspondence, see Cahan, Letters of Hermann von Helmholtz to His Parents (ref. 18), pp. 1–2.

  56. Ernst Schering gave a memorial lecture for Riemann on December 1, 1866; see “Zum Gedächtnis an B. Riemann,” in Riemann, Gesammelte Mathematische Werke (ref. 35), pp. 828–847. Schering sent a copy of his lecture to Helmholtz, which he acknowledged on May 18, 1868; see Koenigsberger, Helmholtz (ref. 18), p. 255. The first publication of the lecture, communicated by Dedekind, was Bernhard Riemann, “Ueber die Hypothesen, welche der Geometrie zu Grunde liegen,” Abhandlungen der Königlichen Gesellschaft der Wissenschaften zu Göttingen 13 (1867), 133–152; reprinted in Gesammelte Mathematische Werke (ref. 35), pp. 304–319. R. Dedekind, “Analytische Untersuchungen zu Bernhard Riemann’s Abhandlungen Über die Hypothesen, welche der Geometrie zu Grunde liegen,” Revue d’histoire des sciences 43 (1990), 239–294. The same issue of the Göttingen Abhandlungen also includes the first publication of two other major unpublished mathematical works by Riemann, “Ueber die Fläche vom kleinsten Inhalt bei gegebener Begrenzung,” his 1854 Habilitationsschrift, and “Ueber die Darstellbarkeit einer Function durch eine trigonometrische Reihe,” in Gesammelte Mathematische Werke (ref. 35), pp. 259–296, 333–369.

  57. Helmholtz to Schering, April 21, 1868, quoted in Koeningsberger, Helmholtz (ref. 18), pp. 254–255.

  58. For instance, Helmholtz’s statement is not discussed in Alfred M. Bork, “The Fourth Dimension in Nineteenth-Century Physics,” Isis 55 (1964), 326–338. For the references to d’Alembert and Lagrange, see R.C. Archibald, “Time as a Fourth Dimension,” Bulletin (New Series) of the American Mathematical Society 20 (1914), 409–412.

  59. Hermann Helmholtz, “Ueber die thatsächlichen Grundlagen der Geometrie,” [1866] in Wissenschaftliche Abhandlungen. Zweiter Band (ref. 4), pp. 610–617, on pp. 610–611; translated as “On the Factual Foundations of Geometry,” in Pesic, Beyond Geometry (ref. 35), pp. 47–52, on p. 47. The date of 1866 given in Helmholtz’s Wissenschaftliche Abhandlungen seems to have been a misprint for 1868, as shown by Klaus Volkert, “On Helmholtz’ Paper ‘Ueber die thatsächlichen Grundlagen der Geometrie’,” Historia Mathematica 20 (1993), 307–309.

  60. Helmholtz, “Ueber die thatsächlichen Grundlagen” (ref. 59), p. 611; “On the Factual Foundations” (ref. 59), pp. 47–48.

  61. In 1868 Helmholtz published another, longer paper that presents the details of the argument he had summarized in his brief 1868 paper and incorporates the corrections that he had learned from Beltrami and Lie; see Hermann von Helmholtz, “Ueber die Thatsachen, die der Geometrie zum Grunde liegen” [1868] in Wissenschaftliche Abhandlungen. Zweiter Band (ref. 4), pp. 618–639; reprinted in Hertz and Schlick, Helmholtz: Schriften zur Erkenntnistheorie (ref. 50), pp. 38–69; newly translated by Malcolm F. Lowe as “On the Facts Underlying Geometry,” in Cohen and Elkana, Helmholtz: Epistemological Writings, (ref. 50), pp. 39–71. For a helpful modern account of Helmholtz’s argument, see Ronald Adler, Maurice Bazin, and Menahem Schiffer, Introduction to General Relativity, Second Edition (New York: McGraw-Hill Book Company, 1975), pp. 7–16; see also B.A. Rosenfeld, A History of Non-Euclidean Geometry: Evolution of the Concept of a Geometric Space. Translated by Abe Shenitzer (New York, Berlin, Heidelberg: Springer-Verlag, 1988), pp. 333–338; Renate Wahsner, “Apriorische Funktion und aposteriorische Herkunft: Hermann von Helmholtz’ Untersuchungen zum Erfahrungsstatus der Geometrie,” in Krüger, Universalgenie Helmholtz (ref. 17), pp. 245–259; Olivier Darrigol, “Number and measure: Hermann von Helmholtz at the crossroads of mathematics, physics, and psychology,” Studies in History and Philosophy of Science 34 (2003), 515–573; idem, “A Helmholtzian approach to space and time,” ibid. 38 (2007), 528–542; Gregor Schiemann, Hermann von Helmholtz’s Mechanism: The Loss of Certainty [Archimedes, Vol. 17] ([Dordrecht]: Springer, 2009), pp. 98–110.

  62. Goethe, Faust, Part I, line 1237. Beside many other references, Helmholtz devoted two major essays to Goethe, “Ueber Goethe’s naturwissenschaftliche Arbeiten” [1853], in Vorträge und Reden. Fünfte Auflage. Erster Band (ref. 7), pp. 23–45; translated by H.W. Eve as “On Goethe’s Scientific Researches” in Popular Scientific Lectures (ref. 7), pp. 1–21, and “Goethe’s Vorahnungen kommender naturwissenschaftlicher Ideen” [1892], in Vorträge und Reden. Fünfte Auflage. Zweiter Band (ref, 19), pp. 335–361; former also translated, as well as the latter as “Goethe’s Presentiments of Major Scientific Ideas,” in David Cahan, ed., Science and Culture: Popular and Philosophical Essays (Chicago and London: The University of Chicago Press, 1995), pp. 1–17, 393–412.

  63. Helmholtz, “Ueber die thatsächlichen Grundlagen” (ref. 59), pp. 614–615; “On the Factual Foundations” (ref. 59), pp. 49–50.

  64. Ibid., p. 617; 51.

  65. Ibid., p. 638–639; 51. For the work of Beltrami, see Jeremy Gray, Ideas of Space: Euclidean, Non-Euclidean, and Relativistic, Second Edition (Oxford: Clarendon Press, 1989), pp. 147–154.

  66. Roberto Torretti, Philosophy of Geometry from Riemann to Poincaré (Dordrecht-Holland and London: D. Reidel Publishing Co., 1978), pp. 155–179; Volkmar Schüller, “Das Helmholtz-Liesche Raumproblem und seine ersten Lösungen,” in Krüger, Universalgenie Helmholtz (ref. 17), pp. 260–275; Klaus Volkert, “Hermann von Helmholtz und die Grundlagen der Geometrie,” in Wolfgang U. Eckart and Klaus Volkert, ed., Hermann von Helmholtz: Vorträge eines Heidelberger Symposiums anlässlich des einhundersten Todestages (Pfaffenweiler: Centaurus-Verlagsgesellschaft, 1996), pp. 177–207.

  67. The non-Euclidean character of color space was emphasized by Erwin Schrödinger, “Outline of a Theory of Color Measurement for Daytime Vision” [1920] and “Thresholds of Color Differences” [1926], in MacAdam, Sources of Color Science (ref. 10), pp. 134–182, 183–193.

  68. Helmholtz, “Ueber die tatsächlichen Grundlagen” (ref. 59), pp. 616–617; “On the Factual Foundations,” (ref. 59), pp. 50–51. Helmholtz goes on to cite his own work on eye movements from the first edition of his Handbuch. See also Gerhard Heinzmann, “The Foundations of Geometry and the Concept of Motion: Helmholtz and Poincaré,” Sci. in Con. 14 (2001), 457–470.

  69. The second edition of Helmholtz’s Handbuch mentions Riemann and describes color perception as a three-dimensional manifold comparable to space; H. von Helmholtz, Handbuch der physiologischen Optik. Zweite umgearbeitete Aufllage (Hamburg und Leipzig: Leopold Voss, 1896), p. 336.

  70. Helmholtz, Sensations of Tone (ref. 25), p. 370. Because this English version is based on the fourth German edition (1877), the reader should compare it with the second and third editions, Helmholtz, Die Lehre von den Tonempfindungen (ref. 31), p. 560; idem, Dritte umgearbeitete Auflage (Braunschweig: Friedrich Vieweg und Sohn, 1870), p. 576. As Vogel notes, Helmholtz’s “exposition of the theory of the tone quality of musical instruments was essentially grounded in mathematics”; see Vogel, “Sensations of Tone,” (ref. 21), p. 273.

  71. Helmholtz, Sensations of Tone (ref. 25), p. 370.

  72. These larger concerns also can be felt in his inaugural address as pro-rector at Heidelberg University; see Hermann von Helmholtz, “Ueber das Verhältnis der Naturwissenschaften zur Gesammtheit der Wissenschaftern” [1862], in Vorträge und Reden. Fünfte Auflage. Erster Band (ref. 7), pp, 157–185; translated as “On the Relation of Natural Science to Science,” in Cahan, Science and Culture (ref. 62), pp. 76–95.

  73. Hermann Helmholtz, “Ueber den Ursprung und die Beudeutung der geometrischen Axioms” [1870], in Vorträge und Reden. Fünfte Auflage. Zweiter Band (ref. 19), pp. 1–31, on pp. 4–5; translated by E. Atkinson as “On the Origin and Significance of Geometrical Axioms,” in Popular Scientific Lectures (ref. 7), pp. 223–249, on p. 224; and as “On the Origin and Meaning of Geometrical Axioms,” in Pesic, Beyond Geometry (ref. 35), pp. 53–70, on p. 53. My translation here differs from Atkinson’s, and elsewhere I sometimes also altered Atkinson’s translation of Mannigfaltigkeit as “aggregate” to the now standard term “manifold.”

  74. Ibid., p. 25; 241; 64.

  75. Ibid., p. 27; 243; 65–66.

  76. Ibid., p. 19; 236; 61.

  77. Ibid., p. 16; 233; 59.

  78. Though Kant had referred to space as a manifold, he had drawn back from using that term about time, guardedly noting that “we represent the time-sequence by a line progressing to infinity, in which the manifold constitutes a series of one dimension only”; see Immanuel Kant, Critique of pure reason. Translated and Edited by Paul Guyer and Allen W. Wood (Cambridge: Cambridge University Press, 1998), B50, p. 180. The word “manifold” may have ascribed more reality to time than Kant felt appropriate for it as “a purely subjective condition of our [human] intuition,” thus “not something which exists of itself”; see ibid., B51, B49, pp. 181, 180. Timothy Lenoir, “Operationalizing Kant: Manifolds, Models, and Mathematics in Helmholtz’s Theories of Perception,” in Michael Friedman and Alfred Nordmann, ed., The Kantian Legacy in Nineteenth-Century Science (Cambridge, Mass. and London: The MIT Press, 2006), pp. 141–210.

  79. Helmholtz, “Ueber den Ursprung” (ref. 73), p. 17; “On the Origin” (ref. 73), p. 234; “On the Origin and Meaning” (ref. 73), p. 59.

  80. In 1878 Helmholtz began to explore the factors he calls “topogenous” and “hylogenous” leading to space and time perceptions; see Helmholtz, “Thatsachen” (ref. 50), pp. 402–403; [Hertz and Schlick,] “Tatsachen” (ref. 50), pp. 149–150; [Kahl,] “Facts” (ref. 50), p. 405; [Cohen and Elkana,] “Facts” (ref. 50), pp. 159–160. David Jalal Hyder, “Helmholtz’s Naturalized Conception of Geometry and his Spatial Theory of Signs,” Philosophy of Science 66 (September 1999), S273–S286.

  81. Helmholtz, “On the Origin” (ref. 73), pp. 246–247; “On the Origin and Meaning” (ref. 73), p. 68.

  82. Ibid., p. 246; 68, See also, for instance, S.P. Fullinwider, “Hermann von Helmholtz: The Problem of Kantian Influence,” Stud. Hist. Phil. Sci. 21 (1990), 41–55.

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  83. Helmholtz, “Ueber den Ursprung” (ref. 73), p. 8; “On the Origin” (ref. 73), p. 227; “On the Origin and Meaning” (ref. 73), p. 54. According to Reichenbach, “Helmholtz was the first to advocate the idea that human beings, living in a non-Euclidean world, would develop an ability of visualization which would make them regard the laws of non-Euclidean geometry as necessary and self-evident, in the same fashion as the laws of Euclidean geometry appear self-evident to us”; see Hans Reichenbach, “The Philosophical Significance of the Theory of Relativity,” in Paul Arthur Schipp, ed., Albert Einstein: Philosopher-Scientist (Evanston, Ill.: The Library of Living Philosophers, 1949), pp. 289–311, on p. 308.

  84. Thomas Hawkins, “The Birth of Lie’s Theory of Groups,” The Mathematical Intelligencer 16 (1994), 6–17; idem, The Emergence of the Theory of Lie Groups: An Essay in the History of Mathematics, 1869–1926 (New York: Springer-Verlag, 2000), pp. 124–130.

  85. William Kingdon Clifford, “The Postulates of the Science of Space” [1873], in Pesic, Beyond Geometry (ref. 35), pp. 73–87.

  86. Hawkins, Emergence of the Theory of Lie Groups (ref. 84), pp. 34–42; David E. Rowe, “Klein, Lie, and the ‘Erlanger Programm’,” in L. Boi, D. Flament, and J.-M. Salanskis, ed., 1830–1930: A Century of Geometry: Epistemology, History and Mathematics (Berlin, Heidelberg: Springer-Verlag, 1992), pp. 45–54.

  87. Felix Klein, “The Most Recent Researches in Non-Euclidean Geometry” [1893], in Pesic, Beyond Geometry (ref. 35), pp. 109–116, on p. 110.

  88. Gerhard Heinzmann, “Helmholtz and Poincaré’s Considerations on the Genesis of Geometry,” in Boi, Flament, and Salanskis, 1830–1930: A Century of Geometry (ref. 86), pp. 245–249; Heinzmann, “Foundations of Geometry” (ref. 68).

  89. Henri Poincaré, “Non-Euclidean Geometries” [1891], in Pesic, Beyond Geometry (ref. 35), pp. 97–105, on p. 100. For discussion of the Poincaré and Klein models, see Rosenfeld, History of Non-Euclidean Geometry (ref. 61), pp. 236–246.

  90. Howard Stein, “Some Philosophical Prehistory of General Relativity,” in John Earman, Clark Glymour, and John Stachel, ed., Minnesota Studies in the Philosophy of Science. Vol. 8. Foundations of Space-Time Theories (Minneapolis: University of Minnesota Press, 1977), pp. 3–49, on pp. 21–25; Martin Carrier, “Geometric Facts and Geometric Theory: Helmholtz and 20th-Century Philosophy of Physical Geometry,” in Krüger, Universalgenie Helmholtz (ref. 17), pp. 276–291; Michael Friedman, “Geometry as a Branch of Physics: Background and Context for Einstein’s ‘Geometry and Experience’,” in David B. Malament, ed., Reading Natural Philosophy: Essays in the History and Philosophy of Science and Mathematics (Chicago and La Salle: Open Court, 2002), pp. 193–229; Josć Ferreirós, “Riemann’s Habilitationsvortrag at the Crossroads of Mathematics, Physics, and Philosophy,” in J. Ferreirós and J. Gray, ed., The Architecture of Modern Mathematics: Essays in History and Philosophy (Oxford and New York: Oxford University Press, 2006), pp. 67–96.

  91. Einstein to Thornton, December 7, 1944, quoted in Don A. Howard, “Albert Einstein as a Philosopher of Science,” Physics Today 58 (December 2005), 34–40, on 34.

  92. Albert Einstein, “The Problem of Space, Ether, and the Field in Physics” [1934], in Pesic, Beyond Geometry (ref. 35), pp. 187–193, on p. 190.

  93. Riemann did not include these details in his 1854 lecture but provided them in his “Commentatio mathematica” [1861], in Gesammelte Mathematische Werke (ref. 35), pp. 423–436. Ruth Farwell and Christopher Knee, “The Missing Link: Riemann’s ‘Commentatio,’ Differential Geometry, and Tensor Analysis,” Hist. Math. 17 (1990), 223–255.

  94. Albert Einstein and Leopold Infeld, The Evolution of Physics: The Growth of Ideas from Early Concepts to Relativity and Quanta (New York: Simon and Schuster, 1936), p. 33. For Einstein’s reading of Riemann, see ref. 2 above; though Einstein surely knew Riemann’s 1854 lecture, it is not clear whether he had read his work on the mechanism of the ear.

  95. Einstein’s review of a wartime (1917) edition of Helmholtz’s, Zwei Vorträge über Goethe, in A.J. Kox, Martin J. Klein, and Robert Schulmann, ed., The Collected Papers of Albert Einstein. Vol. 6. The Berlin Years: Writings 1914–1917 (Princeton: Princeton University Press, 1996), p. 569.

  96. Albert Einstein, “Non-Euclidean Geometry and Physics” [1925], in Pesic, Beyond Geometry (ref. 35), pp. 159–162, on p. 161.

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Acknowledgments

I thank Carol Anderson, for whose conference at Yale University in November 2010 on the “Long Nineteenth Century: Time and Memory” I originally prepared this work. I greatly appreciate her warm hospitality and the ensuing discussions on these issues during and after that conference. I also thank the Modern Science Working Group at Harvard University, organized by Lisa Crystal, for allowing me to present this material. I am especially grateful to the John Simon Guggenheim Memorial Foundation for its support. I particularly thank Roger H. Stuewer for his outstanding editorial care. Finally, I dedicate this paper to Creig Hoyt, a good friend and a worthy successor of Helmholtz as physician, ophthalmological pioneer, lover of music, and polymath.

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Correspondence to Peter Pesic.

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Peter Pesic is Tutor and Musician-in-Residence at St. John’s College, Santa Fe, New Mexico.

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Pesic, P. Helmholtz, Riemann, and the Sirens: Sound, Color, and the “Problem of Space”. Phys. Perspect. 15, 256–294 (2013). https://doi.org/10.1007/s00016-013-0109-1

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Keywords

  • Hermann von Helmholtz
  • Bernhard Riemann
  • Albert Einstein
  • Raumproblem
  • space
  • color
  • music
  • general relativity
  • geometry
  • physiological optics
  • psychological acoustics
  • manifold