308 Notes to pages 248–257
- Stoney here instances the Fourier series, so that “ the n th of these lines is represented by the term Cnxnnsin( +α),
in which C n is the amplitude of the vibration; and consequently C n 2 represents the brightness of the line ” ( Stoney
1871 , 293). - Ibid. ; on the clarinet, see Helmholtz 1954 , 98 – 99.
- Stoney 1871 , 295.
- Ibid. , 296.
- See Stoney and Reynolds 1871 , which explicitly mentions Helmholtz and the violin string on 47.
- There is only a brief mention of this result in Stoney 1880.
- Balmer 1885a , 551.
- Ibid. McGucken (1969, 131) notes that “ certainly Balmer knew of Stoney ’ s earlier work. ”
- Balmer 1885a, 551 – 552. Pais 1986 , 172, though generally useful as a summary, misleadingly translates
Grundton as “ keynote. ” - Balmer 1885a , 553.
- Helmholtz 1954 , 40 – 41. Balmer would have known these as “ cylinder functions [ Zylinderfunktionen ]. ”
- The frequency goes as Cm n( + 2 )p , where C is a coefficient depending on the plate, m and n are integers,
and p is roughly 2 for a circular plate. The coefficient p can vary between 1.4 and 2.4 for other, more complicated
shapes, such as cymbals, hand bells, or church bells. As a mathematician, Balmer would also have known the
approximate expression of Bessel functions involving squares; see Rayleigh 1945 ; Airey 1910. - His second publication ( Balmer 1885b ) extends his first results to several more recently discovered hydrogen
lines, which, compared with his formula for n = 2, m = 5 – 16, he finds “ agreement that must surprise to the
highest degree ” ; his fourth and final paper ( Balmer 1897 ) addresses a number of elemental spectra and, in an
addendum, discusses the work of Johannes Rydberg (1890). Pais (1986 , 173) implies misleadingly that Balmer
did not address other elements besides hydrogen until this publication, whereas he does at least mention his
thoughts in Balmer 1885a , 559 – 560. - Balmer 1885c.
- Rayleigh 1945. In the whole work, the word “ music ” is only used explicitly five times. For the distinction
between discovery and justification, see Reichenbach 2006, 382. - Rayleigh 1945 , vi – vii.
- This is true, for example, of Rydberg 1890.
- Husserl 1970 , 52 – 53.
- Ibid. , 360 – 361. For discussions of this process, see Klein 1985 , 65 – 84; Derrida 1989 , 98 – 107.
- I have discussed this in Pesic 2000a , 2 – 3, and applied it throughout that work.
17 Planck ’ s Cosmic Harmonium - Husserl 1939 , 212. The phrase is not included in the standard edition and translation ( Husserl 1970 ); Klein
(1985 , 372) concludes that “ this sentence is based on Husserl ’ s own words, uttered in conversation with Fink, ”
the editor who first published the essay in 1939. - Planck 1998 , 7.
- For his comments on Helmholtz and energy, see Planck 1998 , 19 – 20, 99 – 107.
- This and the following general information about Planck ’ s musical life come from the excellent account in
Heilbron 1986 , 3, 34. For a superb account of the “ singing savants ” in the earlier part of the century, see Jackson
2006 , 45 – 74. - For Planck ’ s personal recollections of Helmholtz, see Planck 1949 , 15, 24 – 25.
- Planck 1893 , 428; Pesic 2014c.
- Heilbron 1986 , 34.
- Hui 2013b ; Hiebert 2003.
- Planck 1949 , 26.
- Helmholtz 1954 , 316.