306 Notes to pages 231–238
- Helmholtz 1867 , 288 – 297. Young 2002, 4:617.
- Riemann 1990 , 309 – 310; quoted in Pesic 2007 , 26 – 27. In modern notation, Riemann generalized the quadratic
Euclidean line element ds 2 = dx 1 2 +dx 2 2 +dx 3 2 (in terms of three spatial coordinates 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 + ... = Σ (^) μ (^) ν g μ ν dx^ μ^ dx^ ν^ (summed over all n dimensions, μ , ν = 1, ... ,
n ), where g μ ν is the “ metric tensor. ”
- Riemann 1990 , 320 – 325; see also Ionescu-Pallas and Sofonea 1986 ; Archibald 1991 ; Laugwitz 1999 , 254 –
263, 269 – 272. - Maxwell 1868 ; 1890 , 2:125 – 143.
- Riemann 1990 , 168 – 176.
- See Pesic 2007 , 41 – 45.
- Riemann 1990 , 370 – 382; 1984 ; Laugwitz 1999 , 281 – 287.
- Riemann 1990, 587 – 589; Laugwitz 1999, 2 – 3.
- Riemann 1990, 373; 1984, 32 – 33.
- Riemann 1990, 375; 1984, 35.
- See Ritchey 1991 ; Tonietti 2011.
- Riemann 1990 , 376; 1984 , 35.
- Helmholtz 1971 , 366 – 408, at 370 – 377; Hui 2008 , 78 – 79. For Ernst Mach ’ s sign theory of hearing, see Hui
2013a. - Riemann 1990 , 380; 1984 , 37. For the developing concept of attention, see Crary 1999 , 30, 64, 104 – 105;
Steege 2012 , 82 – 155. - Riemann 1990 , 374; 1984, 34.
- Jacob Henle, the friend and editor who published Riemann ’ s paper posthumously, noted that “ Riemann
thought that the mathematical problem to be solved was in fact hydraulic [ hydraulisches ] ” ; quoted in Riemann
1990, 370n. See also Riemann 1990, 807 – 810, and Gallagher 1984. - Helmholtz 1882 , 2:503 – 514, 515 – 581, collected in Helmholtz 1873.
- Koenigsberger 1965 , 267.
- Eventually, Helmholtz ’ s detailed description of the ear was superseded by later anatomical findings, particu-
larly 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. Cf. Sterne 2003 , 62 – 67; Erlmann 2010, 312 – 314. - Riemann 1867 ; Dedekind 1990.
- Koenigsberger 1965 , 254 – 255. “ Monodromous ” means that two congruent bodies remain congruent after
one of them has undergone a complete rotation around any axis. - For d ’ Alembert ’ s and Lagrange ’ s remarks, see Archibald 1914 ; Bork 1964.
- Helmholtz 1868 ; 1882 , 2:610 – 617, at 610 – 611, as translated in Pesic 2007 , 47 – 52, at 47, though the date
of publication of this paper should be listed as 1868, as shown by Volkert (1993 ). - Helmholtz 1868; 1882, 2:611; Pesic 2007, 47 – 48.
- See also Helmholtz ’ s longer 1868 paper on this subject: Helmholtz 1977 , 39 – 58; 1882 , 2:618 – 639. For a
helpful modern account of Helmholtz ’ s argument, see Adler, Bazin, and Schiffer 1975 , 7 – 16; also Rosenfeld
1987 , 333 – 338; Wahsner 1994 ; Darrigol 2003 ; Darrigol 2007. - Goethe, Faust , Part I, line 1237. Helmholtz devoted two major essays to Goethe: Helmholtz 1995 , 1 – 17,
393 – 412. - Helmholtz 1868; 1882, 2:617; Pesic 2007, 51.
- For Beltrami, see Gray 1989 , 147 – 154.
- Torretti 1978 , 155 – 179; Sch ü ller 1994 ; Volkert 1996.