300 Notes to pages 146–161
- “ Elementa doctrina solida, ” E230, I.26.71 – 93; “ Demonstratio nonnullarum insignium proprietatum, quibus
solida hedris planis inclusa sunt praedita ” E231, I.26.94 – 109. For commentary, see Sandifer 2007c , 9 – 18.
Velminski 2009a collects these papers, highlighting their connection with Euler ’ s work on the K ö nigsberg
problem, though Sachs, Stiebitz, and Wilson 1988 point out that Euler did not note this connection. See also
Mahr and Velminski 2010. Later proofs of Euler ’ s formula exploited the analogy with an Euler walk; see
Fajtlowicz and Mathew 2012. - For an excellent presentation of the details of both arguments and their connections, see Richeson 2008 ; see
also Debnath 2010 , 153 – 173. - For a closed, orientable surface of genus g , χ = 2 – 2 g ; for example, M ö bius strips are nonorientable because
their “ inside ” is not distinct from their “ outside. ” See also Blatter and Ziegler 2010. - For instance, comparing superparticular ratios [ n: ( n + 1)] or multiples [1: n ] to other classes of ratios.
- Thus, a quadratic equation has degree 2 because it contains no power of x higher than x 2 ; a cubic equation
has degree 3 and no power higher than x 3 ; and so on.
10 Euler: From Sound to Light - For an excellent overview of the reception of Newton ’ s theory, see Hakfoort 1995 , 11 – 71.
- Ibid. , 79 – 80.
- For the connection with de Mairan ’ s work in 1717, see Hakfoort 1995 , 63, 37 – 42. For Malebranche ’ s use of
the analogy with sound, see 56. See also the excellent treatment in Darrigol 2009 , 115 – 185; 2012, 152 – 161,
treating Malebranche, de Mairan, and Bernoulli at 136 – 152. - Hakfoort 1995 , 80 – 82.
- Quoted in Hakfoort 1995 , 72; regarding Euler ’ s theory, see also Home 1988 ; Pedersen 2008.
- My translation, from the manuscript cited in Hakfoort 1995 , 79 – 80.
- The assessment of the response to the 1744 lecture follows Hakfoort 1995 , 80 – 82; Pedersen (2008 , 393)
considers the analogy between sound and light to be “ the hard core of Euler ’ s optical research program. ” Euler ’ s
extended 1746 presentation of his theory was “ Nova theoria lucis et colorum ” (E88, III.5.1 – 45). - Hakfoort 1995 , 98.
- Quoted in Jean Formay ’ s summary of Euler ’ s 1744 “ Pens é es ” in Hakfoort 1995 , 90 – 91.
- See Pesic 2005 , 42 – 52.
- Hakfoort 1995, 111.
- Ibid. , 113. Note that here f has been substituted for Euler ’ s original notation of α for frequency.
- This quote from Rameau ’ s G é n é ration harmonique (1737) is cited in Cohen 2001 , 68 – 92, at 73 – 74.
- See Rehding 2003 , 15 – 35.
- “ Du v é ritable caractere de la musique moderne ” (E315, III.1.516 – 539). See Knobloch 2008.
- I thank Noam Elkies for pointing out to me these problems in Euler ’ s voice-leading.
- “ De harmoniae veris principiis per speculum musicum repraesentatis ” (E457, III.1.568 – 587). In 1840, F é tis
(1994, 97) noted the priority of Euler ’ s analysis of the dominant seventh, but was in general very critical of
Euler ’ s approach (69 – 84). - Euler 1837 , 2:71, 1:112, 1:109. See also Welsh 2010.
11 Young ’ s Musical Optics - Peacock 1855 , 128.
- Ibid. , 12, and Gurney 1831 , which both rely on Young ’ s autobiographical notes; see also Hilts 1978. The most
complete modern biography is Wood and Oldham 1954 , which notes that Young ’ s father and grandmother “ were
not merely nominal Quakers, but active members of the Society ” and adduces “ a certain affinity between the
Quaker pursuit of truth, with its emphasis on verification in personal experience, and the scientific method ” (3).
More recently, see Robinson 2006. Regarding the Quaker background, see Isichei 1970 ; Cantor 2004 ; 2005 , 64,
82 – 83, 111; Mathieson 2007.