Notes to pages 135–146 299
- Ibid. , 37.
- Ibid. , 42.
- Pelseneer 1951 , 480 – 482, at 482. Recall that superparticular ratios have the form n :( n+1 ); see above, 32.
- Smith 1960 , 119 – 122 (IV.35 – 39).
- Among the very few other attempts, note Birkhoff 1933. For a brief summary, see Newman 1956 , 4:2185 –
- Birkhoff ’ s basic equation, M O
C
= (where M is the aesthetic measure, O the order, and C the complexity),
is consistent with Euler ’ s approach.- Smith 1960 , 27 – 28 (E33, III.1.197 – 427).
- Helmholtz 1954 , 229 – 233.
- As pointed out by Jeans (1968 , 155 – 156).
- Smith 1960 , 68 (II.7).
- Ibid. , 71 (II.12).
- Ibid. , 71 – 72 (II.13).
- Ibid. , 72 (II.14).
- Aristotle, Poetics 1453b10 – 12 ( Aristotle 1984 , 2326); his terms are tragik ē hedon ē and katharsis.
- Smith 1960 , 73 (II.15 – 16).
- Ibid. , 23. See also Tserlyuk-Askadskaya 2007.
- For instance, we learn that the standard musical pitch he knew was a full major second lower than the present
standard (A440); Smith 1960 , 42. - Ibid. , 119 – 122 (IV.35 – 39). Euler seems unaware of earlier work on logarithms in music; see Wardhaugh
2008, 43 – 56; B ü hler 2013, 39 – 41. - Recall (box 4.1) that equal temperament divides the octave into twelve equal semitones, each given by the
irrational factor^122. For instance, J. S. Bach ’ s Well-Tempered Keyboard (1722) required a temperament capable
of playing in all twenty-four major and minor keys, though not necessarily equally; see Duffin 2007. Euler
discusses equal temperament in his early “ Adversaria mathematica ” (1726, fol. 45r), cited in B ü hler 2013, 225,
and reproduced in Bredekamp and Velminski 2010, 53. Euler ’ s Tentamen mentions equal temperament briefly
at Smith 1960 , 204 – 205 (IX.17); the rest of the book uses just intonation. - Ibid. , 121 (IV.38).
- John Wallis devised its name in 1695; see Gowers 2008 , 192 – 193, 315 – 317.
- Euler 1985 , 302 – 305 (E71, I.14.187 – 216).
- For his argument, see Euler 1985 and Sandifer 2007a , 234 – 248; 2007c , 185 – 190.
- Cited in Weil 1984 , 172; Dunham 1999, 7. Calinger 1996, 130 – 133, argues that the Bernoullis were a more
important influence on Euler ’ s turn to number theory than Goldbach. - See Leibniz 1989 , 212; Tserlyuk-Askadskaya 2007.
- Weil 1984 , 267. For Leibniz ’ s work on music, see B ü hler 2010 ; 2013, 130 – 175; for the Euler/Leibniz con-
nection, see also Downs 2012. - The proper divisors of a number exclude that number itself.
- See E152, I.2.86 – 162 ( Dunham 1999 , 7 – 12); Sandifer 2007c , 49 – 62.
- For Euler ’ s work on harmonic progressions, see McKinzie 2007. For the general history of the harmonic
series, see Green 1969. See also Bullynck 2010. - Weil 1984 , 267.
- Cited in Hakfoort 1995 , 60 – 65, at 61. See also Sachs, Stiebitz, and Wilson 1988.
- “ The Seven Bridges of K ö nigsberg, ” in Newman 1956 , 1:573 – 580 (E53, I.7.1 – 10).
- Euler ’ s 1736 paper is generally regarded as the origin of graph theory, a field introduced by J. J. Sylvester
in 1878 whose terminology was codified by George P ó lya and others about 1936; see Biggs, Lloyd, and Wilson