290 Notes to pages 53–63
- G. Galilei 1977 , 71 – 77.
- G. Galilei 1989 , 74 – 75.
- Ibid., 196.
- Quoted from Westman 2011, 489; for Hooke ’ s attempts to provide experimental verification of the Coper-
nican view, see 504 – 510.
4 Hearing the Irrational - Recorde 1557 , sigs. Aiir, Sir, Siv; Neal 2002 , 49 – 55, at 50. See also Van Wymeersch 2008.
- See Boyer 1991 , 304; Dijksterhuis 1970 , 16 – 19, 21 – 22, 38 – 39; Bos 2001 , 119 – 143; Rasch 2008. In 1585,
the Flemish mathematician and engineer Simon Stevin also advocated decimal notation; see Klein 1992,
186 – 197. - The seminal work on Vi è te in relation to Greek mathematics is Klein 1992 , 150 – 185, 321 – 322n10, which
contains a translation of Vi è te ’ s Isagoge (313 – 353). For cryptographic parallels, see Pesic 1997a,b ; 2000a , 59 – 83;
2000b , 73 – 83; and Panza 2006. - Field 1997 , 67; 2005 , 24 – 31, 282 – 284, 312 – 316. See also Moyer 2008 ; Peterson 2011 , 106 – 124.
- Boyer 1991 , 281 – 282; Aubel 2008 ; Neal 2002 , 49.
- Stifel 1544 , fol. 7v, 55r – 58r.
- Boethius 1989 , 4.11; Stifel 1544 , fols. 70r – 75v; for the sources, see Euclid and Porphyry 1991 , props. 3, 16;
Barker 1984 , 2:190 – 208 and Barbera 1984 ; Knorr 1975 , chap. 7; Field 2011 ; Moyer 2011. - Modern convention has an octave comprise 1,200 cents, of which an equal-tempered semitone would be 100
cents, a “ major semitone ” (2,187:2,048 = 3^7 :2^11 ) 114.7 cents, a “ minor semitone ” (256:243 = 2^8 :3^5 ) 90.2 cents,
and a Pythagorean comma (3^12 /2^19 = 531,441/524,288) 24.5 cents. See Fauvel, Flood, and Wilson 2004 ,
13 – 27. - Stifel 1544 , fol. 76r; for the earlier theorists, see de Muris 1992, 292 – 301, at 294; Hentschel 1998 , 39 – 60, at
- As in its 1482 Latin translation: Busard 2005, 1:160 – 161.
- Oresme 1966 , 60 – 65, 304 – 309; 1971 , 78 – 161, 296 – 305, at 297; see also Abdounar 2008.
- Lef è vre d ’ É taples and Jordanus 1496 , fol. g6v, cited in Stifel 1544 , fol. 76v.
- For details of Stifel ’ s computations, see Pesic 2010.
- Stifel 1544 , fol. 79v. I thank William Donahue for his kind help with the translations from Stifel ’ s Latin.
- Ibid. , fol. 103r.
- Cardano 1967 , 10:222, as translated in Cardano 2007 , 1.
- Cardano 1967 , 2:337, cited in Cardano 1973 , 22n36, which also includes the quotation from Miller.
- Cardano 1973 , 45, cites Lef è vre; see also Pesic 2010 ; Barbour 1972 , 7. For a lively introduction to questions
of temperament, see Duffin 2007. - Cardano 1967 , 4:281; 2007 , 204.
- Boethius 1989 , 1.21. Boethius ’ s semitone 243:256 is not an exact equal division of the tone. The trihemitone
is 294.1 cents, slightly smaller than the modern minor third, 300 cents. - Ibid. The ditone is 407.8 cents, slightly larger than the modern major third, 400 cents.
- Palisca 1985 , 88 – 110; he comments on Vincenzo on 10 n 35. For Vicentino, see Berger 1980 ; Cordes
- Palisca 1985 , 119; see also Kaufmann 1966.
- Vicentino 1996 , 302 – 314, on 304, discussed in Moyer 1992 , 168 – 184. Regarding the presence of Lasso, see
Kaufmann 1966 , 24n5, though Hell and Leuchtmann (1982 , 112) think he may have arrived later. In either case,
Lasso ’ s chromatic Prophetiae Sibyllarum dates from 1550 to 1552 and may well show the influence of
Vicentino. - Vicentino 1996 , 313 – 314; see Boncella 1988 ; McKinney 2005.