 Burkert’s arguments (1972a, 238–277), supported by further study (Huffman 1993), have led to a consensus that some 11 fragments are genuine (Frs. 1–6, 6a, 7, 13, 16 and 17) and derive from Philolaus’ book On Nature (Barnes 1982; Kahn 1993 and 2001; Kirk, Raven and Schofield 1983; Nussbaum 1979; Zhmud 2012, Graham 2014). Fragments 1, 6a and 13 are identified as coming from the book On Nature by ancient sources. Stobaeus cites fragments 2 and 4–7 as coming from a work On the Cosmos, but this appears to be an alternate title for On Nature, which probably arose because the chapter heading in Stobaeus under which the fragments are cited is ‘On the Cosmos.’ Testimonia A7a, A9, A10, A16 (part), A17 (part), A18–24, A27–9 are also generally agreed to reflect the contents of On Nature. The authenticity of some fragments and testimonia remains controversial. Fr. 6b and Testimonium A 26, which, in contrast to Fr. 6a, take a numerological approach rather than a properly mathematical approach to music theory, were accepted as authentic by Burkert (1972a, 394–400; followed by Mueller 1997, 292–3), whereas Huffman argues that they are spurious (1993, 364–74; followed by Zhmud 1997, 185). Recent scholarship on harmonic theory argues that the conflict in approach may be less significant than it appears at first sight (see the end of section 2.2 below) and that 6b and A26 may thus be authentic (Barker 2007: 271; Hagel 2009: 143–151). Similarly, Burkert accepts A 14 as genuine and as representing the earliest evidence for horoscopic astrology in Greece (1972a, 350), whereas Huffman argues against authenticity (1993, 381–91).