From the Beginning to Plato

GREEK ARITHMETIC, GEOMETRY AND HARMONICS 263
arithmetic itself. 14 But if, as is generally done, those difficulties are ignored, and
we assume that at least by the time of Book X Euclid includes numbers among
magnitudes, then the only law which Euclid has not proved in Book V and which
he needs to justify X.5–8 is trivial. 15 The assumption that Euclid left the proof of
this law to his readers or students does not seem to me implausible. However,
even if this is true, Aristotle’s remark in the Topics makes it very likely that at
some point before his and Eudoxus’ time an anthuphairetic theory of ratios was
developed to apply to commensurable and incommensurable magnitudes. 16
Whether we should ascribe this theory to Theaetetus seems to me moot.
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Archytas, Harmonics and Arithmetic
In section 4 I described the representation of the fundamental concords fourth,
fifth and octave in terms of the ratios 4:3, 3:2, 2:1. In a fragment from On
Mathematics (Porphyry [8.73], 56.5–57.27, DK 47 B 1), Archytas correlates high
pitches with fast movements and low pitches with slow ones in a piece of
physical acoustics. 17 We find an analogue of this correlation in the prologue of
Euclid’s Sectio Canonis, our earliest text in mathematical harmonics. The
prologue concludes with an argument that it is ‘reasonable’ that:
SC Assumption 1. The concordant intervals are ratios of the form n+1:n or
n:1, i.e. they are either ‘epimorics’ or multiples.
It seems possible that Archytas put forward an argument of the same kind,
although only Euclid makes an explicit correlation between pitch and
frequency. 18 Euclid’s argument for SC Assumption 1 relies only on an analogy
between the idea that concordant notes make a single sound and the fact that
ratios of the two forms are expressed by a single name in Greek: double is
diplasios, triple triplasios, etc., and 3:2 is hēmiolios, 4:3 epitritos, 5:4
epitetartos, etc. There is no question that the argument is a post hoc attempt to
justify previously established correlations; and it fails rather badly as a
foundation for the programme of the Sectio, which is:
1 to establish the numerical representation of the fundamental concords;
2 to use mathematics to disprove apparent musical facts, such as the existence
of a half-tone;
3 to construct a diatonic ‘scale’.
In the treatise Euclid tacitly takes for granted that addition of intervals is
represented by what we would call multiplication of ratios, subtraction of
intervals by what we would call division. To divide an interval represented by
m:n in half is, then, to find i, j, k such that i:k :: m:n, and i:j :: j:k. In addition to
SC Assumption 1 Euclid also relies on the following empirical ‘facts’: