330 reviel netz
scribe’s discretion, and no pattern was assumed at the outset. Th e two –
full word and its abbreviation – acted as allographs. Th is may be seen as a
consequence of the scribal culture within which Diophantus operated. Th e
upshot of this chapter, then, is to situate Diophantus historically in terms of
a precise deuteronomic, scribal culture, and within the context of practices
available to him from elite Greek mathematics.2. Notes on symbolism in D iophantus
We recall Nesselmann’s observation: Diophantus belongs to the category
of ‘syncopated algebra’, where the text is primarily arranged as discursive,
natural language (if of course in the rigid style typical of so much Greek
mathematics), with certain expressions systematically abbreviated. 7 In this,
it is generally understood to constitute a stepping stone leading from the
rhetorical algebra of, say (if we allow ourselves such heresy), Elements Book
ii , to the fully symbolic algebra of the moderns.
As a fi rst approximation, let us take a couple of sentences printed in
Tannery’s edition (prop. I.10, T1893, i. 28.13–15):
(1) Τετάχθω ὁ προστιθέμενος καὶ ἀφαιρούμενος ἑκατέρῳ ἀριθμῷ ςΑ. κἂν μὲν τῷ Κ
προστεθῇ, γίνεται ςΑ Μ ο Κ.
Let the <number> which is added and taken away from each number <sc. of the
two other given numbers> be set down, <namely> ςΑ <:Number 1>. And if it is
added to 20, result: ςΑ Μ ο Κ <:number 1 Monads 20>.
We see here the most important element in Diophantus’ symbolism:
a special symbol for ‘number’, ς. We also see a transparent abbreviation
for ‘monads’, Μ ο. To these should be added especially: two transparent
abbreviations, for ‘dunamis’ (eff ectively, ‘square’), Δ υ , and for ‘cube’, Κ υ.
Symbols for higher powers exist and are made by combining symbols for
the low powers, e.g. ΔΚ υ , dunamis–cube, or the fi ft h power. An appended
χ turns such a power into its related unit fraction: a dunamis, Δ υ , can
become a dunamiston, Δ υχ , or the unit fraction correlated with a dunamis.
(Th e symbol itself is reminiscent in form especially of the standard scribal
symbols for case endings.) Finally we should mention a special symbol for(^7) For a previous, brief characterization of Diophantus’ symbolism in practice, see Rashed 1984 :
lxxxi–lxxxii, whose position I follow here. Heath 1885 : 57–82 may still be read with profi t. In
general, many of the claims made in this section were made by past scholars already, and my
apology for going through this section in detail is that the point is worth repeating – and should
be seen in detail as an introduction to the following and much more speculative discussion.