Reasoning and symbolism in Diophantus 335
Th e symbols for square, and for sides in a right-angled triangle, are the
exception, then. Th ere the trace, and not the sound, becomes the vehicle
of meaning. Th e reason for this is clear, as the trace here has indeed such
an obvious connotation. Th e sign and the signifi ed are isomorphic. Even
so, note that the understanding is that stands not just for the concept
‘square’ but also and perhaps primarily for the sequence ‘tetragon’, as wit-
nessed by the fact that the symbol is oft en followed by case marking: οις
for ‘tetragonois’, ‘by the squares’. Th e most interesting exception is the form
, sometimes used to represent ‘squares’, the plural marked not by the
sound of the case ending, but by the tracing of duplication (compare our
use of ‘pp.’, for instance, for ‘pages’; notice also that the same also happens
occasionally with the ‘number’ symbol).
Speaking generally for Greek writing in manuscripts, the phonological
nature of abbreviation symbolism becomes most apparent through the
rebus principle. To provide an example: there is a standard scribal abbrevia-
tion for the Greek word ‘ara’, ‘therefore’. Th ere is also an important prepo-
sition, ‘para’, meaning, roughly, ‘alongside’. Th e letter pi, followed by the
symbol for ‘ara’, may be used to represent the preposition ‘para’. Such rebus
writing is common in Greek manuscripts and shows that the symbol for
‘ara’ stands not merely for the concept ‘therefore’ but, perhaps more funda-
mentally, for the sound-sequence ‘ara’.
Obviously, Diophantus’ symbolism does not lend itself to such rebus
combinations. One can mention, however, an important close analogue.
We recall Diophantus’ symbol for ‘number’, meaning, eff ectively, the
‘unknown’. Th is may be said to be the cornerstone of Diophantus’ symbol-
ism: on it ride the higher powers; it is the starting point for investigation
in each problem. It is thus, perhaps, not inappropriate that this symbol
is the least transparently phonological. It is, so to speak, Diophantus’
cipher. Crucially, it is also clearly defi ned by Diophantus in his introduc-
tion: ‘Th at which possesses none of these properties [such as dunamis,
cube, etc.] and has in it an indeterminate number of monads, is called a
number and its symbol is ς’ (Tannery 6.3–5). Th us the symbol is, strictly
speaking, only to be used for the indeterminate, or unknown, goal of
the problem. It should be used in such contexts as ‘Let the
which is added and taken away from each number <sc. of the two other
given numbers> be set down,
two occurrences of ‘number’ in this phrase. Th e fi rst is ‘number’ in its
standard Greek meaning (which therefore, one would think, should not
be abbreviable into the symbol ς). In the phrase ‘from each number’,
the word ‘number’ does not stand for an unknown number, but just for