356 reviel netz
(1) And if it is added to twenty, results: ς 1 Μ ο 20. (2) And if it is taken away from
100, results: Μ ο 100 lacking number 1. (3) And it shall be required that the greater
be 4- tms the smaller. (4) Th erefore four- tms the smaller is equal to the greater; (5) but
four- tms the smaller results: Μ ο 400 Ψ ς 4; (6) these equal ς 1 Μ ο 20
(7) Let the subtraction be added <as> common, (8) and let similar <terms> be
taken away from similar <terms>. (9) Remaining: numbers, 5, equal Μ ο 380. (10)
And the ς results: monads, 76.
Th is – the entirety of the argumentative part of the proposition – all
revolves around a single verifi cation, the one connecting the statement of
steps 5–6 taken as a whole, and the statement of step 9. Th e operation to
be verifi ed is contained in steps 7–8; steps 1–5 (which are very simple, but
somewhat convoluted) make sense as soon as their purpose becomes clear:
to bring the two expressions of steps 5–6 into close proximity, in prepara-
tion for the verifi cation of the operation. Finally, step 10 is a very simple
consequence of step 9 and calls for no cognitive eff ort.
Note, then, that steps 7–8 are fully spelled out: they do not include any
of Diophantus’ symbolic terms. Th e operation itself is fully verbal and
semantic: the meaning of the operation is directly told to the reader. On the
other hand, the substratum for the operation – the phrases of steps 5, 6 and
9 – is presented in the bimodal form of abbreviations. One knows through-
out what one talks about: these are not abstract symbols, but ‘numbers’
and ‘monads’. On the other hand, the computations can relatively easily
be carried out: a ‘lacking’ in the one can be translated into an addition to
the other, which easily leads to 5; 400 with 20 taken away easily leads to
380; each result is attached to the correct rubric, ‘number’ in the fi rst case,
‘monads’ in the second. In all of this, the simplifi cation introduced by a fi xed
visual structure to which objects can be added or removed is of obvious help.
Th is, then, is my suggestion for the role of symbolism in Diophantus’ rea-
soning. As Diophantus transformed the lay and school algebra material at
his disposal, into the argumentative form of Greek mathematical analysis,
he added in a tool which served in this analytic form – making the argu-
ment display the rationality of the passage from the terms of the problem
to the terms of its solution. 28 We can see why the transition from lay and
school algebras, to elite literate algebra, would encourage Diophantus to
introduce the type of symbolism he uses. But we should also consider the
second transition leading to Diophantus’ text. His text diff ers not only from
28 An analogous account can perhaps be provided for Diophantus’ fraction symbolism. With
fractions, as well, Diophantus does not develop a symbolic operation that allows him to
calculate directly on fractions (e.g. from a / b * c / d to get a * b / c * d ). Th us the validity of the
operations is left for the reader to verify explicitly. However, the symbolism – whose essence