Reasoning and symbolism in Diophantus 331
‘lacking’, roughly an upside-down Ψ (I shall indeed represent it in what
follows by Ψ, for lack of better fonts. Note that this is to be understood as a
‘minus’ sign followed by the entirety of the remaining expression – as if it
came equipped with a set of following parentheses.) Together with Greek
alphabetic numerals (Α, Β, Ι, Κ, Ρ, Σ for 1, 2, 10, 20, 100, 200.. .) one has the
main system with which complex phrases can be formed of the type, e.g.
(2) Κ υ ΒΔ υ Α ςΒ Μ ο Γ Ψ Κ υ Α Δ υ Γ ςΔ Μ ο Α
Most of all, Diophantine reasoning has to do with manipulation of such
phrases.
Syntactically, note that such phrases have a fi xed order: one goes through
the powers in a fi xed sequence (although in terms of Greek syntax, any
order could be natural). Th e numeral, also, always follows the unit to which
it refers (this, however, can be explained as natural Greek syntax). Finally,
there is a fi xed order relative to the ‘lacking’ symbol: the subtrahend is
always to the right of the symbol. Th is of course follows from the very
meaning of ‘lacking’.
Semantically, we may say that the ‘number’ functions rather like an
‘unknown’, on which the ‘dunamis’ or the ‘cube’ depend as well (a single
‘number’ multiplied by itself results in a single ‘dunamis’ which, once again
multiplied by a ‘number’, yields a ‘cube’). Th e monads, on the other hand,
are independent of the ‘number’.
Let us consider the wider context. When we discuss symbolism in
Diophantus, we need to describe it at three levels. First, there is the symbol-
ism which Diophantus had explicitly introduced in the preface to his trea-
tise. Second, Diophantus has a number of fairly specialized symbols which
he did not explicitly set out. Th ird, we should have a sense of the entire
symbolic regime of the Diophantine page, bringing everything together –
the markedly Diophantine, and the standard symbolism of Greek scribal
practice.
Th e symbols explicitly introduced by Diophantus are those mentioned
above (in the order in which Diophantus introduces them): Δ υ , Κ υ , Δ υ Δ,
ΔΚ υ , Κ υ Κ, ς, Μ ο , χ , Ψ. Th ese then unmistakably belong to the phrases such
as those of example (2), serving further to underline the importance of this
type of expression.
Beyond that, the manuscripts display a variety of further symbols.
Tannery systematically represents symbolically in his edition such symbols
as he feels, apparently, to be markedly Diophantine (on the other hand, he
always resolves standard scribal abbreviations; more on this below). Th e
following especially are noticeable among the markedly Diophantine: