332 reviel netz
Th e alphabetic numerals themselves. While Greek numbers are very
oft en written out by alphabetic numerals, they are more frequently
spelled out in Greek writing as the appropriate number words – just
as we have to decide between ‘5’ and ‘fi ve’. Th e avoidance of number
words and the use of alphabetical numerals, instead, is therefore a
decision involving a numerical code.
, for ‘square’ (used here in the meaning of ‘a square number’).
, for ‘the right sides’, in a right-angled triangle. Here they are studied
as fulfi lling Pythagoras’ theorem and therefore off ering an arena for
equalities for square numbers. Strangely, Tannery does not print this
symbol.
Αʹ, Βʹ, Γʹ, etc. for ‘fi rst’, ‘second’, ‘third’, etc. Th is is used in the important
context where several numbers are involved in the problem, e.g. what
we represent by ‘ n 1 + n 2 =3 n 3 ’ which, for Diophantus, would be ‘the
fi rst and the second are three times the third’, with ‘fi rst’, ‘second’, etc.
used later on systematically to refer to the same object. Of course,
such symbols are not to be confused with their respective numerals
and they are diff erently written out.
Β πλ , Γ πλ , for ‘two times’, ‘three times’, etc. Th is symbolism is based on the
alphabetic numerals, tucking on to them a transparent abbreviation
of the Greek form of ‘times’.
Ε ΙΓ : this is an especially dramatic notation whereby Diophantus refrains
from resolving the results of divisions into unit fractions, and instead
writes out, like in the example above, ‘fi ve thirteenths’ in a kind of
superscript notation. Tannery further transforms this notation into
a sort of upside-down modern notation. As long as we do not mean
anything technical by the word, we may refer to this as Diophantus’
‘fraction symbolism’.Th e last few mentioned symbols (with the possible exception of the frac-
tion symbolism) are not unique to Diophantus, but for obvious reasons the
text has much more recourse to such symbols than ordinary Greek texts so
that, indeed, they can be said to be markedly Diophantine.
One ought to mention immediately that many words, typical to
Diophantus, are not abbreviated. Th ese fall into two types. First, several
central relations and concepts – ‘multiply’, ‘add’, ‘given’, etc. – are written
in fully spelled out forms. In other words, Diophantus’ abbreviations are
located within the level of the noun-phrase, and do not touch the structure
of the sentence interrelating the noun-phrases. ‘Lacking’ is the exception to
the rule that relations are not abbreviated, but it serves to confi rm the rule
that abbreviations are located at the level of the noun-phrase. Th e ‘lacking’