Reasoning and symbolism in Diophantus 333
abbreviation is used inside the noun-phrase of the specifi c form of example
(2) above, when a quantitative value is set out statically. Th e relation of
subtraction holding dynamically between such noun-phrases – when one
engages in the act of subtracting a value from a quantitative term – this
operation is referred to by a diff erent verb, ‘take away’ ( aphairein ), which is
not abbreviated.
Further, the logical signposts marking the very rigid form of the
problem, such as ‘let it be set down’, ‘to the positions’, etc., are fully written
out. In other words, just as symbolism does not reach the level of the sen-
tence, so it does not reach the level of the paragraph. Th e rule is confi rmed:
abbreviations are confi ned to the level of the noun-phrase. I shall return to
discuss the signifi cance of this limitation in Section 4 below. For the time
being, I note the conclusion, that Diophantus’ marked use of symbolism is
not co-extensive with Diophantus’ marked use of language.
Over and above Diophantus’ marked use of symbolism, it should be
mentioned that Greek manuscripts, certainly from late antiquity onwards,
used many abbreviations for common words such as prepositions, con-
nectors, etc.: our own ‘&’, for instance, ultimately derives from such scribal
practices. Th ere are also many abbreviations of grammatical forms, espe-
cially case markings, so that the Greek nominal root is written, followed
by the abbreviation for ‘ον’, ‘οις’, etc. as appropriate. Such abbreviations are
of course in common use in the manuscripts of Diophantus. Most (but not
all) of such symbols were transparent abbreviations and in general they
could be considered as a mere aid to swift writing. Th eir use is as could be
predicted: the more expensive a manuscript was, the less such abbreviations
would be used; they are more common in technical treatises than in literary
works; humanists, proud of their mastery of Greek forms, would tend to
resolve abbreviations, while Byzantine scribes – oft en scrambling to get as
much into the page as possible – would also oft en tend to abbreviate.
We should mention one scribal abbreviation, which is not at all specifi c
to Diophantus, but which is especially valuable to him: the one for the
sound-sequence /is/. It so happens that this common sound-sequence is the
lexical root for ‘equal’ in Greek. Since it is a very common sound-sequence,
it naturally has a standard abbreviation, so that Diophantus has ‘for free’
a symbol for this important relation.
How are such symbols understood? Th at is, what is the relationship
between Diophantus’ symbols, and the alphabetically written words that
they replace? Th e fi rst thing to notice, as already suggested above, is that the
symbols are most oft en a transparent abbreviation of the alphabetical form.
Diophantus’ own strategy of choice in the symbols he had himself coined
was to clip the word into its fi rst syllable (especially when this is a simple,