334 reviel netz
consonant–vowel syllable), which he then turned into a symbol by placing
the vowel as a superscript on the consonant: Κ υ , Δ υ , Μ ο. Th e symbols result
from two reductions – a word into its initial syllable, a syllable into its con-
sonant. All of this makes sense in terms of natural language phonology so
that, in such cases, Diophantus’ symbolism may be tied to the heard sound
and not just to the visible trace. (It may be relevant that in all three words –
monas, dunamis, kubos – the stress falls indeed on the fi rst syllable.) With
arithm- and leipsei this simple strategy fails. Th e symbols, in both cases, are
more complex: perhaps some combination of the alpha and the rho of the
arithmos (but this is a well-known palaeographic puzzle), certainly some
reference to the psi of the leipsei. Th is is in line with the standard symbol-
ism, e.g. for prepositions: these are oft en rendered by a combination of their
consonants (‘pros’, e.g., becoming a ligature of the pi and the rho).
Note also that while alphabetical numerals do not directly represent the
sounds of the number-words they stand for, the system as a whole is iso-
morphic to spoken numerals (two-number words, ‘two and thirty’ become
two number-symbols, ΛΒ). In this, the alphabetical numeral system diff ers
from its main alternative in Greek antiquity, the acrophonic system where
each symbol had, directly, a sound meaning (Π for pente, fi ve, Δ for deka,
ten, etc.: the only exception is the use of a stroke for the unit), but the acro-
phonic number symbolism as a whole was equivalent to the Roman system
with which we are familiar and was no longer isomorphic to spoken numer-
als: not ΛΒ, but ΔΔΔΙΙ. Th e latter clearly is not meant to be pronounced as
‘deka-deka-deka-click-click’. In fact, it is no longer a pronounced symbol:
the trace has become free of the sound. In the alphabetical system, every-
thing can be understood as symbols standing for sounds in natural Greek:
I believe this may be the reason why this system was fi nally preferred for
most ordinary writing.
With this in mind, we can see that Diophantus’ marked symbols are
at least potentially spoken: the numbers, as explained above, as well as
the symbols based upon them. A stroke turns a numeral into its depend-
ent ordinal or unit-fraction (identical in sound, as in symbol: compare
English ‘third’, ‘fourth’, etc.). Further, ordinals are sometimes rendered in an
even more direct phonological system, e.g. Δ ευ , abbreviating δευτερος, for
‘second’. (Th us the system for ordinals has three separate forms: the fully
written-out word, the phonologically abbreviated form and the alphabetic
numeral-based form. Th is is important, given the role of ordinals as a kind
of unknown-mark in expressions such as ‘the fi rst number’.) Th e ×-times
symbolism, too, merely adds the onset consonants of the abbreviated
words: Β πλ for ‘double’.