38
GREEK NUMERICAL NOTATION
up to 5 and 2 units of the same Lind. On the other hand
A + /j. = o did not at all immediately indicate that r + v = \fr.
The new notation had only one advantage over the other,
namely that it took less space. Consider, for instance, 849,
which in the “ Herodianic ” form is P’HHHAAAAPI 111, but
in the alphabetic system is co/xd. The former is more self-
explanatory and, for reckoning with, has most important
advantages.’ Gow follows Cantor, but goes further and says-
that ‘ the alphabetical numerals were a fatal mistake and
hopelessly confined such nascent arithmetical faculty as the
Greeks may have possessed ’ ! 1 On the other hand, Tannery,
holding that the merits of the alphabetic numerals/ could only
be tested by using them, practised himself in their use until,
applying them to the whole of the calculations in Archimedes’s
Measurement of a Circle, he found that the alphabetic nota
tion had practical advantages which he had hardly suspected
before, and that the operations took little longer with Greek
than with modern numerals. 2 Opposite as these two views are,
they seem to be alike based on a misconception. Surely we do
not ‘ reckon with ’ the numeral signs at all, but with the
words for the numbers which they represent. For instance,
in Cantor’s illustration, we do not conclude that the figure 3
and the figure 4 added together make the figure 7 ; what we
do is to say ‘ three and four are seven ’. Similarly the Greek
would not say to himself ‘ y and S — ^ ’ but rpeîs Kal reacrapes
Inrd ; and, notwithstanding what Cantor says, this would
indicate the corresponding addition ‘ three hundred and four
hundred are seven hundred ’, TpiaKocnot Kal rerpaKocrLoi
tirraKoa-LOL, and similarly with multiples of ten or of 1000 or
10000. Again, in using the multiplication table, we say
‘ three times four is twelve ’, or ‘ three multiplied by four =
twelve ’ ; the Greek would say rpls réaaapes, or rpeis hrl
recrcrapas, ScoSeKcc, and this would equally indicate that ‘ thirty
times forty is twelve hundred or one thousand two hundred ’,
or that ‘ thirty times four hundred is twelve thousand or a
myriad and two thousand ’ {TpiaKovTciKLS recraapaKovra yî'Aîoî
Kal SiaKocrLOL, or rptaKovrccKis rerpaKocnoL pvptOL Kal Slo-^l\loi).
1 Glow, A Short History of Greek Mathematics, p. 46.
2 Tannery, Mémoires scientifiques (etl. Heiberg and Zeuthen), i,
pp. 200-1.
