A history of Greek mathematics - Wilbourhall.org

NOTATION AND DEFINITIONS 459
When there are units in addition, the units are indicated by
the abbreviation M
;
a 3 +13# 2 + 5.x+2.
o
thus K Y a A Y iy s e M /3 corresponds to
The sign (A) for minus and its meaning.
For subtraction alone is a sign used. The full term for
wanting is Aen//-*?, as opposed to virap^is, a forthcoming,
which denotes a positive term. The symbol used to indicate
a wanting, corresponding to our sign for minus, is A, which
is described in the text as a ' \jr turned downwards and
truncated (¥ eXXnres ' kcctco vevov). The description is evidently
interpolated, and it is now certain that the sign has nothing
to do with \jr. Nor is it confined to Diophantus, for it appears
in practically the same form in Heron's Metrical where in one
place the reading of the manuscript is uovdScov oS T i'8',
74— y
1^. In the manuscripts of Diophantus, when the sign
is resolved by writing the full word instead of it, it is
generally resolved into Xetyfrei, the dative of Xeiyjris, but in
other places the symbol is used instead of parts of the verb
XeiTTeiv, namely Xinwu or Xeiyjras and once even Xtirctxn ;
sometimes X^iyjr^i in the manuscripts is followed by the
accusative, which shows that in these cases the sign was
wrongly resolved. It is therefore a question whether Diophantus
himself ever used the dative Xetyei for minus at all.
The use is certainly foreign to classical Greek. Ptolemy has
in two places Xnyjrav and Xdnova-av respectively followed,
properly, by the accusative, and in one case he has to dirb
7-779 TA Xeicpdev vnb rod dirb tt]s ZT (where the meaning is
ZT 2 — TA 2 ). Hence Heron would probably have written a
participle where the T occurs in the expression quoted above,
say ixovdBcov 08 Xeiyjrao-coi' T€. On the whole,
therefore, it is
probable that in Diophantus, and wherever else
it occurred, A is a compendium for the root of the v§rb Xei7T€ii>,
in fact a A with I placed in the middle (cf. A, an abbreviation
for rdXavTov). This is the hypothesis which I put forward
in 1885, and it seems to be confirmed by the fresh evidence
now available as shown above.
o
*
1
Heron, Metrica, p. 156. 8, 10.