A history of Greek mathematics Vol.II from Aristarchus to Diophantus by Heath, Thomas Little, Sir, 1921

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.
1
Heron, Metrica, p. 156. 8, 10.
o
*
460 DIOPHANTUS OF ALEXANDRIA
Attached to the definition of minus is the statement that
'a wanting (i.e. a minus) multiplied by a ivanting makes
a forthcoming (i. e. a plus) ; and a wanting (a minus) multiplied
by a forthcoming (a plus) makes a ivanting (a 'minus) '.
Since Diophantus uses no sign for plus, he has to put all
the positive terms in an expression together and write all
negative terms together after the sign for minus ; e.g. for
x z — 5# 2 + 8# — l he necessarily writes K a s ?j A A Y e M a.
The Diophantine notation for fractions as well as for large
numbers has been fully explained with many illustrations
in Chapter II above. It is only necessary to add here that,
when the numerator and denominator consist of composite
expressions in terms of the unknown and its powers, he puts
the numerator first followed by e^ ftopico or uopiov and the
denominator.
Thus A Y i M fi$K kv fiopicp A Y A aM^A Y ^
O
the
= (60#
2
+ 2520)/(a 4 + 900-60a 2 ),
[VI. 12]
and A ie A M A$- kv uopicp A Y A a M A9 A A Y i/3
o
= (15x a -36)/(x* + 36-12x 2 )
[VI. 14].
For a term in an algebraical expression, i.e.
a power of x
with a certain coefficient, and the term containing a certain
number of units, Diophantus uses the word eWo?, 'species',
which primarily means the particular power of the variable
without the coefficient. At the end of the definitions he gives
directions for simplifying equations until each side contains
positive terms only, by the addition or subtraction of coefficients,
and by getting rid of the negative terms (which is done
by adding the necessary quantities to both sides) ; the object,
he says, is to reduce the equation until one term only is left
'
on each side ; but ', he adds, ' I will show you later how, in
the case also where two terms are left equal to one term,
such a problem is solved '.
We find in fact that, when he has
to solve a quadratic equation, he endeavours by means of
suitable assumptions to reduce it either to a simple equation
or a pure quadratic. The solution of the mixed quadratic