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

NOTATION AND DEFINITIONS 457
that the sign was not duplicated for the plural, although such
duplication was the practice of the Byzantines. That the
sign was merely an abbreviation for the word dpid/169 and no
algebraical symbol is shown by the fact that it occurs in the
manuscripts for dpiOfios in the ordinary sense as well as for
dpiOfios in the technical sense of the unknown quantity. Nor
is it confined to Diophantus. It appears in more or less
similar forms in the manuscripts of other Greek mathematicians,
e.g. in the Bodleian MS. of Euclid (D'Orville 301)
of the ninth century (in
the forms 9 99, or as a curved line
similar to the abbreviation for kcli), in the manuscripts of
the Sand -reckoner of Archimedes (in a form approximating
to y), where again there is confusion caused by the
similarity of the signs for dpiOfios and /cat,
in a manuscript
of the Geodaesia included in the Heronian collections edited
by Hultsch (where it appears in various forms resembling
sometimes £ sometimes p, sometimes o, and once £, with
case- endings superposed) and in a manuscript of Theon of
Smyrna.
What is the origin of the sign? It is certainly not the
final sigma, as is proved by several of the forms which it
takes.
I found that in the Bodleian manuscript of Diophantus
it is written in the form '^4,
final sigma.
larger than and quite unlike the
This form, combined with the fact that in one
place Xylander's manuscript read ap for the full word, suggested
to me that the sign might be a simple contraction of the first
two letters of dpiBfios. This seemed to be confirmed by
Gardthausen's mention of a contraction for ap, in the form up
occurring in a papyrus of a.d. 154, since the transition to the
form found in the manuscripts of Diophantus might easily
have been made through an intermediate form < p. The loss of
the downward stroke, or of the loop, would give a close
approximation to the forms which we know. This hypothesis
as to the origin of the sign has not, so far as I know, been
improved upon. It has the immense advantage that it makes
the sign for dp16/169 similar to the signs for the powers of
the unknown, e.g. A Y for Swapis, K Y for icvfios, and to the
o
sign M for the unit, the sole difference being that the two
letters coalesce into one instead of being separate.
458 DIOPHANTUS OF ALEXANDRIA
Signs for the powers of the unknown and their reciprocals.
The powers of the unknown, corresponding to our x 2 , x ?J . . . x
are defined and denoted as follows
x 2 is Svvctfjus and is denoted by A Y ,
x ?> „ kv/3os „ „ „ K Y ,
X 4 ,, BvvajxoSvvajjLLS „ „ A A,
x 5 „ SvvctfioKvPos „ „ AK
,
x G „ KvfioKvffos „ „ „ K K.
Beyond the sixth power Diophantus does not go. It should
be noted that, while the terms from Kvfios onwards may be
used for the powers of any ordinary known number as well as
for the powers of the unknown, Svuafii? is restricted to the
square of the unknown
;
wherever a particular square number
is spoken of, the term is reTpdyoovos dptOfio?. The term
SwanoSyvajiis occurs once in another author, namely in the
Metrica of Heron, 1 where it is used for the fourth power of
the side of a triangle.
Diophantus has also terms and signs for the reciprocals of
the various powers of the unknown, i.e. for 1/x, l/x 2 ....
As an aliquot part was ordinarily denoted by the corresponding
numeral sign with an accent, e.g. /= J> ia!= tt> Diophantus
has a mark appended to the symbols for x, x 2 . . . to denote the
reciprocals; this, which is used for aliquot parts as well, is
printed by Tannery thus, *. With Diophantus then
dpiOjxoa-Tou, denoted by ?*, is equivalent to l/x,
SwafiocrTOv, „ A „ „ 1 / x 2 ,
and so on.
The coefficient of the term in x, x 2 ... or l/x, l/x 2 ... is
expressed by the ordinary numeral immediately following,
e.g. AK Y /c