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 dpiOpos and no 
algebraical symbol is shown by the fact that it occurs in the 
manuscripts for dpidpos in the ordinary sense as well as for 
dpiOpos 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 mathe 
maticians, 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 kul), in the manuscripts of 
the Sand-reckoner of Archimedes (in a form' approximat 
ing to ?), where again there is confusion caused by the 
similarity of the signs for dptdpos and koll, 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 1 ? 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 larger than and quite unlike the 
final sigma. 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 dpiOpos. This seemed to be confirmed by 
Gardthausen’s mention of a contraction for ap, in the form ap 
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 dpidpos similar to the signs for the powers of 
the unknown, e.g. A' for Svvapus, for kv/Sos, and to the 
sign M for the unit, the sole difference being that the two 
letters coalesce into one instead of being separate.