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 ?> ... x [ 
are defined and denoted as follows ; 
x 2 is Svvapis and is denoted by A'', 
X Z „ Kvfdos „ „ 
» K\ 
• 
a: 4 „ SvvapoSvvapLS „ 
„ A y A, 
x 5 „ SvuapÔKv/Sos „ 
<1 
X? „ KvfioKvPoS „ „ 
„ K Y K. 
Beyond the sixth power Diophantus does not go. It should 
be noted that, while the terms from kv(3os onwards may be 
used for the powers of any ordinary known number as well as 
for the powers of the unknown, Svvapis is restricted to the 
square of the unknown; wherever a particular square number 
is spoken of, the term is rerpayoovos dpidpos. The term 
8vua/j.oSvrap.Ls 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, 1/x 2 .... 
As an aliquot part was ordinarily denoted by the corresponding 
numeral sign with an accent, e.g. y'= §, ia = y T , 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 
dpidpoarov, denoted by s*, is equivalent to 1 /x, 
Svvapocrrov, „ A' * „ »1 /, 
and so on. 
The coefficient of the term in x, x 2 ... or 1 /x, \ /x 2 ... is 
expressed by the ordinary numeral immediately following, 
e.g. AK 1 kc; = 26a; 5 , A^ X av — 250/x 2 . 
Diophantus does not need any signs for the operations of 
multiplication and division. Addition is indicated by mere 
juxtaposition; thus K'a A^ iy s e corresponds to ar + 13a: 2 + 5x. 
1 Heron, Metrica, p. 48. 11, 19, Schone.