456 
DIOPHANTUS OF ALEXANDRIA 
literary style, but marked by the use of certain abbreviational 
symbols for constantly recurring quantities and operations. 
To this stage belong Diophantus and, after him, all the later 
Europeans until about the middle of the seventeenth century 
(with the exception of Yieta, who was the first to establish, 
under the name of Logistica speciosa, as distinct from Logistica 
numerosa, a regular system of reckoning with letters denoting 
magnitudes as well as numbers). (3) To the third stage 
Nesselmann gives the name of ‘ Symbolic Algebra ’, which 
uses a complete system of notation by signs having no visible 
connexion with the words or things which they represent, 
a complete language of symbols, which entirely supplants the 
‘rhetorical’ system, it being possible to work out a solution 
without using a single word of ordinary language with the 
exception of a connecting word or two here and there used for 
clearness’ sake. 
Sign for the unknown (= x), and its origin, 
Diophantus’s system of notation then is merely abbrevia 
tional. We will consider first the representation of the 
unknown quantity (our x). Diophantus defines the unknown 
quantity as ‘ containing an indeterminate or undefined multi 
tude of units ’ (ttA fjdo9 povd8a»v dopurrou), adding that it is 
called dpiOpos, i.e. number simply, and is denoted by a certain 
sign. This sign is then used all through the book. In the 
earliest (the Madrid) MS. the sign takes the form *-(, in 
Marcianus 308 it appears as S. In the printed editions of 
Diophantus before Tannery’s it was represented by the final 
sigma with an accent, y', which is sufficiently like the second 
of the two forms. Where the symbol takes the place of 
inflected forms dpidpov, dptdpov, &c., the termination was put 
above and to the right of the sign like an exponent, e.g. y" for 
dpi6pov as r" for tov, y° 5 for dpidpov’, the symbol was, in 
addition, doubled in the plural cases, thus yy°‘, yy 0 " ç , &c. The 
coefficient is expressed by putting the required Greek numeral 
immediately after it ; thus yy° l La = 11 dpi.6p.0L, equivalent 
to 11 x, y' a. = x, and so on. Tannery gives reasons for think 
ing that in the archetype the case-endings did not appear, and