A history of Greek mathematics - Wilbourhall.org

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.
the
Thus A Y i M fi$K kv fiopicp A Y A aM^A Y ^
= (60#
2
+ 2520)/(a 4 + 900-60a 2 ),
[VI. 12]
O
and A ie A M A$- 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