DETERMINATE EQUATIONS 
463 
equation and neutralize negative terms by adding to both 
sides, then take like from like again, until we have one term 
left equal to one term. After these operations have been 
performed, the equation (after dividing out, if both sides 
contain a power of x, by the lesser power) reduces to Ax m — B, 
and is considered solved. Diophantus regards this as giving 
one root only, excluding any negative value as ‘ impossible 
No equation of the kind is admitted which does not give 
a ‘ rational ’ value, integral or fractional. The value x = 0 is 
ignored in the case where the degree of the equation is reduced 
by dividing out by any power of x. 
(2) Mixed quadratic equations. 
Diophantus never gives the explanation of the method of 
solution which he promises in the preface. That he had 
a definite method like that used in the Geometry of Heron 
is proved by clear verbal explanations in different propositions. 
As he requires the equation to be in the form of two positive 
terms being equal to one positive term, the possible forms for 
Diophantus are 
(a) mx 2 + r px = q, (h) mx 2 = px + q, (c) mx 2 + q=px. 
It does not appear that Diophantus divided by m in order to 
make the first term a square; rather he multiplied by m for 
this purpose. It is clear that he stated the roots in the above 
cases in a form equivalent to 
( a , ~hV+ J (ip^+mq) 
iP + ^ (i p 2 + mq) 
[0) 5 
x 7 ri'Yi. 
( c \ iP + ^dp 2 -™g) 
' 7 /m 
The explanations which show this are to be found in VI. 6, 
in IV, 39 and 31, and in Y. 10 and VI. 22 respectively. For 
example in V. 10 he has the equation 17îc 2 + 17 < 72x, and he 
says ‘ Multiply half the coefficient of x into itself and we have 
1296; subtract the product of the coefficient of x 2 and the 
term in units, or 289. The remainder is 1007, the square root 
of which is not greater than 31. Add half the coefficient of x 
and the result is not greater than 67. Divide by the coefficient 
of x 2 , and x is not greater than fv-’ I n IV- 39 he has the