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DIOPHANTUS OF ALEXANDRIA 
The methods of Diophantus. 
It should be premised that Diophantus will have in his 
solutions no numbers whatever except £ rational ’ numbers ; 
he admits fractional solutions as well as integral, but he 
excludes not only surds and imaginary quantities but also 
negative quantities. Of a negative quantity per se, i.e. with 
out some greater positive quantity to subtract it from, he 
had apparently no conception. Such equations then as lead 
to imaginary or negative roots he regards as useless for his 
purpose ; the solution is in these cases dSvvaro?, impossible. 
So we find him (Y. 2) describing the equation 4 = 4 a;+ 20 as 
(xtottos, absurd, because it would give x = — 4. He does, it is 
true, make occasional use of a quadratic which would give 
a root which is positive but a surd, but only for the purpose 
of obtaining limits to the root which are integers or numerical 
fractions ; he never uses or tries to express the actual root of 
such an equation. When therefore he arrives in the course 
of solution at an equation which would give an ‘irrational’ 
result, he retraces his steps, finds out how his equation has 
arisen, and how he may, by altering the previous work, 
substitute for it another which shall give a rational result. 
This gives rise in general to a subsidiary problem the solution 
of which ensures a rational result for the problem itself. 
It is difficult to give a complete account of Diophantus’s 
methods without setting out the whole book, so great is the 
variety of devices and artifices employed in the different 
problems. There are, however, a few general methods which 
do admit of differentiation and description, and these we pro 
ceed to set out under subjects. 
I. Diophantus’s treatment of equations. 
(A) Determinate equations. 
Diophantus solved without difficulty determinate equations 
of the first and second degrees ; of a cubic we find only one 
example in the Arithmetica, and that is a very special case. 
(1) Pare determinate equations. 
Diophantus gives a general rule for this case without regard 
to degree. We have to take like from like on both sides of an