465 
LI A 
DETERMINATE EQUATIONS 
LI A 
;ake the square 
ict of the unit 
s’, we have 45, 
Add half the 
x 2 \; whence x 
served that 31 
earest integral 
ith the radical, 
ether he knew 
evidence of the 
very case, is to 
root would be 
ired as ‘absurd’ 
; second root is 
phantus’s point 
r impossible to 
dstence of two 
the geometrical 
iat contained in 
oo, corresponds 
i the case of the 
te obvious and 
ve the solution 
cs occurring in 
suits stated by 
2. (IV. 22) 
(IV. 31) 
(VI. 6) 
(VI. 7) 
(VI. 9) 
(VI. 8) 
t > 12. (V. 30) 
lot <f|. (V. 10) 
< 21. (V. 30) 
In the first and third of the last three cases the limits are not 
accurate, but are integral limits which are a fortiori safe. 
In the second f f should have been f|, and it would have been 
more correct to say that, if x is not greater than f-7- and not 
less than f|, the given conditions are a fortiori satisfied. 
For comparison with Diophantus’s solutions of quadratic 
equations we may refer to a few of his solutions of 
(3) Simultaneous equations involving quadratics. 
In I. 27, 28, and 30 we have the following pairs of equations. 
I use the Greek letters for the numbers required to be found 
as distinct from the one unknown which Diophantus uses, and 
which I shall call x. 
In (a), he says, let £ — g = 2x(£>g). 
It follows, by addition and subtraction, that | = a + x, 
g = a~x; 
therefore £g = (a + x) (a — x) = a 2 — x 2 = B, 
and x is found from the pure quadratic equation. 
In (/3) similarly he assumes | — g = 2x, and the resulting 
equation is | 2 + g 2 = {a + x) 2 + {a — x) 2 = 2 (a 2 + x 2 ) = B. 
In (y) he puts £ + g = 2x and solves as in the case of (a). 
(4) Cubic equation. 
Only one very particular case occurs. In VI. 17 the problem 
leads to the equation 
Diophantus says simply ‘ whence x is found to be 4 ’. In fact 
the equation reduces to 
Diophantus no doubt detected, and divided out by, the common 
factor x 2 + 1, leaving x = 4. 
;ake the square 
ict of the unit 
s’, we have 45, 
Add half the 
x 2 \; whence x 
served that 31 
earest integral 
ith the radical, 
ether he knew 
evidence of the 
very case, is to 
root would be 
ired as ‘absurd’ 
; second root is 
phantus’s point 
r impossible to 
dstence of two 
the geometrical 
iat contained in 
oo, corresponds 
i the case of the 
te obvious and 
ve the solution 
cs occurring in 
suits stated by 
2. (IV. 22) 
(IV. 31) 
(VI. 6) 
(VI. 7) 
(VI. 9) 
(VI. 8) 
t > 12. (V. 30) 
l0 t <f|. (V. 10) 
< 21. (V. 30) 
1523.2 
h h