A history of Greek mathematics - Wilbourhall.org

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468 DIOPHANTUS OF ALEXANDRIA
substituting in the original equation 1 + x for x and (q
— kx)
for y, where k is some integer.
3. Form Ax 2 + Bx + C=y 2 .
This can be reduced to the form in which the second term is
wanting by replacing x by z — —j
•
2 XL
Diophantus, however, treats this case separately and less
fully. According to him, a rational solution of the equation
Ax' 1 + Bx +C = y
2
is only possible
(a) when A is positive and a square, say a 2 ;
(/?) when C is positive and a square, say c 2 ;
(y)
when ^B 2 — AC is positive and a square.
In case (a) y is put equal to (ax — m), and in case (/3)
equal to (mx — c).
y is put
Case (y) is not expressly enunciated, but occurs, as it
were, accidentally (IV. 31). The equation to be solved is
3 x + 1 8 — x 2 = y
2
. Diophantus first assumes 3 x + 1 8 — x 2 = 4 x 2 ,
which gives the quadratic 3^+18 = 5 sc 2 ;
but this 'is not
rational '. Therefore the assumption of 4 x 2 for y 2 will not do,
'
and we must find a square [to replace 4] such that 1 8 times
(this square + 1 ) 4- (f )
2
may be a square '. The auxiliary
equation is therefore 18(m 2 + 1) + § = y
2
, or 7 2 m 2 + 81= a
square, and Diophantus assumes 72 m 2 + 8 1 = (8 m + 9)
2
, whence
m= 18. Then, assuming 3 x + 18 — x 2 = (1 8) 2 ^ 2 , he obtains the
equation 325
(2) Double equation.
2
— 3x— 18 = 0, whence x — /2t> ^na^ is > 2V
The Greek term is SLTrXoicroT-qs, SlttXtj IcroT'qs or SlttXyj io-cdctis.
Two different functions of the unknown have to be made
simultaneously squares. The general case is to solve in
rational
numbers the equations
mx 2 + oc x + a — u 2 j
nx 2 -\- fix+ b = w 2 )
The necessary preliminary condition is
that each of the two
expressions can be made a square. This is always possible
when the first term (in x 2 ) is wanting. We take this simplest
case first.