A history of Greek mathematics - Wilbourhall.org

1
464 DIOPHANTUS OF ALEXANDRIA
equation 2& 2 >6#+18 and says, 'To solve this, take the square
of half the coefficient of x, i.e.
9, and the product of the unitterm
and the coefficient of x 2 , i.e. 36. Adding, we have 45,
the square root of which is not less than 7. Add half the
coefficient of x [and divide by the coefficient of & 2 ]
whence x
;
is not less than 5.' In these cases it will be observed that 3
and 7 are not accurate limits, but are the nearest integral
limits which will serve his purpose.
Diophantus always uses the positive sign with the radical,
and there has been much discussion as to whether he knew
that a quadratic equation has two roots.
The evidence of the
text is inconclusive because his only object, in every case, is to
get one solution ; in some cases the other root would be
negative, and would therefore naturally be ignored as 'absurd'
or ' impossible '. In yet other cases where the second root is
possible it can be shown to be useless from Diophantus's point
of view. For my part, I find it difficult or impossible to
believe that Diophantus was unaware of the existence of two
real roots in such cases.
It is so obvious from the geometrical
form of solution based on Eucl. II. 5, 6 and that contained in
Eucl. VI. 27-9; the construction of VI. 28, too, corresponds
in fact to the negative sign before the radical in the case of the
particular equation there solved, while a quite obvious and
slight variation of the construction would give the solution
corresponding to the 'positive sign.
The following particular cases of quadratics occurring in
the Arithmetica may be quoted, with the results stated by
Diophantus.
x 2 4x — 4 ; therefore x = 2. (IV. 22)
325a; 2 = 3&+18
« = *7 Aor A. (IV. 31)
8ix 2 + 7x = 7
84x 2 —7x — 7
630o; 2 -73x = 6
630ar+73a? = 6
x = i. (VI. 6)
= 1. (VI. 7)
»=.&• (VI. 9)
x is rational. (VI. 8)
5x < x 2 -60 < 8x; x not < 11 and not > 12. (V. 30)
\7x 2 +\7 < 72&f| and not