Answer:
The field must have contained between 179 and 180
acres—to be more
exact, 179.37254 acres.
Had the measurements been 3, 2, and 4 furlongs
respectively from successive corners, then the field would have been
209.70537 acres in area.
One method of solving this problem is as follows.
Find the
area of
triangle APB in terms of x, the side of the square.
Double the
result=xy.
Divide by x and then
square, and we have the value of y^{2} in terms of x.
Similarly find value of z^{2}
in terms of x;
then solve the equation y^{2}+z^{2}=3^{2},
which will come out in the
form x^{4}20x^{2}=37.
Therefore x^{2}=10+(sqrt{63})=17.937254
square furlongs, very nearly, and as there are ten acres in one square
furlong, this equals 179.37254 acres.
If we take the negative root of
the
equation, we get the area of the field as 20.62746 acres, in which case
the treasure would have been buried outside
the field, as in Diagram 2.
But this solution is excluded by the condition that the treasure was
buried in the field.
The words were, "The document ... states clearly
that the field is square, and that the treasure is buried in it."
