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The Buried Treasure

 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 y2 in terms of x.  Similarly find value of z2 in terms of x; then solve the equation y2+z2=32, which will come out in the form x4-20x2=-37.  Therefore x2=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."

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