The diagram shows how to cut into five pieces to form a square. 
The dotted lines are intended to show how to find the points C and F—the only difficulty. A B is half B D, and A E is parallel to B H. 
With the point of the compasses at B describe the arc H E, and A E will be the distance of C from B. 
Then F G equals B C less A B.

This puzzle—with the added condition that it shall be cut into four parts of the same size and shape—I have not been able to trace to an earlier date than 1835. 
Strictly speaking, it is, in that form, impossible of solution; but I give the answer that is always presented, and that seems to satisfy most people.

We are asked to assume that the two portions containing the same letter—AA, BB, CC, DD—are joined by "a mere hair," and are, therefore, only one piece. 
To the geometrician this is absurd, and the four shares are not equal in area unless they consist of two pieces each. 
If you make them equal in area, they will not be exactly alike in shape.