XII.—Professor Kelland's Problem on Superposition

From a given square one quarter is cut off; to divide the remaining gnomon into four such parts that they shall be capable of forming a square. [See Trans. Roy. Soc. Edin., xxi. 271.] Preliminary problem, viz.: To cut a rectangle into three parts capable of forming a square (Figs. 1, 2). Lay off AX and CY = side of square, join BX, draw YZ ∥ AB, and 1, 2 and 3 will be the required pieces, as is evident. The problem will be impossible when the length of the rectangle is greater than four times its breadth; four or more pieces may then be required. The problem can be solved in four different ways by drawing the sloping line BX from each of the four angles. The above is a particular case of the more general problem, viz., to cut a rectangle or a parallelogram into three pieces so as to form another with a given side, as AB (Figs. 3, 4). In the gnomon (Figs. 5,6), suppose the square EC placed below in the position BK, then the rectangle AK is formed, and can be divided into three pieces by FX and YZ, so as to form a square as before.