1.  Cover a board with the paper on which you want to draw the ellipse. 
2.  Pound two small nails through the paper and into the board.
3.  Tie a piece of string into a loop and place it over the nails.
4.  With a pencil, draw the string up tightly so that it makes a triangle.
5.  Keeping the string tight, move the pencil along the paper.
 

To make an ellipse with specific length and width, i.e., with specific values of M and m in the figure below, we take advantage of a curious relation between these lengths and the distance between the nails.
In terms of the symbols defined below, this relation is written as C2 = M2 - m2. So if you want to make an ellipse for which M = 2 and m = 1, then you must separate the nails by 2c where c2 = 22 - 12 = 3 or c = v3 = 1.72 inches. You would drive your nails in 2(1.72) = 3.44 inches apart. To get the proper length string, just try different lengths until you find one that allows you to make the shaded triangle in the figure below one inch high (m = 1). With these instructions, supplemented perhaps by a little patience and a table of square roots, you can make any size ellipse you want.