The Rule of False Position and Geometric Problems - The Rule of False Position

The rule of false position, regula falsi, rule of one false position, or simple rule of false position was very popular in mathematics texts of the sixteenth century, and was still present in some books of elementary mathematics in the first half of the twentieth century [8]. Lumpkin [2] traces the development of the method of false position from its uses in ancient Egypt, the Hellenistic world, medieval Islam, to its transmission to Europe and subsequent appearance in mathematical texts in the eighteenth century, to Benjamin Banneker’s advancement of the rule of false position through his own efforts.

In general, regula falsi was used to solve equations of first degree with one unknown, without using algebraic symbolic notation. The statements of problems that were solved using the rule of false position can be translated to an equation of the type ax = b or more precisely, a₁x + a₂x + . . . + a_nx = b.

We describe the simple rule of false position with modern algebraic notation:

Suppose we want to solve the equation ax = b (1).

If we make x = c, then ac = b₁ (2).

There are two possibilities:

• If b₁ equals b, then x = c is the solution of the equation.
• If b₁ does not equal b, dividing equation (1) by equation (2) gives x/c = b/b₁, so x = bc/b₁.

Regula falsi can be justified geometrically using the following figure:

Using similar triangles we have c/b₁ = x/b; therefore x = bc/b₁.