The method of "Double False Position" is an arithmetic approach to solving linear equations that pre-dates current algebraic methods by more than 3,000 years. The method applies to problems that, in algebraic notation, would be expressed as y = L(x), where L(x) is a linear function of x. Double False Position works by evaluating the described function with approximations/conjectures used for the input argument. These approximations, together with their resulting output values, are regarded as false positions (i.e., not equal to the solution). The underlying proportionality between changes in input and output in a linear relationship is used to move from the false positions to the solution. (Contains 2 figures.)