Imagine sitting in a classroom 400 years ago. Algebraic notation has not yet been invented. Your math teacher gives the following problem: about Spanish merchants paying customs, or dues:

"Two merchants carry silk, one 70 lbs; the other 200 pounds: and arriving at a port where they had to pay a certain due: the one of the 70 pounds, because he had no money, gives to the due one pound of silk, and the ones of the due [or custom officials] give him back 32 salaries. And the one with the 200 pounds pays of the duty one pound of silk and 20 salaries. I request, how much did the ones of the duty [or customs officers] reckon one pound of the silk, paying the two at one rate?" (Meaville & Flores, 2005, p. 257)

Today we would solve it using algebra and a system of linear equations. The ancients had a simple arithmetic quick method to solve these kinds of problems.

The entire twelfth Chapter of Fibonacci’s work "Liber Abaci" is devoted to a mathematical arithmetic method he called the el-chataym, using the Arabic term for this technique.

This method is widely known as the rule of false position and double false position in English. It relies on the scholar making a convenient guess of what the unknown value could be, then leveraging the guess to find the true value. Students do this today by using 100 as the unknown value in a word problem involving percentages. Indeed it is intuitive enough that it is consistently reinvented by students. In the false position method, a guess is made and the problem calculated. The incorrect answer is then scaled to the correct answer using a ratio, and then that ratio is applied to the guess to arrive at the correct value.

The rule of double false position is slightly more complicated as two guesses and two errors must be cross-multiplied and subtracted from each other, and then divided by the difference of the errors.

The mathematical principle involved can be proved using the properties of similar triangles.

The earliest known examples of False Position are from the Ahmes Papyrus found by the Scotsman Alexander Henry Rhind. This mathematical textbook has been dated to 1650 B.C.. Some scholars claim that the Chinese Tshuifen and Ying Buzu Shu (or methods of surplus and deficiency) predate the Ahmes papyrus, as traditions state it was written in the 27th century B.C. during the reign of the yellow emperor. But as the originals were destroyed during the book burnings in the late 2nd century BC. there is controversy with some favoring the traditional date and others placing it closer to at 3rd century B.C.

Before algebraic notation, word problems were complex and lengthy, and using a guess solved some practicality in working with lengthy word problems, that we can know symbolically represent both the content and the relationships between the numbers in a line or two.

Roberte Spears used to amaze the locals by solving complex mathematical riddles using guesses from his audience in the mid 1500s by using this method.

Although the method was still found in mathematical textbooks in the early 1900s, the use of algebraic notation and ease of simplification of complex problems using the notation, the method fell out of favor.

Today, however, the method of double false position, or regula falsi is still used as a numerical analysis method in computations involving the approximations of zeros, while computationally slower, it can sometimes work where Newton’s method, or the secant method fail, particularly where a function is not differentiable, on a corner or a cusp.
References