The method of false position

Long before equations were written with letters and equals signs, the ancient Egyptians already knew how to solve problems that we would translate into first-degree equations. The most famous witness is the Rhind papyrus (or Ahmes papyrus), a scroll more than five metres long copied around 1650 BC by the scribe Ahmes from an even older original. It is a collection of 84 practical problems — bread, beer, land, fodder — and in many of them an unknown appears, called aha, meaning “heap” or “quantity”.

A problem from the papyrus

Problem 24 of the papyrus asks, in essence:

A quantity and its seventh part, added together, give $19$. What is the quantity?

With our symbolism we would simply write $x + \frac{1}{7}x = 19.$

The Egyptians had no such notation, and instead used the method of false position: one picks a convenient value for the unknown, computes the result it produces, and then corrects it with a proportion.

How it works

It pays to choose a “false” value that makes the arithmetic easy: here $7$, because its seventh part is a whole number. Substituting $x = 7$: $7 + \frac{1}{7}\cdot 7 = 7 + 1 = 8.$

The trial gives $8$, but we want $19$. Since the expression $x + \tfrac{1}{7}x$ is proportional to $x$, it suffices to multiply the false value by the correction factor $\dfrac{19}{8}$: $x = 7 \cdot \frac{19}{8} = \frac{133}{8} = 16 + \frac{5}{8}.$

And indeed $16\tfrac{5}{8} + \tfrac{1}{7}\cdot 16\tfrac{5}{8} = 19$: the answer is exact. The method works because the equation is linear and homogeneous in the unknown — doubling $x$ doubles the left-hand side too — so the ratio between the desired result and the obtained result is exactly the ratio between the true solution and the false value.

Why it still matters

False position shows that the idea of “isolating the unknown” with the principles of equivalence is not the only way to reason: for centuries, from the Egyptians to medieval algebra, problems were solved by guesses guided by proportionality. Only with modern symbolism did the procedure become the mechanical manipulation we know today. It is worth noting that the method, as it stands, works only for equations of the type $ax = b$: for equations with an additive constant term (such as $ax + c = b$) one needs the double false position variant, developed later.

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Topics: First-degree equations
Concepts: Equation · Unknown
People: Ahmes