# A piece of the Berlin Papyrus

Photography by Carlos Dorce

The Berlin Papyrus 6619 (1800 BC) is one of the only surviving witness which demonstrates that the Egyptian escribes knew how to solve certain quadratic equations.

The first problem in the papyrus says: You are told the area of a square of 100 square cubits is equal to that of two smaller squares, the side of one square is 1/2 + 1/4 of the other. What are the sides of the two unknown squares? That is:

x2 + y2 = 100
4x – 3y = 0

There also is a second similar problem equivalent to the quadratic system:

x2 + y2 = 400
4x – 3y = 0

The solving method is the rule of false position. The escribe assumed that x = 0,75 and y = 1 so x2 + y2 = 1,5625. But the result should be 100 = 64 · 1,5625! Therefore, our two squares must be 64 times bigger and their sides must be 8 times bigger. So the result is x = 0,75 · 8 = 6 units and y = 1 · 8 = 8 units, and  x2 + y2 = 100.

This papyrus becames unnotices in the Neues Museum of Berlin due to its close position to the famous bust of Nefertiti:

Source: Wikimedia Commons

But dont’t leave the museum without giving attention to this important mathematical document.

Location: Neues Museum (map)