# Ancient Babylonian Use of the Pythagorean Theorem and Its Three Dimensions

*By Luis Teia, contributing writer for Ancient-Origins.net*

Very much like today, the Old Babylonians—20th to 16th centuries BC—had the need to understand and use what is now called the Pythagoras’ (or Pythagorean) theorem. They applied it in very practical problems such as to determine how the height of a cane leaning against a wall changes with its inclination. This sounds trivial, but it was one of the most important problems studied at the time.

A remarkable Old Babylonian clay tablet, commonly referred to as Plimpton 322, was found to store combinations of three positive integers that satisfy Pythagoras’ theorem. Today we call them primitive Pythagorean triples where the term primitive implies that the side lengths share no common divisor.

### Why was the tablet built?

Unlike what one may imagine, the reason behind the tablet was not an interest in the number-theoretical question, but rather the need to find data for a ‘solvable’ mathematical problem. It is even believed that this tablet was a ‘teacher’s aide’ for setting up and solving problems involving right triangles. This sounds like an environment not so different from our classrooms today.

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### How is this related to us?

As human beings we share the same nature as the Old Babylonians – in solving problems to live and evolve. The problems nowadays are normally more exotic and elaborate than a cane against a wall, but they share the same legacy. Right-angles are everywhere, whether it is a building, a table, a graph with axes, or the atomic structure of a crystal. While these are our contemporary challenges, we, like the Babylonians, strive to deepen our understanding of the Pythagoras’ theorem, and on the various triples that generate these useful right-angles for our everyday practical applications.

### Were the Babylonians so different from us?

Babylonians may have used algorithms to compute side lengths of right-angled triangles into areas, and vice versa, similar to our contemporary numerical methods of analysis. These areas were farming fields, while the side lengths were canals for irrigation. Maybe the canals were structured to distribute a certain amount of water per canal.

Nowadays, in a very similar manner, computers are used to find the distribution of properties (e.g., stress, deflection, etc.) along a material (e.g., a metal beam), or even the displacement of fluids through volumes (i.e., computational fluid dynamics).

To find a solution to a problem, analytical solutions are often not available. Hence, numerical methods are employed. These consist of splitting the volume that is being analyzed (say the material of a beam, or the air in a room) into small elements (typically Platonic solids like prisms or tetrahedral). It is interesting to think that this so-called “meshing” in the engineering world, or splitting a calculation into small portions, was already applied by the old Babylonians.

Overall, one could say that the tendency to split a problem into pieces, and solve them individually to find the answer, is a human characteristic that we share with the people of Babylon. From this perspective, the Babylonians were not so different from us.

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