# Babylonian Mathematics/Examples/Pythagorean Triangle whose Side Ratio is 1.54

## Example of Babylonian Mathematics

Consider a Pythagorean triangle whose hypotenuse and one leg are in the ratio $1.54 : 1$.

What are the lengths of that hypotenuse and that leg?

## Solution

The lengths in question are $829$ and $540$.

## Proof

Let $a$, $b$ and $c$ be positive integers such that $a^2 + b^2 = c^2$ and such that $1.54 \times a = c$.

Without loss of generality, suppose $a$ is even.

From Solutions of Pythagorean Equation, there exist positive integers $p$ and $q$ such that:

\(\ds a\) | \(=\) | \(\ds 2 p q\) | ||||||||||||

\(\ds b\) | \(=\) | \(\ds p^2 - q^2\) | ||||||||||||

\(\ds c\) | \(=\) | \(\ds p^2 + q^2\) |

Hence it follows that:

- $\dfrac c a = \dfrac 1 2 \paren {\dfrac p q + \dfrac q p}$

The Babylonians would then consult the various standard tables of reciprocals which they used for multiplication.

Without these tables, we set:

- $\dfrac p q = t$

and solve the quadratic equation:

\(\ds \dfrac 1 2 \paren {t + \dfrac 1 t}\) | \(=\) | \(\ds 1.54\) | ||||||||||||

\(\ds \leadsto \ \ \) | \(\ds t^2 + 1\) | \(=\) | \(\ds 3.08 t\) | |||||||||||

\(\ds \leadsto \ \ \) | \(\ds t\) | \(=\) | \(\ds \pm 2.711 \text { or } 0.369\) |

We can discard $0.369$ because we are after $p > q$.

Hence:

\(\ds \dfrac p q\) | \(=\) | \(\ds \dfrac {27} {10}\) | as a rough approximation | |||||||||||

\(\ds \leadsto \ \ \) | \(\ds p\) | \(=\) | \(\ds 27\) | |||||||||||

\(\ds q\) | \(=\) | \(\ds 10\) | ||||||||||||

\(\ds \leadsto \ \ \) | \(\ds a\) | \(=\) | \(\ds 2 p q\) | |||||||||||

\(\ds \) | \(=\) | \(\ds 540\) | ||||||||||||

\(\ds c\) | \(=\) | \(\ds p^2 + q^2\) | ||||||||||||

\(\ds \) | \(=\) | \(\ds 829\) | ||||||||||||

\(\ds \leadsto \ \ \) | \(\ds \dfrac c a\) | \(=\) | \(\ds 1.535\) |

which is what is found in the original Babylonian clay tablet.

$\blacksquare$

## Historical Note

This result appears in *Plimpton $\mathit { 322 }$* as figures $3452$ and $2291$.

## Sources

- 1945: O. Neugebauer and A. Sachs:
*Mathematical Cuneiform Texts*: pp. $38 - 41$ - 1976: Howard Eves:
*Introduction to the History of Mathematics*(4th ed.): p. $37$ - 1992: David Wells:
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