Babylonian Mathematics/Examples/Division of Triangular Field

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Examples of Babylonian Mathematics

A triangular field is to be divided between $6$ brothers by equidistant lines parallel to one of the sides.

Expressed in Babylonian notation:

the length of the marked side is $6; 30$
the area of the triangle is $11, 22; 30$.

What is the difference between the brothers' shares?


Solution

The difference between each successive share is:

$37; 55$ in Babylonian notation
$37 \frac {11} {12}$ in mixed fractions.


Proof

Division-of-Triangular-Field.png

Let $\triangle ABC$ be the triangular field in question.

Let $d$ be the marked side.

Let $a$ be the side which is parallel to the dividing lines.

Let $\AA$ be the total area of $ABC$.


Let $\AA_1, \AA_2, \ldots, \AA_6$ be the areas of each of the divisions of $ABC$ such that $\AA_1 > \AA_2 > \cdots > \AA_6$.

Let $a_1, a_2, \ldots, a_5$ denote the dividing lines such that $a_1 > a_2 > \cdots > a_5$.


From Area of Triangle, we have that:

$\AA = \dfrac 1 2 k d a$

where $k = \sin \angle CAB$.

From Area of Trapezoid:

\(\ds \AA_1\) \(=\) \(\ds \dfrac {\paren {a + a_1} k d} {2 \times 6}\) Area of Trapezoid
\(\ds \AA_j\) \(=\) \(\ds \dfrac {\paren {a_j + a_{j - 1} } k d} {2 \times 6}\) for $j = 2$ to $5$
\(\ds \AA_6\) \(=\) \(\ds \dfrac {a_5 k d} {2 \times 6}\) Area of Triangle

We have:

\(\ds a_j\) \(=\) \(\ds a \paren {1 - \dfrac j 6}\) for $j = 1$ to $5$

and so:

\(\ds \AA_{j - 1} - \AA_j\) \(=\) \(\ds \dfrac {\paren {a_{j - 1} + a_{j - 2} } k d} {2 \times 6} - \dfrac {\paren {a_j + a_{j - 1} } k d} {2 \times 6}\)
\(\ds \) \(=\) \(\ds \dfrac {\paren {a_{j - 2} - a_j} k d} {2 \times 6}\)
\(\ds \) \(=\) \(\ds \dfrac {\paren {a \paren {1 - \dfrac {j - 2} 6} - a \paren {1 - \dfrac j 6} } k d} {2 \times 6}\)
\(\ds \) \(=\) \(\ds \dfrac {a k d} {36}\) after tedious algebra
\(\ds \) \(=\) \(\ds \dfrac {\AA} {18}\) substituting for $\AA$

We have that:

\(\ds \AA\) \(=\) \(\ds 11, 22; 30\)
\(\ds \) \(=\) \(\ds 11 \times 60 + 22 + \dfrac 1 2\) expressing in decimal notation
\(\ds \leadsto \ \ \) \(\ds \dfrac {\AA} {18}\) \(=\) \(\ds \dfrac {682} {18} + \dfrac 1 {36}\)
\(\ds \) \(=\) \(\ds 37 + \frac 8 9 + \dfrac 1 {36}\)
\(\ds \) \(=\) \(\ds 37 + \frac {32} {36} + \dfrac 1 {36}\)
\(\ds \) \(=\) \(\ds 37 + \frac {33} {36}\)
\(\ds \) \(=\) \(\ds 37 + \frac {11} {12}\)
\(\ds \) \(=\) \(\ds 37 + \frac {55} {60}\)
\(\ds \) \(=\) \(\ds 37; 55\)


Note that it is not necessary to know how long the marked side is.

$\blacksquare$


Sources

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