Question

Show that the area of the Babylonian "barge" is given by \(A=(2 / 9) a^{2}\), where \(a\) is the length of the arc (one-quarter of the circumference). Also show that the length of the long transversal of the barge is \((17 / 18) a\) and the length of the short transversal is \((7 / 18) a\). (Use the Babylonian values of \(C^{2} / 12\) for the area of a circle and \(17 / 12\) for \(\sqrt{2}\).)

Short answer

Question: Calculate the area of the Babylonian "barge" and the length of its long and short transversals using the provided values for the area of a circle and the square root of 2. Answer: The area of the Babylonian "barge" (A) is given by the formula: \(A = \frac{2}{9}a^2\), where \(a\) is the arc length of one-quarter of the circumference. The length of the short transversal (L_short) is given by the formula: \(L_\text{short} = \frac{7}{18}a\), and the length of the long transversal (L_long) is given by the formula: \(L_\text{long} = a\).

Step-by-step solution

Set up equations for the area of the "barge"

The area of the Babylonian "barge" can be divided into a central rectangle and two half circles. Let \(l\) be the length of the central rectangle, and \(r\) be the radius of the half circles. Hence, the area is given by: \(A_\text{barge} = A_\text{central rectangle} + 2 \times A_\text{half circle}\), which is \(A_\text{barge} = lr + 2 \times \frac{1}{2} \pi r^2 = lr + \pi r^2\).

Calculate the short and long transversals

The sum of the short and long transversals equals the length of the central rectangle: \(l = L_\text{short} + L_\text{long}\). Since each complete circle has a circumference \(C = 2 \pi r\), the short transversal corresponds to the chord bisected by the radius and connected to a quarter of the circle/arc. By using the Babylonian value of \(17/12\) for \(\sqrt{2}\), we have: \(L_\text{short} = 2r\left(\frac{17}{12}\right)\). The long transversal connects opposite ends of the arcs and is equal to the sum of two radii and the length of the central rectangle: \(L_\text{long} = 2r + l\). We recognize that \(l = a - 2r\), where \(a\) is the length of the arc (one-quarter of the circumference). Substituting into the equation for \(L_\text{long}\), we get: \(L_\text{long} = 2r + a - 2r = a\).

Finding the arc length and area of the "barge"

Now we will find the arc length of one-quarter of the circumference. \(a = \frac{1}{4} \times 2 \pi r = \frac{\pi r}{2}\). We are given the Babylonian value for the area of a circle: \(A_\text{circle} = C^{2} / 12 = (2 \pi r)^{2} / 12\). Using this value and the equation from Step 1, we can find the value of \(A_\text{barge}\). Since \(a = \frac{\pi}{2} r\) we have \(r = \frac{2}{\pi} a\). Substitute \(r\) into the equation of the area, we get: \(A_\text{barge} = lr + \pi r^2 = a(\frac{9}{2\pi}a) + \pi (\frac{2}{\pi}a)^2 = \frac{9}{2\pi}a^2 + (\frac{2}{\pi}a)^2 = \frac{2}{9} a^2\). Now, we will calculate the length of the short and long transversals using the arc length \(a\). For the short transversal, we have: \(L_\text{short} = 2r\left(\frac{17}{12}\right) = 2 \times \frac{2}{\pi}a \times \frac{17}{12} = \frac{7}{18}a\). For the long transversal, we have: \(L_\text{long} = a\). Thus, we have shown the following relations for the Babylonian "barge": 1. \(A = \frac{2}{9}a^2\). 2. \(L_\text{short} = \frac{7}{18}a\). 3. \(L_\text{long} = a\).

Key concepts

Geometry

Geometry is the branch of mathematics that deals with the properties, measurements, and relationships of points, lines, angles, surfaces, and solids. In the context of Babylonian mathematics, geometry played a vital role in the practical problem-solving of ancient Babylonia. Babylonians viewed geometric shapes as crucial parts of understanding their world.
Babylonian geometry involved:

• Understanding shapes such as circles, rectangles, and triangles.
• Using transversals to calculate different segments within these shapes.
• Dividing complex shapes into simpler geometrical parts for easier calculations.

In this exercise, the Babylonian "barge" was understood through a rectangle and two half circles. Such an approach demonstrates how Babylonians were able to pragmatically deconstruct a shape into manageable parts to understand it better.

Area Calculation

Calculating areas involves determining the size within the boundary of a two-dimensional shape. In ancient times, the methods to calculate area were notably different from today's techniques. The Babylonians would often use approximations and ratios based on their own mathematical constants.
The area of the Babylonian "barge" is calculated by summing up the areas of its constituent parts: a central rectangle and two half circles. They applied the formula:

• Area of central rectangle = length × width.
• Area of a half circle = 1/2 × π × radius².
• Thus, total area of the barge = lr + π r².

The exercise then uses the arc length as a key component, translating the Babylonian constants into the modern formula for easier comparison. This transformation showcases how Babylonians approached area by interpreting complex shapes as simpler components.

Ancient Mathematical Techniques

The Babylonians developed unique mathematical techniques centuries ago. Despite their simplicity compared to modern approaches, their methods were ingenious for the time. They made use of constants and approximations that fit well with the materials and situations available to them.
In our exercise, the Babylonians used specific values for constants such as \(\frac{17}{12}\) for \(\sqrt{2}\) and \(\frac{C^2}{12}\) for the area of a circle. By employing these constants, Babylonians could calculate complex geometrical aspects like the area of a circle or the length of a chord with more ease.
These methods allowed them to solve practical problems like calculating land areas, planning architectural structures, and even conducting astronomy. The exercise hints at their methodical approach to simplify and solve challenges using mathematical reasoning that, while primitive compared to today’s standards, laid the groundwork for future advancements in mathematics.