Problem 22 - A History of Mathematics: An Introduction

Question

Show that the area of the Babylonian "bull's-eye" is given by \(A=(9 / 32) a^{2}\), where \(a\) is the length of the arc (one-third of the circumference). Also show that the length of the long transversal of the bull's-eye is \((7 / 8) a\), whereas the length of the short transversal is \((1 / 2) a\). (Use the Babylonian values of \(C^{2} / 12\) for the area of a circle and \(7 / 4\) for \(\sqrt{3}\).)

Step-by-Step Solution

Answer

Answer: The area of the Babylonian bull's-eye is given by \(A =\frac{9}{32}a^2\), where \(a\) is the length of the arc. The length of the long transversal is approximately \((\frac{7}{8})(1.154\sqrt{A})\), and the length of the short transversal is approximately \((\frac{1}{2})(1.154\sqrt{A})\).

Step 1: Find the area of the Babylonian bull's-eye

To find the area of the bull's-eye, we will use the Babylonian formula \(A = \frac{9}{32}a^2\). Since \(a\) is the length of the arc (one-third of the circumference), we find the area using the given values. \(A = \frac{9}{32}a^2\)

Step 2: Use the Babylonian value for the area of a circle to find the circumference

Using the Babylonian value for the area of a circle \(A = \frac{C^2}{12}\), where \(A\) is the area and \(C\) is the circumference, we can express the circumference in terms of the area: \(C = \sqrt{12A}\)

Step 3: Find the length of the arc in terms of the circumference

Since the length of the arc \(a\) is one-third of the circumference, we can express \(a\) in terms of \(C\): \(a = \frac{1}{3}C\)

Step 4: Substitute the circumference to find the length of the arc in babylonaining terms of the area

Substitute the expression for \(C\) from Step 2 into the expression for \(a\) from Step 3 to find the length of the arc in terms of the area: \(a = \frac{1}{3}\sqrt{12A}\)

Step 5: Find the length of the long transversal

We will use the Babylonian value for \(\sqrt{3} = \frac{7}{4}\): Length of the long transversal \(= (\frac{7}{8}) a = (\frac{7}{8})(\frac{1}{3}\sqrt{12A}) = (\frac{7}{8})(\frac{1}{3}(3.46\sqrt{A})) \approx (\frac{7}{8})(1.154\sqrt{A})\)

Step 6: Find the length of the short transversal

Using the same value for \(\sqrt{3}\) and the arc length, we can find the length of the short transversal: Length of the short transversal \(= (\frac{1}{2})a = (\frac{1}{2})(\frac{1}{3}\sqrt{12A}) = (\frac{1}{2})(\frac{1}{3}(3.46\sqrt{A})) \approx (\frac{1}{2})(1.154\sqrt{A})\) #Summary# The area of the Babylonian bull's-eye is given by \(A =\frac{9}{32}a^2\), where \(a\) is the length of the arc (one-third of the circumference). The length of the long transversal is approximately \((\frac{7}{8})(1.154\sqrt{A})\), and the length of the short transversal is approximately \((\frac{1}{2})(1.154\sqrt{A})\).

Key Concepts

Babylonian Geometry

Babylonian geometry is both fascinating and innovative. Ancient Babylonians were known for their arithmetical approach to geometry. They used practical methods and approximations that simplified complex calculations. Instead of pi, which is crucial in today’s geometry, they used simpler ratios and values to make calculations feasible.

• The Babylonians' deep understanding of squares and rectangles was applied in various geometrical contexts.
• They introduced the concept of a "bull's-eye," which pushed the boundaries of their geometrical explorations.

This clever approach, although approximate, laid significant groundwork for future developments in mathematical geometry.

Area Calculation

Calculating the area of shapes is crucial in geometry, and Babylonians had their own methodology. To calculate the area of their "bull's-eye," they used a formula that differed from the modern method.

• Their bull's-eye formula used the Babylonian value of \(\frac{C^2}{12}\) for the area of a full circle.
• By manipulating this formula, the area of the "bull's-eye" could be computed as \(A = \frac{9}{32}a^2\).

This formula highlights the Babylonian's innovative adaptation of known formulas to suit their geometric figures and concepts.

Transversal Lengths

Understanding transversal lengths involves special calculations, particularly in Babylonian geometry. A transversal is a line or system of lines that cross at different points.

• In the context of the "bull's-eye," the long transversal is calculated as \((\frac{7}{8}) a\).
• The short transversal measures \((\frac{1}{2}) a\).

These lengths were derived using their estimate of \(\sqrt{3}\) as \(\frac{7}{4}\), showcasing their ability to break down complex geometric figures into understandable parts. This system reflects their knack for practical yet profound mathematical solutions.