Al-B?r?ni devised a method for determining the radius r of...

Al-B?r?ni devised a method for determining the radius $r$ of the earth by sighting the horizon from the top of a mountain of known height $h$. That is, al-B?r?ni assumed that one could measure $\alpha$, the angle of depression from the horizontal at which one sights the apparent horizon (Fig. 9.38). Show that $r$ is determined by the formula $$ r=\frac{h \cos \alpha}{1-\cos \alpha} $$ Al-B?r?n? performed this measurement in a particular case, determining that $\alpha=0^{\circ} 34^{\prime}$ as measured from the summit of a mountain of height $652 ; 3,18$ cubits. Calculate the radius of the earth in cubits. Assuming that a cubit equals $18^{\prime \prime}$, convert your answer to miles and compare to a modern value. Comment on the efficacy of al-Bir?ni's procedure.

Al-B?r?ni devised a method for determining the radius $r$ of the earth by sighting the horizon from the top of a mountain of known height $h$. That is, al-B?r?ni assumed that one could measure $\alpha$, the angle of depression from the horizontal at which one sights the apparent horizon (Fig. 9.38). Show that $r$ is determined by the formula
$$
r=\frac{h \cos \alpha}{1-\cos \alpha}
$$
Al-B?r?n? performed this measurement in a particular case, determining that $\alpha=0^{\circ} 34^{\prime}$ as measured from the summit of a mountain of height $652 ; 3,18$ cubits. Calculate the radius of the earth in cubits. Assuming that a cubit equals $18^{\prime \prime}$, convert your answer to miles and compare to a modern value. Comment on the efficacy of al-Bir?ni's procedure.

Key Concepts

Unit Conversion and Dimensional Analysis

Unit conversion is essential when measurement units differ from the ones used in modern contexts. Converting cubits to inches and then to miles requires careful dimensional analysis to ensure consistency and accuracy. This step is necessary to compare historical measurements with contemporary values of the Earth’s radius.

Historical Measurement and Efficacy Analysis

This concept involves evaluating the precision and limitations of al-Biruni’s method in measuring the Earth’s radius. It encompasses a discussion of measurement accuracy, potential sources of error, and the inherent challenges of observing very small angles (e.g., 0° 34') from high altitudes. This analysis provides insight into the ingenuity of early scientific methods and compares them with modern measurement techniques.

Angle of Depression

The angle of depression is defined as the angle between a horizontal line through the observer’s eye and the line of sight down to the horizon. In the context of determining the Earth’s radius, this small angle is measured from a known altitude and is used in trigonometric relationships to connect the observer’s height and the Earth's curvature.

Tangent Line Geometry

This concept involves understanding that when an observer looks at the horizon from an elevated point, the line of sight is tangent to the Earth’s surface modeled as a circle. At the point of tangency, the radius of the circle is perpendicular to the tangent line. This geometric property is crucial for establishing the relationship between the height above the Earth and the angle at which the horizon is observed.

Trigonometric Relationships

Trigonometry provides the tools to relate angles and distances in the circle geometry of the Earth. In this procedure, the cosine of the angle of depression is directly related to the ratio of the observer's height and the Earth’s radius. This relationship is captured in the formula r = (h cos ?) / (1 - cos ?), which is derived using properties of right triangles and the circle’s geometry.

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