Eratosthenes, the fifth librarian of Alexandria ( 230 B.C.),...

Eratosthenes, the fifth librarian of Alexandria ( 230 B.C.), was able to estimate the radius of the earth from the following data: The distance between the Egyptian cities of Alexandria and Syrene was 5000 stadia ( 520 miles). Syrene was located directly south of Alexandria. One summer, at noon, the sun was directly overhead at Syrene, whereas at the same time in Alexandria, the sun was at a $7.5^{\circ}$ angle from the zenith. (IMAGE CAN'T COPY). Eratosthenes reasoned that because the sun is far away, the rays of sunlight that reach the earth must be nearly parallel. From this assumption he concluded that the measure of $\angle A O S$ in the accompanying figure must be $7.5^{\circ} .$ Use this information to estimate the radius (to the nearest 10 miles) of the earth. $3970 \mathrm{mi}$

Eratosthenes, the fifth librarian of Alexandria ( 230 B.C.), was able to estimate the radius of the earth from the following data: The distance between the Egyptian cities of Alexandria and Syrene was 5000 stadia ( 520 miles). Syrene was located directly south of Alexandria. One summer, at noon, the sun was directly overhead at Syrene, whereas at the same time in Alexandria, the sun was at a $7.5^{\circ}$ angle from the zenith.
(IMAGE CAN'T COPY).
Eratosthenes reasoned that because the sun is far away, the rays of sunlight that reach the earth must be nearly parallel. From this assumption he concluded that the measure of $\angle A O S$ in the accompanying figure must be $7.5^{\circ} .$ Use this information to estimate the radius (to the nearest 10 miles) of the earth. $3970 \mathrm{mi}$

Key Concepts

Assumption of Parallel Rays

This key concept relies on the idea that, because light (or other distant radiation) travels in nearly parallel rays, the angles of incidence on different parts of a spherical surface can be used to deduce geometric relationships. This simplifies the analysis by ensuring that the difference in shadow or sun angle at different locations is due solely to the curvature of the sphere, rather than variations in the light’s direction.

Angle Measurement and Conversion

Understanding how to measure and interpret angles, typically in degrees or radians, is essential when relating local angular observations to the geometry of a circle. This concept enables the conversion of a measured angle into a fraction of the whole circle, thus facilitating the calculation of other parameters like arc length and radius.

Spherical Geometry

This concept involves modeling objects such as the Earth as a sphere, which allows one to apply the properties of circles and spheres. In this context, understanding that a circle has a constant radius and that the arc length is proportional to the central angle is crucial for estimating dimensions like circumference and radius based on local measurements.

Proportional Reasoning in Circles

In a circle, any arc length is directly proportional to its central angle relative to 360°. This means that if a small segment of the circle (an arc) and its corresponding central angle are known, one can determine the entire circumference by scaling the measured segment based on the fraction of the circle that the angle represents.

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