How did Eratosthenes measure the size of the Earth?

Ancient Greek astronomy stands as one of the remarkable precedents to modern scientific practices. This era was marked by intellectual giants who made significant strides in understanding celestial motions, often using simple tools but brilliant deductive reasoning. For instance, the ancient Greeks knew about the spherical nature of the Earth, track planetary paths, and developed theories about the cosmos that would persist for centuries.Astronomers like Eratosthenes, who lived in the 3rd century BCE, embodied the spirit of inquiry and scientific method in their time. He was a polymath, mastering in fields like geography, mathematics, and, crucially, astronomy. Detailed records and empirical observations were at the heart of his methodologies. By extrapolating data from phenomena such as solar positions and using geometry as a tool for understanding the universe, figures like Eratosthenes ushered in new ways of thinking about our world and beyond.

Geometry has been a cornerstone in the practice of astronomy since ancient times. The Greeks, in particular, were adept at applying geometric principles to unveil the mysteries of the cosmos. They used shapes, angles, and mathematical relationships to make accurate predictions and build a coherent understanding of spatial relationships in the sky.Eratosthenes demonstrated this by employing basic geometric knowledge to estimate the Earth's circumference. He realized that the angle of the sun's rays at different locations could give insights into the spherical shape of the planet. By drawing imaginary lines from the Sun to different points on Earth and recognizing that his observations suggested a spherical geometry, Eratosthenes used the principles of circles and angles to make his revolutionary measurements. This ingenious method proved the power of geometry in comprehending the astronomical phenomena that were visible to the naked eye.

The quest to determine the Earth's size has long captivated scientists, but Eratosthenes' estimation stands as one of the earliest and most ingenious attempts. His approach was based on a blend of empirical observation and geometric reasoning. He noted the sun's rays were vertical in Syene at noon during the summer solstice, whereas in Alexandria, there was a notable angular difference of about 7.2 degrees.This observation was the linchpin to his methodology, which relied on the assumption that a consistent curvature existed over the Earth's surface. By associating the angle difference to the fraction of the Earth's full 360-degree rotation, Eratosthenes determined that the distance between the two cities was 1/50th of the planet's total circumference. With the known distance between Syene and Alexandria, he calculated the full circumference to be approximately 40,000 kilometers—a figure remarkably close to modern measurements. His work demonstrated an exceptional use of simple geometry to solve a complex problem and is celebrated for its elegance and accuracy.