Introduction to Eratosthenes and His Ingenious Calculations

Eratosthenes, born in Cyrene, Libya, in 276 BC, was a polymath who lived during the Hellenistic period. He is often referred to as one of the greatest minds of antiquity, known for his contributions to various fields, including astronomy, geography, mathematics, and music. One of his most famous achievements, however, was the calculation of the Earth's circumference using trigonometry and simple observations. This article explores how Eratosthenes used the principles of trigonometry to make his groundbreaking measurement in the 3rd century BC.

The Origin of Trigonometry and Eratosthenes' Inspiration

Trigonometry, the branch of mathematics dealing with the relationships among the sides and angles of triangles, has its roots in the study of astronomical phenomena. Eratosthenes was well-aware of the importance of these principles, as they provided a powerful tool for understanding the spatial relations in the cosmos. Specifically, the concept of the Tropic of Cancer played a crucial role in his calculations.

The Tropic of Cancer and its Significance

The Tropic of Cancer marks the northernmost latitude where the Sun can be directly overhead. This line of latitude is one of the five major circles of latitude that are visible in celestial maps and serve as crucial reference points in the study of the Earth's geometry. In the 3rd century BC, the Tropic of Cancer was an essential marker for Eratosthenes in his experiment to determine the Earth's circumference.

Eratosthenes' Experiment: The Genesis of His Measurement

Eratosthenes conducted his experiment in Alexandria, on the 21st of June, around 240 BC. He observed that the Sun reached the highest point in the sky at noon, or what he measured as the summit of the noonday Sun, which is the zenith angle. He knew that at the same time in a city called Syene (modern-day Aswan, Egypt), the Sun was directly overhead.

At Syene, Eratosthenes noticed that the Sun's rays illuminated the bottom of a well, indicating that the Sun was directly overhead. This observation provided Eratosthenes with the angle of elevation, which was crucial for his calculations. By comparing the sun's angle at midday between these two locations, Eratosthenes made a significant inference about the Earth's curvature and circumference.

The Calculation: Utilizing Trigonometry

Knowing that the Sun was directly overhead in Syene, and the angle at which it shone in Alexandria, Eratosthenes could accurately deduce the angle at which the Sun's rays struck the Earth. This angle was 7.2 degrees. Given that a full circle is 360 degrees, Eratosthenes reasoned that the distance between Alexandria and Syene represented 1/50th of the Earth's circumference.

With the knowledge that the distance between Alexandria and Syene was approximately 5,000 stadia (an ancient Greek measurement), Eratosthenes concluded that the Earth's circumference was roughly 250,000 stadia. Converting this into modern units, the calculation yields a circumference of around 40,000 kilometers, which is remarkably close to the actual circumference of the Earth, estimated to be around 40,075 kilometers.

Implications and Impact of Eratosthenes' Work

The genius of Eratosthenes lies in his ability to apply the principles of trigonometry to a practical and accurate measurement of the Earth. This method not only demonstrates the power of mathematical and observational skills but also establishes the concept of the Earth's curvature on a grand scale. His work laid the foundation for modern geodesy and reinforced the spherical shape of the Earth.

Legacy: Continuation of Eratosthenes' Method

Although Eratosthenes' method was advanced for his time, the principles he discovered and employed are still used in modern geodesy and astronomy. Modern technology can now measure the Earth's circumference with even greater precision, but the core concept of using trigonometry to understand the celestial sphere remains unchanged.

Conclusion: A Remarkable Scientist and Mathematician

Eratosthenes' masterful use of trigonometry to calculate the Earth's circumference stands as a testament to his genius and the enduring value of mathematics in understanding the natural world. From the Tropic of Cancer to the complex geometry of the Earth, Eratosthenes' work remains a cornerstone in the history of science and an inspiration for future generations of scientists and mathematicians.