Imagine a time, over two millennia ago, when the shape and size of our world were largely mysteries. While many philosophers and thinkers had posited that the Earth was a sphere, proving it and, more astonishingly, measuring it, seemed an insurmountable task. Yet, one brilliant mind, armed with little more than observation, basic geometry, and an insatiable curiosity, managed to achieve just that. This is the story of Eratosthenes of Cyrene and his remarkably accurate calculation of Earth’s circumference.
Who Was This Ancient Genius?
Eratosthenes, born around 276 BCE in Cyrene (a Greek colony in modern-day Libya), was a true polymath – a scholar excelling in many fields. He was a mathematician, geographer, poet, astronomer, and music theorist. His intellectual prowess eventually led him to one of the most prestigious positions in the ancient world: chief librarian at the Great Library of Alexandria in Egypt. This library was a beacon of knowledge, housing countless scrolls and attracting the brightest minds. It was in this vibrant intellectual environment that Eratosthenes conceived of his ingenious experiment.
His nickname was “Beta” (the second letter of the Greek alphabet), supposedly because he was considered the second best in many fields, though some argue it was “Pentathlos,” like an athlete who excels in five events. Regardless of the exact meaning, it points to a man of vast and diverse talents, perfectly positioned to connect disparate pieces of information to solve a grand puzzle.
The Spark of an Idea: A Well Without Shadows
The seed for Eratosthenes’ groundbreaking calculation was planted by a curious observation. He learned of a well in Syene (modern-day Aswan, Egypt), a city located south of Alexandria. It was reported that on a specific day, the summer solstice (around June 21st), at noon, the sun’s rays shone directly down into this deep well, illuminating its bottom. This meant that at that precise moment, the sun was directly overhead in Syene, and vertical objects, like obelisks or even a simple stick, cast no shadow.
This piece of information, perhaps gleaned from travelers or texts within the library, intrigued Eratosthenes. He knew that in Alexandria, on the very same day and at the same time, vertical objects did cast shadows. This difference was the key. If the Earth were flat, the sun’s rays, which he assumed were parallel due to the sun’s immense distance, would strike all points at the same angle. Vertical objects everywhere would either cast no shadow or shadows of the same length. The fact that they didn’t indicated that the Earth’s surface was curved.
The Alexandria Experiment
Eratosthenes decided to conduct an experiment. He waited for the summer solstice. While the sun blazed directly down into the well at Syene, he, in Alexandria, set up a gnomon – a simple vertical rod or pillar – and measured the angle of the shadow it cast at noon. The gnomon acted as one arm of a giant triangle, with the sun’s ray forming the hypotenuse and the shadow on the ground forming the base.
By measuring the length of the shadow and knowing the height of the gnomon, he could use basic trigonometry (or a geometric equivalent, as formal trigonometry was still developing) to determine the angle of the sun’s rays relative to the vertical gnomon. He found this angle to be approximately 1/50th of a full circle. A full circle is 360 degrees, so 1/50th of that is 7.2 degrees (360 / 50 = 7.2).
Eratosthenes’ method relied on a few key assumptions. The most crucial were that the Earth is a sphere, the sun’s rays are parallel when they reach Earth, and that Syene was directly south of Alexandria and on the Tropic of Cancer. While not all these assumptions were perfectly accurate, they were close enough for a remarkably good estimate, showcasing incredible ingenuity for his time.
Unlocking the Geometry
Now, how did this shadow angle in Alexandria relate to the Earth’s circumference? This is where Eratosthenes’ genius truly shone. He reasoned as follows:
1. Parallel Sun Rays: Because the sun is so far away, its rays arriving at Earth are virtually parallel to each other.
2. Earth’s Curvature: If you imagine a line extending from the center of the Earth up through Syene (where there’s no shadow, so the sun is directly overhead), and another line from the Earth’s center up through Alexandria, these two lines form an angle at the Earth’s center.
3. Alternate Interior Angles: The sun’s ray at Syene is perpendicular to the ground (hitting the bottom of the well). The gnomon in Alexandria is also perpendicular to the ground (locally). The angle measured by Eratosthenes in Alexandria (between the gnomon and the sun’s ray) is, due to the parallel nature of the sun’s rays and the principles of geometry (specifically, alternate interior angles when a transversal line intersects two parallel lines), equal to the angle formed at the Earth’s center by the arc connecting Syene and Alexandria.
So, the 7.2-degree angle he measured in Alexandria was also the angle separating Syene and Alexandria as viewed from the Earth’s core. This meant that the distance between the two cities represented 7.2/360, or 1/50th, of the Earth’s total circumference.
Measuring the Distance: A Stroll Between Cities
The next crucial piece of the puzzle was the distance between Syene and Alexandria. How did Eratosthenes determine this? Precise surveying methods as we know them didn’t exist. Historical accounts suggest a few possibilities:
- Bematists: The most commonly cited method involves bematists. These were professional surveyors trained to walk in exceptionally regular, measured paces. Armies and administrators often used them to measure distances for mapping and logistical purposes. Eratosthenes might have commissioned such a measurement or used existing records.
- Caravan Times: Another possibility is that the distance was estimated from the average time it took caravans to travel between the two cities, a known trade route along the Nile.
Regardless of the exact method, the distance was determined to be approximately 5,000 stadia. The “stadion” (plural: stadia) was an ancient Greek unit of length, but its exact value varied depending on the region and time. This uncertainty in the length of the stadion is one of the main reasons for the range in estimates of how accurate Eratosthenes’ final calculation truly was.
The Grand Calculation
With the angle (1/50th of a circle) and the distance (5,000 stadia) in hand, the final calculation was straightforward:
If 1/50th of the Earth’s circumference is 5,000 stadia, then the total circumference is 50 times that amount.
Circumference = 50 * 5,000 stadia = 250,000 stadia.
Some sources suggest he later refined this to 252,000 stadia. This refinement might have been to make the number easily divisible by 60 (or 360), common units in ancient astronomy and mathematics, leading to a neat 700 stadia per degree of latitude.
How Accurate Was He?
To assess the accuracy, we need to convert stadia to modern units. The most commonly accepted value for the stadion Eratosthenes likely used is the “Attic” or “Olympic” stadion, which is about 185 meters (or about 607 feet).
Using 250,000 stadia:
250,000 stadia * 185 meters/stadion = 46,250,000 meters = 46,250 kilometers.
Using the refined 252,000 stadia:
252,000 stadia * 185 meters/stadion = 46,620,000 meters = 46,620 kilometers.
The actual, modern-day accepted value for the Earth’s polar circumference is about 40,008 kilometers, and the equatorial circumference is about 40,075 kilometers. Eratosthenes’ figure of 250,000 stadia (using 185m/stadion) would be around 46,250 km. If a shorter stadion of about 157.5 meters was used (the “Egyptian” stadion), his 250,000 stadia would be 39,375 km, and 252,000 stadia would be 39,690 km. This latter interpretation puts his result astonishingly close – within about 1-2% of the true value!
The debate over which stadion Eratosthenes used continues, but even with the longer Attic stadion, his result was within about 15% of the correct figure. Given the tools and knowledge available, this was an absolutely monumental achievement.
Acknowledging the Imperfections
Eratosthenes’ method, while brilliant, relied on several assumptions that weren’t perfectly true, and faced practical limitations:
- Syene’s Location: Syene is not exactly on the Tropic of Cancer. It’s slightly off, meaning the sun wouldn’t have been perfectly, mathematically overhead.
- North-South Alignment: Alexandria is not directly north of Syene. It’s a bit to the west. This would introduce a small error in the arc distance being purely latitudinal.
- Distance Measurement: The 5,000 stadia distance was an estimate, likely with some margin of error, whether paced by bematists or derived from travel times. The terrain itself isn’t perfectly flat.
- Earth’s Shape: The Earth is not a perfect sphere; it’s an oblate spheroid, slightly flattened at the poles and bulging at the equator. However, for the scale of his measurement, a sphere is a very good approximation.
- Sun’s Rays: While very nearly parallel, the sun’s rays do diverge ever so slightly. But this effect is negligible for this calculation.
- Defining Noon/Solstice: Precisely determining local noon and the exact moment of the solstice with ancient instruments would have been challenging.
Despite these potential sources of error, many of them were minor, or fortuitously, some may have even cancelled each other out to some extent. The sheer elegance of the method and the remarkably close result speak volumes.
The Lasting Legacy of a Simple Measurement
Eratosthenes’ calculation was more than just a number. It was a profound demonstration of the power of human reason, observation, and the application of mathematics to understand the world. It showed that the cosmos was not just a realm of gods and myths, but a physical reality that could be measured and understood through scientific inquiry.
His work provided the first reasonably scientific basis for the size of our planet, a fundamental piece of knowledge that would underpin future developments in geography, navigation, and astronomy. It stood as a testament to the intellectual heights achieved in the Hellenistic period, particularly at the vibrant hub of learning that was the Library of Alexandria.
While his exact figures were debated and sometimes lost or supplanted by less accurate ones during the Middle Ages in Europe, the core idea and the astonishing feat of measuring the world with such simple tools remained an inspiration. Eratosthenes didn’t just measure the Earth; he expanded the horizons of human understanding and set a benchmark for scientific endeavor that echoes down to our own age of satellites and GPS.
