What Did Eratosthenes Use To Calculate Earth\’s Circumference

Recreate the ancient experiment that measured the world.

Compute Circumference Like Eratosthenes

Fraction of Circle

1/50

Converted to km

40,250 km

Accuracy (vs Modern)

99.6%

Figure 1: Comparison of your calculation against Eratosthenes’ historical value and modern measurements.

Metric	Your Calculation	Historical (Eratosthenes)	Modern Value

Table 1: Detailed breakdown of the experiment variables compared to historical data.

What is the Eratosthenes Earth Circumference Experiment?

When asking what did Eratosthenes use to calculate Earth’s circumference, we delve into one of the most elegant experiments in the history of science. Performed around 240 B.C., this experiment allowed Eratosthenes, the chief librarian at the Great Library of Alexandria, to estimate the size of our planet with remarkable accuracy using simple geometry and shadows.

The experiment relies on the observation that the Earth is spherical and that the sun’s rays arrive parallel to each other. Eratosthenes noticed that in the city of Syene (modern-day Aswan), at solar noon on the summer solstice, vertical objects cast no shadow, indicating the sun was directly overhead. However, in Alexandria, situated north of Syene, a vertical obelisk did cast a shadow. By measuring the angle of this shadow and the distance between the two cities, he could extrapolate the circumference of the entire globe.

This method is ideal for students of geometry, geography enthusiasts, and history buffs who want to understand how humanity mapped the world long before satellites.

Eratosthenes’ Formula and Mathematical Explanation

The mathematics behind what did Eratosthenes use to calculate Earth’s circumference is grounded in the geometry of circles and parallel lines. The core formula relates the ratio of the shadow angle to a full circle (360°) with the ratio of the distance between the cities to the Earth’s full circumference.

The Derivation

1. Step 1: Assume the sun’s rays are parallel.
2. Step 2: Measure the angle of the shadow in Alexandria ($\theta$). Due to the Alternate Interior Angles theorem, this angle is equal to the angle subtended at the Earth’s center between Alexandria and Syene.
3. Step 3: Calculate the ratio of this angle to the full circle: $\text{Ratio} = \frac{\theta}{360^{\circ}}$.
4. Step 4: Equate this ratio to the distance ratio: $\frac{\text{Distance}}{\text{Circumference}} = \frac{\theta}{360^{\circ}}$.
5. Step 5: Solve for Circumference:
   
   Circumference = Distance × (360° / Angle)

Variables Table

Variable	Meaning	Unit	Typical Historical Value
$\theta$ (Theta)	Shadow Angle at Alexandria	Degrees (°)	7.2° (approx. 1/50th of circle)
$D$ (Distance)	Arc length between cities	Stadia or km	5,000 Stadia (~800 km)
$C$ (Circumference)	Total perimeter of Earth	Stadia or km	250,000 Stadia

Table 2: Key variables used in the Eratosthenes formula.

Practical Examples (Real-World Use Cases)

Example 1: The Historical Recreation

Let’s simulate exactly what did Eratosthenes use to calculate Earth’s circumference in ancient times.

• Input Distance: 5,000 Stadia
• Input Angle: 7.2°
• Calculation: Since 7.2° fits into 360° exactly 50 times (360 / 7.2 = 50), the Earth’s circumference must be 50 times the distance between the cities.
• Result: 5,000 × 50 = 250,000 Stadia.

Financial/Scientific Interpretation: Assuming 1 Stadion is approximately 160 meters (an Egyptian stadion), this yields 40,000 km, which is astonishingly close to the actual polar circumference of 40,008 km. The error margin is less than 0.2%.

Example 2: A Modern Amateur Measurement

Imagine two schools collaborating today. One is in Paris, the other in Barcelona (roughly on the same longitude).

• Measured Distance: 830 km
• Difference in Solar Angle: 7.5°
• Calculation: Circumference = 830 × (360 / 7.5)
• Result: 830 × 48 = 39,840 km.

Interpretation: This result is 39,840 km. Comparing this to the actual 40,075 km (equatorial), the result is off by only roughly 235 km, demonstrating that simple tools can still yield high-precision geographic data.

How to Use This Eratosthenes Calculator

This tool is designed to help you explore what did Eratosthenes use to calculate Earth’s circumference by allowing you to modify the variables he used.

1. Enter the Distance: Input the measured distance between the two observation points. The default is 5,000, representing the distance between Syene and Alexandria.
2. Select Unit: Choose “Stadia” for historical accuracy, or “Kilometers” for a modern context.
3. Enter the Shadow Angle: Input the angle of the shadow measured at the northern location (in degrees). Eratosthenes measured 7.2°.
4. Review Results: The calculator instantly computes the total circumference, converts it to kilometers (if Stadia is used), and compares it to the modern accepted value of Earth’s circumference.
5. Analyze the Chart: Use the visual bar chart to see how your inputs compare to historical data.

Key Factors That Affect Results

When discussing what did Eratosthenes use to calculate Earth’s circumference, it is critical to understand the factors that influence the accuracy of the result.

• 1. Measurement of Distance: In ancient times, professional walkers (bematists) paced out the distance. Even a 1% error in pacing translates to a 1% error in the final circumference (approx. 400 km).
• 2. Precision of the Angle: Measuring the shadow angle requires a perfectly vertical gnomon (pole). A tilt of just 0.1 degrees can skew the final result by hundreds of kilometers.
• 3. Definition of the Stadion: One of the biggest debates in history is the length of the “Stadion” unit. It varies between 157 meters and 185 meters. The choice of conversion unit drastically changes the perceived accuracy of Eratosthenes’ result.
• 4. Solar Time Synchronization: Measurements must be taken at exactly solar noon at both locations. If the measurement is taken even a few minutes off, the shadow length changes, distorting the angle.
• 5. Longitudinal Alignment: The formula assumes both cities lie on the same meridian (longitude). Syene and Alexandria are not perfectly aligned; they differ by about 3 degrees longitude, which introduces a geometric error.
• 6. Atmospheric Refraction: The atmosphere bends sunlight. While negligible for rough estimates, for high-precision geodesy, refraction near the horizon or high sun angles can alter the apparent shadow length.

Frequently Asked Questions (FAQ)

1. What exactly did Eratosthenes use to calculate Earth’s circumference?

He used a vertical rod (gnomon) to measure shadow length in Alexandria, a deep well in Syene to confirm the sun was overhead, and the distance between these two cities.

2. How accurate was Eratosthenes?

Depending on the definition of a “stadion,” his calculation was between 1% and 16% accurate. If he used the Egyptian stadion (157.5m), his error was less than 2%.

3. Why did he choose Syene and Alexandria?

Syene (Aswan) was located near the Tropic of Cancer, meaning the sun was directly overhead at the solstice. Alexandria was a known distance north, making the geometry solvable.

4. Can I do this experiment today?

Yes. You need a partner in a different city roughly north or south of you. Measure shadow lengths at solar noon on the same day and share data to find the angle difference.

5. What is the modern value of Earth’s circumference?

The meridional circumference (through the poles) is approximately 40,008 km, and the equatorial circumference is 40,075 km.

6. Did Eratosthenes prove the Earth was round?

No, he assumed it. Greek scholars largely accepted the Earth was spherical by the 3rd century B.C.; Eratosthenes was trying to measure its size, not prove its shape.

7. What if the Earth was flat?

If the Earth were flat and the sun was close, the shadows would also behave differently based on triangulation. However, the angles Eratosthenes observed are consistent only with a distant sun and a curved Earth.

8. What unit is a Stadia in modern terms?

There is no single standard, but the most common conversions are the Attic stadion (~185m) and the Egyptian stadion (~157.5m).