Al-Biruni: The forgotten genius who measured the Earth

Source: FB Mathematics Learning

5 January 2025

Al-Biruni (about 1,000 years ago) measured the circumference of the Earth with an accuracy of 99.7% compared to today’s accepted value.

Al-Biruni was a Persian polymath who lived from 973 to 1048. He was a scholar of many fields, including astronomy, mathematics, geography, physics, and history.

In 1030, al-Biruni used trigonometry to measure the circumference of the Earth. His estimate was 6339.6 kilometers, which is within 0.3% of the modern accepted value of 6378.1 kilometers.

Al-Biruni's method was based on the principle that the Earth's curvature causes the horizon to appear lower from a mountaintop than it does from sea level. He measured the angle between the horizon and a plumb line at two different locations, and used this information to calculate the Earth's radius.

Al-Biruni's measurement of the Earth's circumference was one of the most accurate of its time. It was not surpassed until the 17th century, when the French mathematician and astronomer Jean Picard used a more precise method to measure the Earth's circumference.

"Eratosthenes' experiment to measure the Earth's circumference is a classic example of early scientific measurement that can be replicated by anyone today.

Here's a simplified overview of how it worked:

Eratosthenes knew that at local noon on the summer solstice in Syene (now Aswan, Egypt), the Sun was directly overhead, as evidenced by the fact that it illuminated the bottom of a deep well, something that only happens when the Sun is at the zenith. There were no shadows cast by vertical objects.

At the same time in Alexandria, which is north of Syene, vertical objects did cast shadows. Eratosthenes measured the angle of the shadow cast by a stick and found it to be approximately 7.2 degrees, or 1/50 of a full circle.

The distance between Syene and Alexandria was known to be approximately 5,000 stadia (the exact length of a stadion is not known but is often taken to be about 185 meters based on the typical length used in the Hellenistic Mediterranean world).

Eratosthenes reasoned that if a stick in Alexandria cast a shadow with an angle of 7.2 degrees, then, in a full 360-degree circle, the distance from Alexandria to Syene must be 1/50 of the Earth's total circumference. So, he multiplied the distance between the two cities by 50 to get the Earth's circumference.

Using modern units for the stadion, the calculation would be:

5,000 stadia x 185 meters/stadion = 925,000 meters

925,000 meters x 50 = 46,250,000 meters

Thus, Eratosthenes calculated the Earth's circumference to be about 46,250 kilometers. The actual circumference of the Earth at the equator is about 40,075 kilometers, so although his method was sound, the accuracy of his result was off due to the inexact value of the stadion and possibly the accuracy of his measurements. Nevertheless, Eratosthenes' experiment was remarkably precise for his time and remains a powerful demonstration of the scientific method applied to the natural world.

If the Earth were flat and to reconcile Eratosthenes' observations with a flat Earth model, the sun would have to be much closer and smaller than it is in the heliocentric model. The idea would be that the sun is moving in circles above the flat plane of the Earth, creating the different angles of shadows at different points.

To calculate the size and proximity of the sun under this flat Earth scenario, you would use the angle of the shadow measured by Eratosthenes (about 7.2 degrees) and the distance between Syene and Alexandria. Assuming this distance is about 800 kilometers (the figure is usually given as 5000 stadia, with the exact length of a stadion varying), basic trigonometry would give you the height of the sun above the plane of the Earth.

Using the tangent of the angle, which is the ratio of the opposite side (the height of the sun) to the adjacent side (the distance between the two cities), you would get:

tan(7.2°) = Height of the sun/800km

Solving for the height of the sun, you would find that it would have to be about:

Height of the sun = 800 km x tan(7.2°)

This would place the sun at a height of approximately 100 kilometers above the Earth's surface—a vastly different scenario than the actual average distance to the sun of about 150 million kilometers.

If the Sun were hypothetically placed at a height of only 100 kilometers above the Earth's surface, and we needed to adjust its size to match the actual observed angles of sunlight as measured by Eratosthenes, we would have to perform a calculation based on the angular diameter of the Sun as seen from Earth.

The angular diameter of the Sun as observed from Earth is about 0.5 degrees. To maintain this angular diameter from a distance of only 100 kilometers, we would need to calculate the actual diameter (D) that the Sun would need to have using the formula for the angular diameter:

θ = 2 arctan (D/2)

where:

θ is the angular diameter in radians,

D is the diameter of the Sun,

d is the distance to the Sun.

Rearranging this formula to solve for D and converting θ to radians for a 0.5-degree angular diameter gives us:

D = 2d tan (θ/2)

Let's do the calculation.

If the Sun were only 100 kilometers above the Earth's surface and we wanted to maintain the same angular diameter of 0.5 degrees as we observe from Earth's actual surface, the Sun's diameter would have to be approximately 872.67 meters.

At this size, in order to maintain the solar constant of 1361 watts per square meter, the hypothetical sun would need a total energy output of about 1.74 x 10^17 watts. Nuclear fusion requires incredibly high pressures and temperatures that are only possible within the cores of stars due to their immense gravitational forces, which are a product of their massive size.

Therefore, nuclear fusion as the source of energy would not be possible with the described dimensions of a sun only 872.67 meters in diameter and 100 kilometers from Earth.

Another proven explanation of the source of its power would be required.

Additionally, this model would have difficulties explaining many other observations, such as the lack of change in the angular size of the sun throughout the day, the way the sun sets and rises, and the phenomenon of night and day across different time zones, among other issues.

I challenge any FE believer to present a mathematical model - with data and figures commensurate that what I have presented above - to support your claim. The information above strongly refutes the idea of a flat Earth and local sun. It's simply not possible without additional explanations.

Copy and paste:

10/1/2025: 2.17 p.m (Jumaat Al Barakah)