How was the radius of earth discovered at early time?

How was the radius of earth discovered at early time? by Unnikrishnan Menon

Answer by Unnikrishnan Menon:

The seemingly impossible feat of measuring the size of the Earth was first accomplished by a genius greek guy named Eratosthenes.

He was born in about 276 BC in Cyrene, in modern-day Libya. Google his name if you wanna know more about him 😛

He spent many years as the Chief Librarian at Alexandria. While at the library, he learned about a Well with a remarkable property…

At noon on [math]21^{st}[/math] June each year (the day of the Summer Solstice), the Sun shone directly into the well and illuminated it all the way down to the bottom! This well was situated in Syene. So Eratosthenes deduced that the Sun must be directly overhead on that particular day in Syene. This was something that never happened in Alexandria which was a few hundred kilometers North of Syene! The distance between these [math]2[/math] cities was known at that time.

He was aware that the Earth’s curvature was the reason why the Sun could not be overhead at both places simultaneously. Soon he wondered if he could somehow exploit this fact to measure the Circumference of the Earth!

And to do that, he waited for the perfect day, [math]21^{st}[/math] June! And the perfect time, [math]12[/math] noon!

Click on this image to zoom in.

At exactly the same moment that Sunlight was falling straight down the well at Syene, Eratosthenes stuck a stick vertically on the ground at Alexandria. Then without wasting any time, he immediately measured the angle between the Sun’s rays and the stick. This angle’s measurement blended with the perfect timing was the main ingredient for his recipe to cook up the circumference of our planet!

That’s because this angle is equivalent to the angle between [math]2[/math] radial lines drawn from Alexandria and Syene to the Center of the Earth!

He measured this angle between the Sun’s rays and the stick to be [math]7.2^{o}[/math]

He argued that if the Earth was a sphere, then if he sent a person to walk straight from Syene until he circumnavigates the Earth and returns back to Syene, this person would have traversed a complete circle by covering [math]360^{o}[/math]

So, if the angle between Alexandria and Syene is only [math]7.2^{o}[/math] then the distance between Alexandria and Syene represents [math]\dfrac{7.2}{360}[/math] of the Earth’s circumference!

At that time, the distance between Alexandria and Syene was known to be about [math]785[/math] km.

So, that simply means, the Circumference of our planet [math]=\dfrac{360}{7.2}\times 785=50\times 785=\boxed {39250 km}[/math]

And guess what! This value he found out was accurate to [math]2[/math]%. The reason for this is the fact that the Earth is not a perfect sphere in the first place! It slightly bulges outward at the Equator. I think that’s because of maximum centrifugal force at the Equator due to Earth’s rotation. This teeny-weeny error could also be attributed to…

  • Poor angular Measurement
  • Error in timing of noon on the day of Summer Solstice
  • An error in the distance between Alexandria and Syene

Now that the Circumference of our Planet is know, we can easily find out the Radius as follows…

To make the math a little good-looking, I’ll round it off to [math]40000[/math] km

[math]\begin{equation}\begin{split}2 \pi R = 40000\\&\implies R = \dfrac{40000}{2\pi}\\&\implies \boxed {R=6350 \text{ km}}\end{split}\end{equation}\tag*{}[/math]

The cool thing is that all we need to Measure the Size of our Planet is just a stick and a man with a brain 🙂

PS: Ignore my drawing skills on MS Paint 😛

How was the radius of earth discovered at early time?

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