SO, HOW BIG IS THE EARTH?

Hiya, Sean Lally Physics Guy here. Today on “How do we know that?” we are looking at the Earth yet again.

So, now we know that the Earth is a sphere. How big is that sphere?

Quick, how would you determine that? Think about it – ask a friend.

Here are a couple of possibilities:

Get in an amphibious vehicle (!) and drive/sail around the equator, arriving back where you started. Clearly, you should set your odometer (trip counter) to zero and note the reading upon your return to the starting point.

How about this? Get a gigantic piece of string and wrap it around the equator until it reaches the starting point. Alternately, you could wrap the string around adjoining lines of longitude, passing over the north and south poles. This would give you a slightly different value, but the principle is the same.

What’s wrong with these methods? Anything?

Neither of these methods has been utilized, but the size of the Earth has been known with great accuracy for nearly 2500 years! How did someone know this?

As you might expect, it has something to do with mathematics. Let’s go back to our friends, the ancient Greeks – namely, Eratosthenes of Cyrene (276 – 295 BCE). It was known by the locals that Syene (Aswan, in Egypt) was located at or near the Tropic of Cancer – that’s a circle of latitude around the Earth close to 23.5-degrees North. This location is important because the Sun is directly overhead at noon on the day of the Summer Solstice, and no object will cast a shadow then. (The Earth is inclined at an angle of 23.5-degrees, and yes, this has a lot to do with why there will be no shadown here on the Summer Solstice.) With this in mind, Eratosthenes devised a clever experiment. But first, how would you use this information?

How about if you knew of another location roughly along the same line of longitude a certain known distance from Syene? Any clues?

This other location, Alexandria, was north of Syene by some 5000 stades (close to 800 kilometers). At noon on the day of the Summer Solstice, Eratosthenes set up a large stick – it certainly did cast a shadow at noon (when the shadow was shortest), and he determined the size of this shadow and the angle from the top of the stick to the far side of the base. See diagram above, which is clearly NOT TO SCALE.

Think of this – the distance from Syene to Alexandria was known and the shadow and angle were easy to measure very accurately. How would you use this information to determine the size around the Earth? It’s a geometry problem. Think about it for a moment and discuss with your friend.

This part is not so obvious – the two rays of sunlight drawn above are very nearly parallel to each other. If the two sticks were sufficiently long enough, they would intersect at the center of the Earth – creating an angle between them. But this angle is the SAME as the angle between the top of the stick and the ray of sunlight. Again, it looks that way in this diagram, but the proof is a classic one in geometry. If you don’t know the geometry here, you may need to suspend disbelief.

Once you see this, the problem is a simple proportion. How many degrees are in a complete trip around a circle? 360, right? Consider this:

The angle of shadow (measured by Eratosthenes to be 7.2-degrees) is to the angle completely around a circle as the distance between Syene and Alexandria is to the complete distance around the Earth (circumference).

Wow, that’s a mouthful. Symbolically it is so much simpler:

7.2-degrees / 360-degrees = 800-km/ Circumference

I have taken the liberty of writing this as a proportion, though Eratosthenes thought of this just a little differently. Can you solve this for the circumference of the Earth?

So, is this a good estimate? Well, our current value for the approximate circumference of the Earth is 40,030 km (around 25,000 miles). Would you consider this technique to be pretty accurate?

So, what are sources of error in this experiment? And what assumptions did Eratosthenes make. There is a lot to think about here – talk this over with your friend.



Further Reading

Kitty Ferguson – Measuring the Universe

Nicholas Nicastro - Circumference



ALL TEXT AND IMAGES COPYRIGHT SEAN LALLY 2009