Two "Brute Force" Methods

Method 1 - Circumnavigating the globe and measuring it!
Let's assume for the moment that Eratosthenes actually considered making such a trip (or hiring someone to do it for him.) He would face some serious obstacles. Traveling across land would be hard enough, but what about the problem of crossing oceans? Needless to say, I think Eratosthenes ruled this out as a possibility. 

Let's take a look at a third possibility.

I think we can safely assume that Eratosthenes did not use either of these two methods. 
Method 2 - Digging to the other side of the earth.  He knew how to figure the distance around the earth if he could measure the diameter of its cross section. From his readings and contemplation, he knew that the circumference of an circle is a little more than three times the diameter. Here the diameter is the distance through the center of the earth. This means that Eratosthenes would need to dig and measure a very long tunnel.  Sounds somewhat unrealistic.
Even if he had access tothe papyrus version of this book it probably would not have helped him.

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