Erathostenes' method to measure the circumference of the earth
Erathostenes was a Greek mathematician and philosopher. He was born 275 BC. He studied at Athens and Alexandria where he worked as the director of the Alexandrian Library. He got blind 195 BC and because of that starved himself to death a year later. Today he is known by two things: the method to find the prime numbers (The Sieve of Erathostenes) and measuring for the first time the circumference of the earth.In brief, this is the method called the sieve of Erathostenes:
Say we want to find all the prime numbers not exceeding 200. We make a list from 1 up to 200. Strike over number 1 because it is not a prime, by definition. The smallest prime number is 2. Now strike over all the multiples of the number 2. The smallest number which is not struck over is 3 which is a prime number. Now you strike over all the multiples of number 3 and we find that the next prime number is 5. By repeating this until we reach the integer part (14) of the square root of the biggest number (200). Now all the numbers not struck over are prime numbers.
How Erathostenes measured the circumference of the earth
The idea that the Earth was a sphere was enunciated by Pitagoras and his disciplines. Erathostenes found that on summer solstice day the sun rays were zenithal in Syene (now Assuan). At the same time in Alexandria the rays deviated 7.2 degrees from the vertical direction. Erathostenes approximated that the distance of those two cities was 5 000 stadia ( 800 km). His assumption was based on the time a camel caravan covered that distance.
Thus he derived the terrestial circumference : 360o/7.5o * 800 km = 39 000 km.
This brilliant result actually is a consequence of lucky coincidences. In Syene the sun is not exactly in Zenith on summer solstice day. Besides, Syene and Alexandria are not in the same meridian as the method required. Futhermore, his measurement of the distance was random.
On autumnal equinox we measured the length of the shadow casted by a vertical stick at the moment when the sun was at the highest point. Then we calculated the angle between the sun rays and the vertical direction. The angle was 69.74o.
The length of the arc of one degree is 111.13 km. If we suppose the other measure point was on the equator we derive the circumference x from the equation 
where 61.69o is the latitude of Mikkeli.
Our result is x = 39970 km.
If we count the result from the measurements Sint Niklaas - Mikkeli we get x = 41 400 km. The difference is causedmostly by the fact that these cities are not on the same meridian.
