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Κυριακή, 15 Ιανουαρίου 2012

The Earth's Circumference, Ερατοσθένης, Πτολεμαϊκή Αίγυπτος

Eratosthenes
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The Earth's Circumference
Or, how far is Paris from Ceyenne, French Guiana? This was a critical question for Giovanni Cassini in 1672 in his determination of the Mars Parallax.
The earliest Greek philosophers and mathematicians considered the sphere to be among the most perfect of all geometric figures. With a few exceptions such as Homer and Anaximenes, practically all Greek Era thinkers beginning with Pythagoras ( 569 - 475 BC ) and Aristotle ( 384 - 322 BC ) believed that Earth was God's creation of the perfect celestial body and hence was created as the perfect spherical object in the heavens. Plato guessed Earth's circumference as ≈ 40,000 miles while Archimedes ( 287 - 212 BC ) estimated it as ≈ 30,000 miles. Only Hellenistic philosopher and mathematician Eratosthenes ( 276 - 194 BC ), born in Cyrene ( modern Libya ), but working, studying and dying in Alexandria, Ptolemaic Egypt, first did devise the mathematics for determining Earth's circumference and hence its diameter.  Besides studying in Athens and Alexandria, Eratosthenes was the 2nd appointed head in 236 BC of the Great Library at Alexandria succeeding Zenodotos. Cleomedes's "On the Circular Motion of the Celestial Bodies" gives the first reasonable description of Eratosthenes's Earth circumference mathematics where in 240 BC at the summer solstice on June 21st, the sun shone directly overhead in Syene ( now Aswan, southern Egypt ), but yet cast a shadow upon the ground at noontime by the Alexandria Spire equal to ≈ 1/8th the height of the large tower. 

Eratostenes.png
Now,
tangent_angle.png
Further, the walked distance from Syene to Alexandria was ≈ 5000 Stadia ( ≈ 500 miles ), therefore, the following simple equation is true: 
simple_equation.png
The modern accepted circumference of the Earth at the equator is 24,901.55 miles and hence Earth's true diameter at the equator is 7,926.41 miles. 
Pretty good, Eratosthenes!! In 240 BC! 
By the way, Archimedes obtained the first theoretical calculation of Greek letter pi from an approximation of 
pi.png
which is known as Archimedes' Constant.
And, thus, Giovanni Cassini in 1672 could successfully determine the distance between Paris and Ceyenne, French Guiana as approximately 7,200 km (4,500 miles) for the Mars Parallax with only a 7% error! 

http://relativitycalculator.com/earth_circumference.shtml

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