Greek Geometry
Geo (the Earth) and Metry (to measure)  Geometry then means to measure the Earth
In spite of various myths about the Earth being flat, many practical people had long know the Earth was actually round.
Geo (the Earth) and Metry (to measure)  Geometry then means to measure the Earth
In spite of various myths about the Earth being flat, many practical people had long know the Earth was actually round.
Famous Question 1: How can we estimate the circumference of the Earth?
A Greek guy living in Egypt, named Eratosthenes, made a calculation around 2200 years ago. Here's how.
A Greek guy living in Egypt, named Eratosthenes, made a calculation around 2200 years ago. Here's how.
Here's one way to tell the story.

Carl Sagan, wellknown physicist in his famous astronomy series Cosmos, also tells the story.

Short Biography of Erathosthenes
Eratosthenes (276 BC  194 BC) was a Greek mathematician, geographer and astronomer with (probably) Chaldean origins.
He was born in Cyrene (now Shahhat, Libya) and he died in Ptolemaic Alexandria. He is noted for devising a system of latitude and longitude and computing the size of Earth.
Eratosthenes studied at Alexandria and some years in Athens. In 236 BC he was appointed by Ptolemy III Euergetes I as a head and the third librarian of the Alexandrian library. He made several important contributions to mathematics and science. He was a good friend to Archimedes. Circa 255 BC he invented the armillary sphere, which was used till 17th century.
He calculated the earth's circumference circa 240 BC, using trigonometry and information on the altitude of the Sun at noon in Alexandria and Syene (now Aswan, Egypt). The calculation is based on the assumption that the Sun is so far away that its rays can be taken as parallel.
Eratosthenes knew that on the summer solstice at local noon in Syene, the Sun would appear at the zenith. He also knew that in his hometown of Alexandria, the position of the Sun would be 7° south of the zenith at that time. He knew that this angle was 7/360 of a full circle and thus concluded that the distance from Alexandria to Syene must be 7/360 of the total circumference of Earth. The actual distance between the cities was known from caravan travellings to be about 5,000 stadia. He established a final value of 700 stadia per degree, which implies a circumference of 252,000 stadia. The exact size of the stadion he used is no longer known (the common Attic stadion was about 185 m), but it is generally believed that Eratosthenes' value corresponds to between 39,690 km and 46,620 km. The actual circumference of the Earth around the poles is 40,008 km. Eratosthenes' method was used by Posidonius about 150 years later.
Circa 200 BC Eratosthenes adopted a word geography, which means a description of the Earth.
Eratosthenes' other contributions include:
from: http://www.biographybase.com/biography/Eratosthenes.html
He was born in Cyrene (now Shahhat, Libya) and he died in Ptolemaic Alexandria. He is noted for devising a system of latitude and longitude and computing the size of Earth.
Eratosthenes studied at Alexandria and some years in Athens. In 236 BC he was appointed by Ptolemy III Euergetes I as a head and the third librarian of the Alexandrian library. He made several important contributions to mathematics and science. He was a good friend to Archimedes. Circa 255 BC he invented the armillary sphere, which was used till 17th century.
He calculated the earth's circumference circa 240 BC, using trigonometry and information on the altitude of the Sun at noon in Alexandria and Syene (now Aswan, Egypt). The calculation is based on the assumption that the Sun is so far away that its rays can be taken as parallel.
Eratosthenes knew that on the summer solstice at local noon in Syene, the Sun would appear at the zenith. He also knew that in his hometown of Alexandria, the position of the Sun would be 7° south of the zenith at that time. He knew that this angle was 7/360 of a full circle and thus concluded that the distance from Alexandria to Syene must be 7/360 of the total circumference of Earth. The actual distance between the cities was known from caravan travellings to be about 5,000 stadia. He established a final value of 700 stadia per degree, which implies a circumference of 252,000 stadia. The exact size of the stadion he used is no longer known (the common Attic stadion was about 185 m), but it is generally believed that Eratosthenes' value corresponds to between 39,690 km and 46,620 km. The actual circumference of the Earth around the poles is 40,008 km. Eratosthenes' method was used by Posidonius about 150 years later.
Circa 200 BC Eratosthenes adopted a word geography, which means a description of the Earth.
Eratosthenes' other contributions include:
 The Sieve of Eratosthenes as a way of finding prime numbers.
 The measurement of the SunEarth distance, now called the astronomical unit (804,000,000 stadia).
 The measurement of the distance to the Moon (780,000 stadia).
 The measurement of the inclination of the ecliptic with an angle error 7'.
 He compiled a star catalogue containing 675 stars, which was not preserved.
 A map of the Nile's route to Khartoum.
 A map of the entire known world, from the British Isles to Ceylon, and from the Caspian Sea to Ethiopia. Only Hipparchus, Strabo, and Ptolemy were able to make better maps than this.
from: http://www.biographybase.com/biography/Eratosthenes.html
Famous Question 2: How far can we see?
Greek, Portuguese, Italian, and Citizens of Earth Geometry
Merchants and seamen from various countries often wanted an answer to the questions about how far away were the incoming ships and when would they arrive. The opportunity to make money off the contents of the arriving ships was of great importance! Obviously, telescopes used in high buildings would help. That's one reason Gallileo in Italy tried (and succeeded) in making a wonderful telescope. And he also wanted to look at objects in our solar system and beyond too.

Make an estimate of how far way these ships are from you, standing on the beach. Assume your eye level to be 1.5 meters.
Guess (in kilometers or in miles)? __________________ Answer: See the chart below. 
Now let our mathematics, combined with some creativity and problem solving, step in to help.
One book from the Murderous Math series, Easy Questions: Evil Answers by Kjartan Poskitt is one place to look. We will need some additional help from another old, dead Greek guy  Pythagoras and the famous Pythagorean Theorem to guide us. If you need to review this theorem check out the webpages here on this website under the tab Shapes => 2Dimensions => Pythagorean Theorem.
Remember the difference between a Conjecture and a Theorem. A Conjecture is an idea that we think works all the time, but has not yet been proven to be true necessarily. A Theorem, by contrast, has been proven to be true for forever more, and thus is accepted as such by the entire mathematical community worldwide.
Remember the difference between a Conjecture and a Theorem. A Conjecture is an idea that we think works all the time, but has not yet been proven to be true necessarily. A Theorem, by contrast, has been proven to be true for forever more, and thus is accepted as such by the entire mathematical community worldwide.
Well, the Murderous Math guys give away the answers first. If you are about 1.5 meters tall you can see about 4.37 km or 2.71 miles. (Note, the dot in the numbers represents a decimal point, not multiplication).
Now, to extend our original question a little,
How Far Can You See on the Moon? (hint: change the number for the radius, as the moon is smaller) 
How about a second extension to our original question?
How Far Can You See Standing in a Tall Building? Obviously farther, but how far? Enter Sir Isaac Newton!
Around 1650 Isaac Newton, an Englishman, wondered about the speed of falling objects, a force he called "gravity" and many other questions that he put to himself as puzzles to investigate. One result, among many, that he came up with was a formula to calculate the height of building or tower or whatever that only required a person to keep track of time to get the solution. Amazing!
The formula is a simple one: d(distance) = 1/2 gt^2 On Earth the g number is 32 ft/sec/sec, so plug in 32 for g and you get a simpler version of d = 16t^2. So, the trick to calculate you height from a tall building, like our math classroom on the third floor, can be done simply holding a ball at eye level, setting your iphone timer, dropping the ball and stopping your timer when the ball hits the ground. Of course, if someone helps you with this, everything becomes much easier. Experiment #1: I dropped a ball from eye level from my classroom on the third floor, and measured the time it took to hit the ground, AND ... it took 2.1 seconds. Next, simple multiply 2.1 times 2.1 time 16 and what did I get? 70.56 feet! Wow!. How far can I see (all the way to the ocean, I might add, turns out to be d = 3.57 times the square root of 70.56. Using my calculator, this gives me 29.99 kilometers 