Wednesday, July 25, 2012

Case Study No. 0458: Eratosthenes

Excerpt from the IBM film "Mathematics Peepshow".
Tags: Eratosthenes sieve history
Added: 1 year ago
From: jbmathfunnew
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(c) 1961 Charles and Ray Eames
(c) 1993 Lucia Eames Demetrios dba Eames Office

[scene opens in "modern" times, with a young boy (drawn all in beige) standing next to his bicycle]
NARRATOR: How would you set about to measure the Earth, with the mathematical knowledge and tools you already possess?
[cut to ancient times, as Eratosthenes (drawn all in blue) is walking through the Library of Alexandria]
NARRATOR: A Greek named Eratosthenes did it two hundred forty years before Christ. He was head of the Great Library of Alexandria in Egypt, a city built by the Greeks.
[Eratosthenes stops and stares at an ancient map of the universe, as he looks to be deep in thought]
NARRATOR: He, like others, suspected that the world was round. After all, the sun and the moon were round.
[cut to a still image of light coming through the library's windows]
NARRATOR: He had also noticed that the sun's rays fell in parallel lines.
[cut to a picture of a circle]
NARRATOR: Greeks before him had divided the circle into degrees, and had measured angles.
[camera zooms out to reveal Eratosthenes (now purple) standing next to the circle]
NARRATOR: With this information, and a logical mind, he measured the Earth. Here is how ...
[cut to a map of Egypt]
NARRATOR: While visiting the city of Syene one midsummer's day, Eratosthenes noticed that the reflection of the sun could be seen in the bottom of a deep well.
[cut to Eratosthenes (now yellow) looking down a well]
NARRATOR: The sun was overhead, and the rays pointed to the center of the Earth.
[cut to Eratosthenes looking at a calendar reading "June 21 B.C."]
NARRATOR: He remembered this, and on the next midsummer's day in Alexandria, he measured the shadow cast by an obelisk.
[cut to Eratosthenes standing on a ladder next to an obelisk]
NARRATOR: Sun beams travel in parallel lines, so the difference in angle had to result from the curvature of the Earth.
[cut to an image of the Earth as a circle, and the two versions of Eratosthenes (one looking down the well and the other next to the obelisk) forming an angle]
NARRATOR: If the angle was one-fiftieth of a circle, then the distance around the world must be fifty times the distance from Alexandria to Syene.
["500 x 50 = 25,000" appears on screen]
NARRATOR: With these simple tools, Eratosthenes made this almost-exact measurement of the world, more than seventeen hundred years before Magellan sailed around it.
[cut to Eratosthenes (now purple again) smiling and standing next to Archimedes (drawn all in blue)]
NARRATOR: He was a friend of Archimedes. He was a mathematician and a poet. Invented the sieve for finding prime numbers. Was the first geographer, and corrected the calendar to the one we use today.

The End

made by Charles and Ray Eames
drawing by Glen Fleck



The IBM Mathematica Peep Shows (also known as "IBM Mathematics Peep Shows") were five short films commissioned by IBM and created by Charles and Ray Eames for inclusion in the 1961 "Mathematica" exhibition that took place at the California Museum of Science and Industry and the Museum of Science and Industry in Chicago.

At least when it comes to their films, Charles and Ray Eames are probably best known for the short Powers of Ten. Like this film, each of the five Mathematica Peep Shows was a succinct and poignant presentation of a single mathematical concept, communicated through the medium of film, mostly through animation with voice-over narration.

The five films included are:

Eratosthenes: A description of the method used by Greek mathematician Eratosthenes to accurately measure the circumferance of the world.
Topology: An introduction to the field of topology, focusing on the division of a plane into regions defined by closed curves.
Symmetry: An examination of what is meant by "symmetry," how it is defined mathematically, and various theories of how to measure an object's degree of symmetry.
Something about functions: A film that uses analogies from everyday experience to convey the notion of what a function is, in laymen's terms.
2 [to the power of] n: An illustration of an exponential progression, a sort of "powers of two" film that preceded powers of ten, which draws its inspiration from a classical anecdote relating to the origin of chess.



Eratosthenes of Cyrene (c. 276 BC – c. 195 BC) was a Greek mathematician, geographer, poet, athlete, astronomer, and music theorist.

He was the first person to use the word "geography" and invented the discipline of geography as we understand it. He invented a system of latitude and longitude.

He was the first person to calculate the circumference of the earth by using a measuring system using stades, or the length of stadiums during that time period (with remarkable accuracy). He was the first to calculate the tilt of the Earth's axis (also with remarkable accuracy). He may also have accurately calculated the distance from the earth to the sun and invented the leap day. He also created the first map of the world incorporating parallels and meridians within his cartographic depictions based on the available geographical knowledge of the era. In addition, Eratosthenes was the founder of scientific chronology; he endeavored to fix the dates of the chief literary and political events from the conquest of Troy.

According to an entry in the Suda (a 10th century reference), his contemporaries nicknamed him beta, from the second letter of the Greek alphabet, because he supposedly proved himself to be the second best in the world in almost every field.

Eratosthenes was born in Cyrene (in modern-day Libya). He was the third chief librarian of the Great Library of Alexandria, the center of science and learning in the ancient world, and died in Alexandria, then the capital of Ptolemaic Egypt.

Eratosthenes studied in Alexandria, and claimed to have also studied for some years in Athens. In 236 BC he was appointed by Ptolemy III Euergetes I as librarian of the Alexandrian library, succeeding the second librarian, Apollonius of Rhodes. He made several important contributions to mathematics and science, and was a good friend to Archimedes. Around 255 BC he invented the armillary sphere. In On the Circular Motions of the Celestial Bodies, Cleomedes credited him with having calculated the Earth's circumference around 240 BC, using knowledge of the angle of elevation of the sun at noon on the summer solstice in Alexandria and on Elephantine Island near Syene (now Aswan, Egypt).

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