greek mathematicians

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Quotations Attributed to Euclid

Euclid replied to King Ptolemy, There is no royal

road to geometry.

A youth who had begun to read geometry with Euclid,when he mastered the rst proposition, asked,

What do I get by learning propositions? So Euclid

called a slave and said, Give him three coins, since he

must make a prot out of what he learns.

Strobaeus,Extracts.

Alexander the Great conquered Egypt, in 332 B.C.

He made it the capital of his new empire. Hissuccessors, King Ptolemy I and his son,Ptolemy II, built the famous Pharos lighthouse,

the Museum, and Library.

After the conquest of Egypt in 332 B.C., Alexander the Great decreed the construction ofAlexandria on the Mediterranean coast of Egypt to be the capital of his vast empire; but it wasPtolemy I Soterand his son Ptolemy II who made the city into the glory of the Hellenic world.They built the Pharos lighthouse, one of the Seven Wonders of the World, and the fabuloustemple of Serapeion, museum and library.

At the museum and library, the Ptolemies founded the greatest university of the ancientworld. The library held over 500,000 papyrus scrolls from all fields of knowledge. Itwas here the best Greek scholars, scientists, and philosophers came to study, teach, andconduct scientific research; and here we find Euclid, about 300 B.C., teaching and writinghis monumental study of geometry in thirteen books, known as the Elements.

While only a few anecdotal comments have survived relating to Euclids personal life, theElements, in annotated copies, endures as the most widely studied mathematical texts ofall time. Euclid draws upon the previous Greek mathematicians from the Pythagorean andPlatonic schools to compose a beautiful unified synthetic development of plane geometry,

number theory, and solid geometry. Euclid sets down 23 definitions, 5 postulates,and 5 common notions. Using Aristotelean logic, Euclid builds an impressive theoryproving and connecting the fundamental discoveries of Greek mathematics. It requiresdetailed study to fully appreciate Euclids Elements because Greek mathematics was abranch of philosophy, defined by philosophical discussion. Unlike modern plane geometry,the concepts ofpoint, line, and plane are defined; line segments, angles, and boundedfigures are never associated with the real number system.

In Elements [I, 47] Euclid proves the Pythagorean theorem without using proportion toavoid the Greek inconsistencies using incommensurable line segments or irrationalnumbers. In block A, Euclids proof is rendered in modern outline. Notice how the proofdepends on equivalent planar figures, but not on numerical area. It is comparable todissecting the squares on the two legs of a right triangle and transforming them into thesquare on the hypotenuse.

According to the Greek commentator, Proclus, 5th century A.D., Euclid extended thePythagorean theorem in Elements [VI, 31] to include similar polygons constructed on the

three sides of a right triangle. In block B, Euclids proposition is rendered in modern outline.R

AR

Band R

Crepresent any three similar polygons constructed on the sides of a right

triangle. Can you supply the reasons in the proof?

Euclid completes the Elements in Book XIII by constructing the five Platonic solids andexploring the relationships between their circumscribed spheres. Euclid provides a connectionbetween the edge of each of these regular polyhedra to the radius of its circumscribedsphere. The five Platonic solids are illustrated on the poster.

Euclids Elements was used as a text to study geometry for more than twenty centuries.Moreover, its study generated new discoveries, new approaches to geometry, and newgeometries. The new geometries, know as non-Euclidean geometries, recognize thehistorical importance of the Elements of Euclid in the history of mathematics.


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greek mathematicians

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