Greek Mathematics Part II

Alexandria was founded in about 332 BC by Alexander the Great, and remained the capital city of Egypt until 641 AD, when Arabs captured it. Soon after its founding, Alexander returned home to Macedonia and Ptolemy, one of his generals, became ruler over Egypt in about 323 BC. Ptolemy founded the Library at Alexandria around 300 BC, and it became the largest library in the world and an important seat of learning in the Hellenistic period.

It is to this Library that Euclid (323 - 285 BC) came soon after its founding. Euclid’s contribution was not in the discovery or creation of original mathematics, but in setting down known mathematics with proofs based on definitions, common notions, and postulates—that is, he introduced what has come to be known as the axiomatic method. The 13 books of his Elements became for hundreds of years the standard in mathematical exposition. Summaries of these books are provided in Journey Through Genius.

Discussing for a moment Euclid’s geometry from Book I, it is interesting to note that by moderns standards there is something to be desired. Euclid used unstated assumptions in his proofs, including notions of betweenness of points and continuity of circles; these ideas were added to his axioms by later mathematicians, including Hilbert. That being said, Euclid’s proofs remain classics, and his Book I develops much of the usual geometry of the triangle found in a modern high school geometry course, with many of the proofs essentially the same.

Perhaps most important is Euclid’s fifth postulate, which he avoided using for as long as possible. In the earlier portion of the class, we discussed the crisis in mathematics that arose from efforts to prove it from the other postulates. From these efforts emerged non- Euclidean geometries in the 19th century.

Archimedes (287 – 212 BC) lived in Syracuse and studied in Alexandria. He had an unparalleled ability to focus his mind and is considered the greatest mathematician of antiquity. He spent most of his life in Syracuse under the protection of his friend and patron, King Heron (it was the problem of the gold in Heron’s crown that gave rise to the “Eureka!” story). Journey Through Genius provides an adequate discussion of Archimedes’ life and mathematical contributions, so I’ll leave it at that.

Alexandria After Archimedes

Eratosthenes (about 230 BC): Librarian at Alexandria. Remembered for his Sieve that filtered out non-prime numbers, and for measuring with remarkable accuracy the circumference of the Earth (some believe this to be a happy accident, although the methods used were essentially correct).

Apollonius (262 – 190 BC) : Wrote Conics, for a long time the authoritative treatise on conic sections.

Heron (about 75 AD): Wrote Metrica, and work on finding areas and volumes of various objects, and containing his formula for the area of triangle. Also invented the world’s first vending machine (for dispensing holy water).

Ptolemy (85 – 160 AD): Wrote the Amlagest, the definitive work on astronomy until the time of Copernicus. Also did work in geography, summarizing what was known at that time in eight volumes called Geographike Syntaxis.

Diophantus (about 250 AD): Next to Pappus, the last great mathematician of the ancient Greek era. He wrote Arithmetica, dealing with equations having rational roots. His work represents some movement away from geometry as the focus of mathematical study. He is considered by some to be the father of algebra (see separate notes).

Pappus (about 350 AD): Wrote a Mathematical Collection, a consolidation of all the geometric knowledge of the time. Wrote mainly commentary.

Theon (about 375 AD): Commentaries on Elements, Amlagest, etc.

Hypatia (400 AD ): (daughter of Theon): Wrote commentaries. Was killed by a mob of overzealous Christians in 415 AD.

The Rise of Rome and Decline of the Greek Culture

In 312 Constantine the Great took power, converted to Christianity and in 330 founded Constantinople (now Istanbul, as all fans of They Might Be Giants know). By 395, the Empire was officially Christian. In the fifth century the western half of the empire (centered in Rome) fell before Saxons, Vandals, Visigoths and Huns. The Eastern Roman Empire (often called the Byzantine Empire) survived for another thousand years, and kept a spark of learning alive. Part of their activity was copying ancient books.

Exactly when the Library at Alexandria was destroyed is open to question. It has been blamed alternately on Romans, Christians, and Muslims. It is likely that parts of the library collection were lost or destroyed over time as various social and political groups came to power in Alexandria. That being said, it seems to have stopped functioning as a library by the eighth century AD.

Western culture took a bit of a hiatus between the barbarian invasions of the fifth century and the 1000’s, which period of time is known as the Dark Ages. Much of what we know of classic Greek culture, including mathematics, we owe to the preservation of books by the Islamic world.