Greek Mathematics

Home , Archimedes, Greek mathematics, Hippocrates of Chios, Thales of Miletus

We’ve Read a Lot. Now, Can We Make a Good Review? What’s New and Important?

• Compared to Mesopotamian and Egyptian Mathematics, Greek Mathematics:

– Used proof, logic, demonstration. – Was much less “practical” in its orientation. – Was more geometrical and less arithmetical. – Was considered good for the mind, soul. What About our Sources?

• No primary sources remain. • Secondary sources (or worse). • Mainly mentions in books written much later. The First Greek Mathematician We Know about Was: • Thales of Miletus (600 BC) – Father of Demonstrative logic – One of the Seven Sages – Proved five basic geometry theorems • Vertical angles, triangle inscribed in a semicircle, isosceles triangle theorem, ASA theorem, circle bisected by diameter – Not very nice; predicted eclipse; predicted olive crop; walked into holes. Next in Line is:

• Pythagoras (572 BC) – Founded Pythagorean Brotherhood – Pythagorean Theorem – Incommensurability of segments – Connections to music – Mystical; believed in transmigration of souls; cult‐ like following. Next in Line is:

• Hippocrates of Chios – Original Elements (lost) – Quadrature of 3 lunes – Lost his money before becoming a scholar. Next in Line is:

• Eudoxus – Theory of Proportions that didn’t depend on commensurability – Method of exhaustion Next in Line is:

• Euclid (300 BC) – Compiled and Organized – Clever and timeless proofs – Still held in honor. About the Elements:

• Why has it been so influential? – Axiomatic approach provided foundation. – Seeing how it was done has inspired countless students. About the Elements:

• What’s in the Elements? – Plane Geometry – Ratios and proportions – Number theory – Geometric Algebra – Solid Geometry About the Elements:

• What are it’s weaknesses from our modern viewpoint? – Definitions are not “careful.” – Unstated assumptions are used After Euclid, we have the great. . .

• Archimedes (287 BC) – 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). Archimedes

• Proved: – Areas and volumes by exhaustion – Area of circle, approximation to pi – Hydrodynamics – Also not a bad practical mathematician and creator of war machines. Alexandria After Archimedes

• Eratosthenes (284 BC) – Famous for the Seive and for measuring the circumference of the earth Alexandria After Archimedes

• Apollonius (262 BC) – Famous for his book on Conics. Alexandria After Archimedes

• Heron (Hero) (75 AD ish) – Famous for his formula – Given a triangle with sides a, b, and c, let ଵ ݏൌ ܽ൅ܾ൅ܿ . Then the area K of the triangle is ଶ given by : ܭൌ ݏ ݏെܽ ݏെܾ ሺݏെܿሻ. Alexandria After Archimedes

• Ptolemy (150 AD) – Famous for The Almagest; astronomy, and his table of chords that allowed for accurate calculations of triangles.

– The Algamest’s predictions and explanations of astronomical phenomena was the standard for 1400 years. Alexandria After Archimedes

• Diophantus (250 AD) – Famous for study of equations, systematic algebraic notation. The Decline of Greek Mathematics The Decline of Greek Mathematics

• Roman Rule from about 150 BC.

• In 312 AD 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. The Decline of Greek Mathematics

• 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. The Decline of Greek Mathematics

• Exactly when the Library at Alexandria was destroyed is open to question. It has been blamed alternately on Romans, Christians, and Muslims. The Decline of Greek Mathematics

• 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. The Decline of Greek Mathematics

• Pappus (about 350 AD):

– Wrote a Mathematical Collection, a consolidation of all the geometric knowledge of the time. Wrote mainly commentary. The Decline of Greek Mathematics

• Theon (about 375 AD):

– Commentaries on Elements, Amlagest, etc. The Decline of Greek Mathematics

• Hypatia (daughter of Theon, 400 AD ):

– Wrote commentaries. Was killed by a mob of overzealous Christians in 415 AD. The Next Chapters

• 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.