The proofs of the Early Greeks 2800 B.C. – 450 B.C.
Greek Mathematicians Believed numbers had mystical meaning Four was a significant number to them because they knew 4 regular solids: tetrahedron, cube, octahedron, dodecahedron. They also had 4 elements of nature: Air, Earth, Water, and Fire. They actually discovered a fifth solid, the Icosahedron, which initially freaked them out because there were more solids than elements. Aristotle solved this by adding Aether as the quintessence (incorruptible) fifth element of nature.
Plato (circa 423 – 348 BCE) One of our main Greek “mathematicians.” Created his Academy and dedicated it to the goddess Athena, which was outside Athens. Plato had the entrance inscribed with the warning: “Let None But Geometers Enter Here”
Circa 624 BC – Circa 546 BC Thales Circa 582 BC – 507 BC Pythagoras 480 BC – 411 BC Antiphon the Sophist Circa 450 BC – Circa 370 BC Democritus Circa 410 BC/408 BC – Circa 355 BC/347 BC Eudoxus of Cnidus Circa 423 BC – 348 BC Plato 384 BC – 322 BC Aristotle Circa 323 BC – Circa 283 BC Euclid Circa 287 BC – 212 BC Erathosthenes Circa 262 BC – Circa 190 BC Apollonius of Perga Circa 190 BC – Circa 120 BC Hipparchus Circa 160 BC – Circa 100 BC Theodosius of Bithynia Circa 200/214 – Circa 284/298 Diophantus Circa 290 - Circa 350 Pappus of Alexandria
Thales – 624 B.C. – 546 B.C. Started as a Merchant Well-traveled and curious Spent time in Egypt Why instead of How? First known Greek Philosopher Credited with first 5 theorems of Geometry
Thales 5 theorems The circle is bisected by its diameter The vertical and opposite angles are equal Equality of triangles (by two angles and a side) The angle is a semi-circle is a right angle If a straight line is drawn parallel to one of the sides of a triangle, then it cuts the other sides of the triangle, or these produced, proportionally; and, if the sides of the triangle, or the sides produced, are cut proportionally, then the line joining the points of section is parallel to the remaining side of the triangle.
Mathematics of Thales Vertical angle Theorem Proof Angle a plus angle c make a straight line Angle c plus angle b make a straight line All straight lines are equal Angle a equals angle b
Mathematics of Thales A circle is bisected by its diameter The base angles of an isosceles triangle are equal Two triangles are congruent if the have two angles and one sides which are respectively equal
Mathematics of Thales An angle inscribed in a semicircle is a right angle. Thales is credited with the proof. Babylonia recognized this 1400 years earlier
A Greek Tragedy
The Sources of Greek Mathematics Euclid Euclid’s Elements in 300 B.C. Trumped all math before it The commentary on the first book of Euclid was extremely influential Eudemian Summary by Proclus around 500 A.D Had access to works lost to us Byzantine Greek Codics : 500 – 1500 yrs after Greek works Proclus
Pythagorus: 572 B.C. – 500 B.C Possible student of Thales Founded Pythagorean School Mathematics Natural Sciences Philosophy Strange cultish rules Abstain from Beans Do not touch a white rooster Do not pick up what has fallen Not to stir the fire with iron Do not look in a mirror beside a light Do not eat meat
Pythagoreans It was the Pythagoreans who named the study of numbers “mathematics” Their motto was “All is Number.” They continually claimed that Mathematics was the basis of ALL knowledge and understanding
Arithmetica vs Logistic Arithmetica – Study of relationships between numbers Logistic – Computation Pythagoras started modern Number Theory Rafael: School of Athens
Works Cited http://www.cut-the-knot.org/pythagoras/ThalesTheorems.shtml University of Arizona, Math History Course