1. The proofs of the Early Greeks
2800 B.C. – 450 B.C.

2. 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.

3. 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”

4. 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/ Diophantus
Circa Circa Pappus of Alexandria

5. 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

6. 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.

7. 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

8. 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

9. Mathematics of Thales
An angle inscribed in a semicircle is a right angle.
Thales is credited with the proof.
Babylonia recognized this years earlier

10. A Greek Tragedy

11. 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

12. 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

13. 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

14. Arithmetica vs Logistic
Arithmetica – Study of relationships between numbers
Logistic – Computation
Pythagoras started modern Number Theory
Rafael: School of Athens

15. Works Cited
University of Arizona, Math History Course