Ancient Geometry

1. Ancient Geometry • “Geometry” from Greek γεωμετρία(“measuring earth”). • Study of shape, size, relative position of figures, and the properties of space. • Person doing geometry is a geometer. • Early usage was in land surveying for calculation of taxes, designing architecture. • Geometry was done everywhere in the world. E.g. China, Egypt, India, the Middle East (largely responsible for preserving and refining legacy of Greek mathematics and algebra)

2. Ancient Geometry

3. Ancient Greek Geometry • Thales of Meletus Miletus • Pre-Socrates philosopher / mathematician / astronomer, 624–546 BCE, from Asia Minor (Turkey). • Practical work: measure pyramid height • First-recorded to use deductive reasoning to geometry (and mathematics, arguably) • Μέγιστον τόπος· ἄπαντα γὰρ χωρεῖ.Megiston topos: hapanta gar chorei“Space is the greatest thing, as it contains all things.”

4. Ancient Greek Geometry • Thales’ Theorem showed his understanding of triangle similarity and right-angles.

5. Ancient Greek Geometry • Thales’ Intercept Theorem was about parallel lines and similar triangles

6. Ancient Greek Geometry • Pythagoras • Philosopher / mathematician, 570 – 495 BCE, from Samos Island, Greece. • That Theorem has his name, even though the idea of it has been around before him (as seen last week) • Formed a Secret Society of nerds number theorists who had stars tattooedon them. (Inspired Freemasons) https://youtu.be/AJgkaU08VvY

7. Ancient Greek Geometry • Pythagoras And The Disciples introduced the idea that mathematical propositions need to be supported with a strong proof by reasoning / arguments which are precise & logical. I.e. rigor. • “The so-called Pythagoreans, who were the first to take up mathematics, not only advanced this subject, but saturated with it, they fancied that the principles of mathematics were the principles of all things.”– Aristotle, Metaphysics 1–5, 350 BCE

8. Ancient Greek Geometry • They were also into music. • And had quite a seriousthing for the Tetractys

9. Ancient Greek Geometry • While numbers could be defined by counting them, ancient Greek mathematics would also render them as geometric objects. In this sense, numbers like √2 and π can be shown to exist in real life. • Even though solving √2 lead to The Theorem,Pythagoras hated the idea of irrational numbers. The Egyptian proof he got from Thales haunted him greatly.

10. Euclid’s Elements • Euclid • ~ 300 BCE • University of Alexandria,but from Athens. • Wrote Elements, which islike a super-textbook of Western mathematics. • Collected and codified knowledge of mathematics up to that point (a bit too well, original sources largely gone now) • 13 volumes, up to 465 propositions

11. Euclid’s Elements • Primary / Secondary school’s “Matematik Moden” is described in Volumes 1, 3, 4, 6, 11, 12. • Elements included propositions on geometry, algebra, and number theory. • These are argued/proven without modern tools and conventions such as algebraic operations, cartesian coordinates. • The geometry is synthetic (propositions proven from axioms) rather than analytic (propositions proven on a coordinate space)

12. Euclid’s Elements • A similar modern attempt to codify mathematical foundations is thePrincipia Mathematica (Alfred North Whitehead, Bertrand Russell. 1910.)

13. Axiomatic System • A collection of elements used to derive mathematical facts as proven theorems • Includes: • Undefined terms • Axioms • Proof procedures • Theorems (already proven) • A bit like language: • Ideas → (Letters → Words → Sentences) → Stories

14. Axiomatic System • Un-defined terms • Elements of theorem accepted without further definition. • Marked by words that may / may not refer to real objects. • Some terms remain undefined to prevent circular definitions. • In geometry: “point”, “line”, “surface”, “angle” • Terms may be equipped with definitions depending on a particular system. • “derivative” has different definitions in Calculusand English Literature

15. Axiomatic System • Axioms • From Greek ἀξίωμα (axioma,“that which is thought worthy as evident”) • Statements constructed from terms accepted as true without an attached proof. • Axioms (or Assumptions, Postulates) give meaning to (un)defined terms, showing how they relate to each other. • Sometimes these statements can be shown to be not true or can be proven using other axioms and theorems. Then they are no longer axioms. • Euclid’s First Postulate: • For every two points, you can draw exactly one straight line between them.

16. Axiomatic System • Proof • A procedure (non-unique of course) using terms, definitions, axioms and logical deduction in order to show a statement is true or false. • Theorem • Mathematical statements constructed from axioms, other theorems and a proof. • Mathematicians prove statements by carefully telling stories about how they are true, and that is the artistic aspect of our job.(Because some stories are short and convincing, and other stories are boring and questionable.)

17. Axiomatic System (2) • Properties of an Axiomatic System • Consistent • No two statements contradict each other. • Independent • Axioms cannot be proven/derivedfrom other axioms. • Complete • Impossible to add/remove new(independent & consistent) axiomsinto the system.

18. Axiomatic System (2) • Several Axiomatic Systems can interact by equipping them with definitions. • E.g. “points” and “lines” in geometry can be mapped to “numbers” and “intervals” in number theory.

19. Euclid’s Postulates of Geometry • [P1] There is one and only one straight line through two distinct points. • [P2] Lines can be extended indefinitely from a segment. (Straightedge) • [P3] For any point and positive number, there exists a circle centered at the point with the number as a radius. (Compass) • [P4] All right angles are equal to each other. • [P5] Given two straight lines, if a third straight line is drawn through them making non-equal angles, then the initial two lines will meet on one side of the third.i.e. Two lines that are not parallel will meet exactly once.Postulate 5 was subject of an argument, whether it can or can not be derived from P1–P4?

20. Euclid’s Common Notions • [C1] Things both equal to one thing, are equal.A=C, B=C → A=B • [C2] The result of adding the same amount to two equal things, is two things with equal amounts.A=B → A+C=B+C • [C3] The result of taking away the same amount to two equal things, is two things with equal amounts.A=B → A–C=B–C • [C4] Things which coincide with one another are equal. • [C5] The whole is greater than the part.C exists, A+C=B → B>A

21. Propositions (Book I) • The Propositions are arranged in such a way that later ones are built using definitions, axioms and propositions before. • Axiom 5, the Parallel Postulate, is first used in proof for Proposition 29.

22. Propositions (Book I) • http://aleph0.clarku.edu/~djoyce/elements/elements.htmlDavid E. Joyce (1996)

23. Propositions (Book I) • Proposition 1: To construct an equilateral triangle on a given finite straight line.Which means: Equilateral triangles exist and this is how you make one.

24. Propositions (Book I) • Proposition 1: To construct an equilateral triangle on a given finite straight line. • Weaknesses: • Is C guaranteed to exist? • Didn’t specify circles on same plane.

25. Propositions (Book I) • Proposition 2: To place a straight line equal to a given straight line with one end at a given point.Which means: How do we move (actually copy-pasting) a line?

26. Propositions (Book I) • Proposition 4: If two triangles have two sides equal to two sides respectively, and have the angles contained by the equal straight lines equal, then they also have the base equal to the base, the triangle equals the triangle, and the remaining angles equal the remaining angles respectively, namely those opposite the equal sides.Which means: Two triangles are the same if two of their sides’ lengths and the angle between are equal.

27. Propositions (Book I) • Proposition 4: If two triangles have two sides equal to two sides respectively, and have the angles contained by the equal straight lines equal, then they also have the base equal to the base, the triangle equals the triangle, and the remaining angles equal the remaining angles respectively, namely those opposite the equal sides. • The proof is by “superposition” of one figure to another, which is unclear, but you can use Proposition 2 to “move” lines.

28. Playfair’s Axiom • By John Playfair (1748–1819) of Scotland. Re-formulation of Euclid’s 5th Postulate so it’s easier to use in proofs. • [P5*] In a plane, given a line and a point not on it, at most one line parallel to the given line can be drawn through the point. • Other reformulations have been considered. (see pg. 22 of euclid.pdf)