History of Euclidean Geometry and Axiomatic Systems.pptx

This presentation covers the history and development of Euclidean geometry, exploring its origins in ancient Greece and the contributions of mathematicians like Euclid. It delves into the structure of Euclid's axiomatic system, highlighting the fundamental axioms and postulates that form the basis of classical geometry. Additionally, the presentation discusses the evolution of axiomatic systems over time and their impact on the development of modern mathematical theories.

1. History of Euclidean Geometry and Axiomatic Systems Dr Shaymaa Darwish

2. Early origins of geometry, Thales, Pythagoras, rise of the axiomatic method The word geometry comes from the ancient Greek geo (Earth), and metros (measure). Measurement (of distance, area, height, angle)

3. History of Geometry Applications: construction : The pyramids were built primarily as tombs, homes,… Surveying : The land of ancient Egypt lay close to the Nile and was divided into plots leased to local Egyptians to farm. Tax collection : To re-calculate how much land was left in order to levy the appropriate amount of rent. Another application : religious practice, astronomy, navigation. Limitations: Typically worked examples without general formulæ /abstraction. They did not delineate between values that were approximations and those that were exact. For example, they knew that certain triples of integers, say 3, 4, and 5, would form the lengths of a right triangle, but they had no notion of what the relationship between the sides of a right triangle were in general. Ancient times (pre-500 BC) : Start in Egypt: basic rules for measuring lengths, areas and volumes of simple shapes.

4. Ancient Greece (from c. 600 BC) In the period between 900 and 600 BC, while the Egyptian empire was waning, a new seafaring and trading culture arose in Greece. The Egyptians study of land measurement was passed on to the Greeks through trading and is evidenced by the word geometry itself, which in Greek means “earth measure.” Philosophers such as Thales and Pythagoras began the process of abstraction. General statements (theorems) formulated and proofs attempted.

5. The mathematical results should be justified and proved by a method of reasoning and argument that was precise and logical. Based on system of deductive reasoning. A method which very useful to prove theorems from axiom A method of proving that the results are correct and organizing them into logical structure The greatest achievement of the Greeks was the development of a precise and logical way of reasoning called the deductive method Deductive method

6. Thales of Miletus (c. 624–546 BC) Thales is known for being the first mathematician, the first to use deductive reasoning to prove mathematical results The five geometric theorems attributed to Thales are A circle is bisected by a diameter. 2. The base angles of an isosceles triangle are equal.

7. 3. The pairs of vertical angles formed by two intersecting lines are equal. 4. Two triangles are congruent if they have two angles and the included side equal.

8. 5 . An angle inscribed in a semicircle is a right angle. “ The Theorem of Thales .” The radius of the circle splits the large triangle into two isosceles triangles. Theorem 2 says that these have equal base angles (labelled), Check that α + β is half the angles in a triangle, namely a right-angle Proof:

9. Philosopher / mathematician, 570 – 495 BCE, from Samos Island, Greece was a pupil of Thales. Pythagoras And The Disciples ( Pythagoreans ): introduced the idea that mathematical propositions need to be supported with a strong proof by reasoning / arguments which are precise & logical . Numbers rule the universe” was their motto. They believed that all of nature could be explained by properties of the natural numbers 1, 2, 3, . . Pythagoras of Samos

10. The theorem historically attributed to Pythagoras, and the Pythagoreans were the first to provide a logical proof of this result: “in a right triangle the square on the hypotenuse is equal to the sum of the squares on the two sides”

11. Euclid Euclid of Alexandria (c. 300 BC) Collected and expanded earlier work, especially that of the Pythagoreans. His compendium the Elements is one of the most important books in Western history The Elements is an early exemplar of the axiomatic method at the heart of modern mathematics. 13 volumes, up to 465 propositions

12. Definitions in mathematics are numerous, but attempting to define everything is impractical because definitions rely on other concepts. This leads either to an endless chain of definitions or circular definitions. In some cases, it's more practical to accept certain terms as fundamental and intuitively understood, such as in set theory. Euclid, however, aimed to define all objects in his geometry, although this approach might not always be advisable. Despite this, historical reasons may necessitate the reproduction of certain definitions. Notes

13. 1.3 The Rise of the Axiomatic Method Starting from a base of undefined terms and agreed upon axioms, we can define other terms and use our axioms to argue the truth of other statements. These other statements are called the theorems of the system. Thus, our deductive system consists of four components: Undefined Terms Axioms (or Postulates) Defined Terms Theorems

14. Undefined Terms Geometry : Point, Line, Incident, Between, Congruent Elements of theorem accepted without further definition. Set theory : Set , Belonging to a set, being a member of a set , intersect, included

15. Axioms Statements constructed from terms accepted as true without an attached proof. Axioms (or Assumptions, Postulates) give meaning to undefined terms, showing how they relate to each other . Considered the starting point of reasoning. Used to prove other statements

16. Defined Terms Mathematical statements which can be constructed/derived from and need proof. Can be constructed/derived from axioms, other/previous theorems and definitions using proof procedure. Theorems Starting from a base of undefined terms and agreed upon axioms, we can define other terms

17. No two statements contradict each other 1.4 PROPERTIES OF AXIOMATIC SYSTEMS Consistent Impossible to add/remove new (independent & consistent) axioms into the system . We can verify that an axiomatic system is complete by showing that there is essentially only one model for it Independent Axioms cannot be proven/derived from other axioms . We can verify that a specified axiom is independent of the others by finding two models - one for which all of the axioms hold, and another for which the specified axiom is false but the other axioms are true Complete

18. Consider the following axiomatic system: A1 There are exactly three points. A2 Two distinct points belong to one and only one line. A3 Not all of the points belong to the same line. A4 Two separate lines have at least one point in common . Example 1.5… Three-Point Geometry Consistent, Independent, Complete

19. Example 1.6… System 1: A1 There are exactly three points. A2 There are exactly two points. Consider the following axiomatic systems: System 2: A1 There are exactly three points. A2 Two distinct points belong to one and only one line. Not Consistent Not Complete

20. System 3: A1 There are exactly three points. A2 Two distinct points belong to one and only one line. A3 Not all of the points belong to the same line. A4 Two separate lines have at least one point in common. A5 A line has exactly two points. Not Independent

21. Consider the following axiomatic systems: System 1: System 2: Excersice 1.5… System 4: System 3:

22. Example 1.7 : Consider the following axiomatic system, prove theorems 1 and 2?

23. 1.5 Euclid’s Axiomatic Geometry System of deductive reasoning (Thales/ Phythagoras ) was codified and put into definitive form by Euclid around 300 BC in Elements (13 vol). Elements: so comprehensive that superseded all previous textbooks in geometry. Euclid’s exposition is important because its clarity and rigor. Comprehensive description and explanation of an idea or theory.

24. Theorem 2: There exist at least four distinct points. Proof: By Axiom 2, there exist at least two distinct points p and q By Axiom 3, there exist a line L containing p and q . By Axiom 4, there exists a point not on L , say x . By Axiom 5, there exists parallel line ! " containing x and parallel to L .

25. By Definition 2, two lines L1 and L are called parallel if there is no point which is on both L1 and L . Thus, L1 must contains other points other than x since every line is a collection of points by Axiom 1, say on of these points is x’ . Hence, p , q , x and x’ are 4 distinct points. Thus, there exist at least four distinct points at two lines

26. Euclid V : 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 Euclid’s Postulates Euclid I : There is one and only one straight line through two distinct Points Euclid II: Lines can be extended indefinitely from a segment. (Straightedge) Euclid III: For any point and positive number, there exists a circle centered at the point with the number as a radius. (Compass) Euclid IV: All right angles are equal to each other.

27. Euclid’s Common Notions Notation 1: Things both equal to one thing, are equal. A=C, B=C → A=B Notation 2: The result of adding the same amount to two equal things, is two things with equal amounts. A=B → A+C=B+C Notation 3: The result of taking away the same amount to two equal things, is two things with equal amounts. A=B → A–C=B–C

28. Notation 4: Things which coincide with one another are equal. A ⸦ B , B ⸦ A → A = B Notation 5: The whole is greater than the part. C exists, A+C=B → B>A