Math 251 Towson University

1. Math 251 Towson University

2. About the Course • What is geometry?

3. History of Geometry – Early Civs • One of the earliest branches of mathematics • Ancient Egyptians, Babylonians, and Indians used some form of geometry as early as 3000 BC (5000 years ago!) • How do you think they might have used geometry?

4. History of Geometry • Ancient cultures used geometry for: • Measuring land and distances • Measuring angles for building structures and planning cities • Drawing circles for wheels and artistic designs • Use of geometric shapes for altar designs • All three civilizations discovered the Pythagorean Theorem at least 1000 years before Pythagoras himself • Why is it called the Pythagorean Theorem then???

5. Greek Geometry • Greek mathematicians, starting with Thales (“Thay-lees”) of Miletus, proposed that geometric statements should be proved by deductive logic rather than trial and error. • What is the difference between proving a statement by a deductive proof rather than a series of examples? Why might someone prefer a deductive proof? • No matter how many examples you provide, you can never be sure that an example exists that disproves your statement • Even more important, proofs often tell us “why” a statement is true

6. Greek Geometry – Pythagoras • Thales’ student, Pythagoras, continued and expanded on the method of deductive proofs. Pythagoras and his disciples used these methods to prove many geometric theorems. • The most famous -- the Pythagorean Theorem: • The sum of the squares of the two sides of a right triangle equals the square of its hypotenuse a2 + b2 = c2

7. Greek Geometry – Pythagoras • Pythagoras and his disciples also discovered a number of other geometric theorems and mathematical ideas: • Area of a circle • Square numbers and square roots • Irrational numbers • Pythagoras believed that everything was related to mathematics and that numbers were the ultimate reality and, through mathematics, everything could be predicted and measured. • They developed a curriculum for students which divided mathematics into four subjects: Arithmetic, Geometry, Astronomy, and Music

8. Greek Geometry – The Liberal Arts How is this the same as / different from the liberal arts at a university today?

9. Greek Geometry – Euclid • While Euclid did not discover many new theorems, he contributed greatly to the advancement of geometry by collecting known theorems and presenting them in a single, logically coherent book – possibly the first textbook? • Euclid’s goal was to start with a few axioms and use these to prove other geometric statements, thus creating a logical system. • What is an axiom??? Why do we even need them? • An axiom is “a statement that is assumed to be true without presenting any reasoning” • Euclid’s goal was for his axioms to be self-evident • These then serve as a starting point for proving other statements.

10. Euclid’s Axioms • First Axiom: For any two points, there is a unique line that can be drawn passing through them. • Second Axiom: Any line segment can be extended as far as desired. • Third Axiom: For any two points, a circle can be drawn with one point as its center and the other point lying on the circle. • Fourth Axiom: All right angles are congruent to one another. • Fifth Axiom: For every line, and for every point that does not lie on that line, there is a unique line (only one!) through the point and parallel to the line.

11. Euclid’s Axioms • First Axiom: For any two points, there is a unique line that can be drawn passing through them.

12. Euclid’s Axioms • Second Axiom: Any line segment can be extended as far as desired.

13. Euclid’s Axioms • Third Axiom: For any two points, a circle can be drawn with one point as its center and the other point lying on the circle.

14. Euclid’s Axioms • Fourth Axiom: All right angles are congruent to one another. • Angle CAB is congruent to Angle FDE

15. Euclid’s Axioms • Fifth Axiom: For every line, and for every point that does not lie on that line, there is a unique line (only one!) through the point and parallel to the line.

16. Euclid’s Axioms • Does the fifth axiom seem different from the first four? • Euclid himself put off using this axiom for as long as possible, proving his first 28 propositions without using it. • For over 2000 years, mathematicians attempted to deal with this axiom by proving it based on the first four axioms, or replacing it with a more self-evident one. • In the 1800s, mathematicians discovered new systems of geometry that could be created by using a different fifth axiom (“Non-Euclidean Geometry”). We will talk more about this later in the course.