Math 251 Towson University

About the CourseWhat is geometry?

History of Geometry – Early CivsOne of the earliest branches of

mathematicsAncient 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?

History of GeometryAncient cultures used geometry for:

Measuring land and distancesMeasuring angles for building structures and planning citiesDrawing circles for wheels and artistic designsUse of geometric shapes for altar designsAll three civilizations discovered the

Pythagorean Theorem at least 1000 years before Pythagoras himself

Why is it called the Pythagorean Theorem then???

Greek GeometryGreek 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

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

b

c

a

a2 + b2 = c2

Greek Geometry – Pythagoras Pythagoras and his disciples also discovered a

number of other geometric theorems and mathematical ideas:Area of a circleSquare numbers and square rootsIrrational 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

Greek Geometry – The Liberal Arts

How is this the same as / different from the liberal arts at a university today?

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-evidentThese then serve as a starting point for proving other

statements.

Euclid’s AxiomsFirst 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.

Euclid’s AxiomsFirst Axiom: For any two points, there is a unique

line that can be drawn passing through them.

BA

A B

Euclid’s AxiomsSecond Axiom: Any line segment can be

extended as far as desired.

A B

A B

Euclid’s AxiomsThird Axiom: For any two points, a circle

can be drawn with one point as its center and the other point lying on the circle.

A BA B

Euclid’s AxiomsFourth Axiom: All right angles are

congruent to one another.

Angle CAB is congruent to Angle FDE

A B ED

C F

Euclid’s AxiomsFifth 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.

A B

C

A B

C

Euclid’s AxiomsDoes 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.