Chapter 1: Euclid's Elements - users.miamioh.edu

Chapter 1: Euclid’s Elements

MTH 411/511

Foundations of Geometry

MTH 411/511 (Geometry) Euclid’s Elements Fall 2020

It’s good to have goals

Goals for today:• Discuss historical foundations for geometry.• Examine Euclid’s axioms and their consequences.• Give an example of a formal axiomatic proof in geometry.



Goals for today:

• Discuss historical foundations for geometry.• Examine Euclid’s axioms and their consequences.• Give an example of a formal axiomatic proof in geometry.



Goals for today:• Discuss historical foundations for geometry.

• Examine Euclid’s axioms and their consequences.• Give an example of a formal axiomatic proof in geometry.



Goals for today:• Discuss historical foundations for geometry.• Examine Euclid’s axioms and their consequences.

• Give an example of a formal axiomatic proof in geometry.



Goals for today:• Discuss historical foundations for geometry.• Examine Euclid’s axioms and their consequences.• Give an example of a formal axiomatic proof in geometry.


Euclid’s Elements

Figure: Euclid (source:Wikipedia, public domain)

Euclid is responsible for setting math on a path to for-malism.

His books consist of four main ingredients:• Definitions (defined terms)• Common notions (accepted facts)• Postulates (assumed facts)• Propositions (statements proved from the above).


Euclid’s Elements


Euclid is responsible for setting math on a path to for-malism. His books consist of four main ingredients:

• Definitions (defined terms)• Common notions (accepted facts)• Postulates (assumed facts)• Propositions (statements proved from the above).


Euclid’s Elements



• Definitions (defined terms)

• Common notions (accepted facts)• Postulates (assumed facts)• Propositions (statements proved from the above).


Euclid’s Elements



• Definitions (defined terms)• Common notions (accepted facts)

• Postulates (assumed facts)• Propositions (statements proved from the above).


Euclid’s Elements



• Definitions (defined terms)• Common notions (accepted facts)• Postulates (assumed facts)

• Propositions (statements proved from the above).


Euclid’s Elements



• Definitions (defined terms)• Common notions (accepted facts)• Postulates (assumed facts)• Propositions (statements proved from the above).


Euclid’s Elements

Euclid’s Five Postulates

1. To draw a straight line from any point to another point.2. To produce a straight line continuously in a straight line.3. To describe a circle with any center and distance.4. That all right angles are equal to one another.5. (Parallel Postulate) That, if a straight line falling on two straight lines makes the

interior angles on the same side less than two right angles, the two straight lines,if produced infinitely, meet on that side which are the angles less than the tworight angles.


Euclid’s Elements

Euclid’s Five Postulates1. To draw a straight line from any point to another point.

2. To produce a straight line continuously in a straight line.3. To describe a circle with any center and distance.4. That all right angles are equal to one another.5. (Parallel Postulate) That, if a straight line falling on two straight lines makes the



Euclid’s Elements

Euclid’s Five Postulates1. To draw a straight line from any point to another point.2. To produce a straight line continuously in a straight line.

3. To describe a circle with any center and distance.4. That all right angles are equal to one another.5. (Parallel Postulate) That, if a straight line falling on two straight lines makes the



Euclid’s Elements

Euclid’s Five Postulates1. To draw a straight line from any point to another point.2. To produce a straight line continuously in a straight line.3. To describe a circle with any center and distance.

4. That all right angles are equal to one another.5. (Parallel Postulate) That, if a straight line falling on two straight lines makes the



Euclid’s Elements

Euclid’s Five Postulates1. To draw a straight line from any point to another point.2. To produce a straight line continuously in a straight line.3. To describe a circle with any center and distance.4. That all right angles are equal to one another.

5. (Parallel Postulate) That, if a straight line falling on two straight lines makes theinterior angles on the same side less than two right angles, the two straight lines,if produced infinitely, meet on that side which are the angles less than the tworight angles.


Euclid’s Elements

Euclid’s Five Postulates1. To draw a straight line from any point to another point.2. To produce a straight line continuously in a straight line.3. To describe a circle with any center and distance.4. That all right angles are equal to one another.5. (Parallel Postulate) That, if a straight line falling on two straight lines makes the



Euclid’s Elements

Euclid’s Five Postulates (translated)

1. There is a line joining any two points.2. Any line segment can be extended to a line segment of any desired length.3. For every line segment L, there is a circle that has L as its radius.4. All right angles are congruent to each other.5. (Parallel Postulate) If P is a point and L is a line not passing through P, then

there is exactly one line through P and parallel to L.

P

L

QuestionDoes the Parallel Postulate follow from the other four postulates?

This question bothered geometers for centuries after Euclid. Euclid himself tried(unsuccessfully) to prove this. In the early-to-mid nineteenth century, this questionwas finally resolved.


Euclid’s Elements

Euclid’s Five Postulates (translated)1. There is a line joining any two points.

2. Any line segment can be extended to a line segment of any desired length.3. For every line segment L, there is a circle that has L as its radius.4. All right angles are congruent to each other.5. (Parallel Postulate) If P is a point and L is a line not passing through P, then


P

L




Euclid’s Elements

Euclid’s Five Postulates (translated)1. There is a line joining any two points.2. Any line segment can be extended to a line segment of any desired length.

3. For every line segment L, there is a circle that has L as its radius.4. All right angles are congruent to each other.5. (Parallel Postulate) If P is a point and L is a line not passing through P, then


P

L




Euclid’s Elements

Euclid’s Five Postulates (translated)1. There is a line joining any two points.2. Any line segment can be extended to a line segment of any desired length.3. For every line segment L, there is a circle that has L as its radius.

4. All right angles are congruent to each other.5. (Parallel Postulate) If P is a point and L is a line not passing through P, then


P

L




Euclid’s Elements

Euclid’s Five Postulates (translated)1. There is a line joining any two points.2. Any line segment can be extended to a line segment of any desired length.3. For every line segment L, there is a circle that has L as its radius.4. All right angles are congruent to each other.

5. (Parallel Postulate) If P is a point and L is a line not passing through P, thenthere is exactly one line through P and parallel to L.

P

L




Euclid’s Elements

Euclid’s Five Postulates (translated)1. There is a line joining any two points.2. Any line segment can be extended to a line segment of any desired length.3. For every line segment L, there is a circle that has L as its radius.4. All right angles are congruent to each other.5. (Parallel Postulate) If P is a point and L is a line not passing through P, then


P

L




Euclid’s Elements



P

L


This question bothered geometers for centuries after Euclid.

Euclid himself tried(unsuccessfully) to prove this. In the early-to-mid nineteenth century, this questionwas finally resolved.


Euclid’s Elements



P

L


This question bothered geometers for centuries after Euclid. Euclid himself tried(unsuccessfully) to prove this.

In the early-to-mid nineteenth century, this questionwas finally resolved.


Euclid’s Elements



P

L




The Fifth Postulate

Figure: Janos Bolyai (source:Wikipedia, by FerencMarkos)

Figure: Carl Friedrich Gauss(source: Wikipedia, byChristian Albrecht Jensen)

Figure: Nikolai Lobachevsky(source: Wikipedia, by LevKryukov)

These men, independently, came to the realization that the parallel postulate was bothnecessary and unnecessary. On one hand, Euclidean Geometry requires the fifthpostulate to accomplish much of what Euclid did. On the other hand, there arecompletely consistent geometries in which this postulate does not hold.


Definitions

Figure: David Hilbert(source: Wikipedia, publicdomain)

Euclid’s definitions are sometimes vague. For example,he defines a point as “that which has no part”, whichis difficult to interpret.

Hilbert’s goal was to reduce ge-ometry to a level of abstraction where “point” could bereplaced by “frog” and it would not matter.

= •

Our goal will be rigor and building geometry from its foundations. Let’s now take alook at one of Euclid’s arguments using his axioms.


Definitions

Figure: David Hilbert(source: Wikipedia, publicdomain)

Euclid’s definitions are sometimes vague. For example,he defines a point as “that which has no part”, whichis difficult to interpret. Hilbert’s goal was to reduce ge-ometry to a level of abstraction where “point” could bereplaced by “frog” and it would not matter.

= •

Our goal will be rigor and building geometry from its foundations. Let’s now take alook at one of Euclid’s arguments using his axioms.


An example

PropositionOn a given finite straight line to construct an equilateral triangle.

(Given a linesegment, there exists an equilateral triangle having that segment as a side.)

A B

C

D

Why does C exist?

Proof.Let AB be the given finite straight line. With center A and distance AB, let circleBCD be described (Post. 3). With center B and distance BA let the circle ACE bedescribed (Post. 3). From the point C in which the circles cut one another, to thepoints A and B, let the straight lines CA and CB be joined (Post. 1).

Since the point A is the center of the circle CBD, AC is equal to AB (Defn 15: radii)Similarly, BC is equal to BA. As AC is equal to AB and AB is equal to BC , then ACis equal to BC (CN 1: transitivity). Thus, AC , AB, and BC are all equal to oneanother so the triangle ABC is equilateral and AB is a side. Q.E.D.


An example

PropositionOn a given finite straight line to construct an equilateral triangle. (Given a linesegment, there exists an equilateral triangle having that segment as a side.)

A B

C

D

Why does C exist?

Proof.Let AB be the given finite straight line. With center A and distance AB, let circleBCD be described (Post. 3). With center B and distance BA let the circle ACE bedescribed (Post. 3). From the point C in which the circles cut one another, to thepoints A and B, let the straight lines CA and CB be joined (Post. 1).

Since the point A is the center of the circle CBD, AC is equal to AB (Defn 15: radii)Similarly, BC is equal to BA. As AC is equal to AB and AB is equal to BC , then ACis equal to BC (CN 1: transitivity). Thus, AC , AB, and BC are all equal to oneanother so the triangle ABC is equilateral and AB is a side. Q.E.D.


Next time

Before next class: Read Sections 2.1-2.2.

In the next lecture we will:• Define an axiomatic system.• Define incidence geometry and consequences of the axioms.• Consider several examples (and non-examples) of incidence geometries.


Next time


In the next lecture we will:

• Define an axiomatic system.• Define incidence geometry and consequences of the axioms.• Consider several examples (and non-examples) of incidence geometries.


Next time


In the next lecture we will:• Define an axiomatic system.

• Define incidence geometry and consequences of the axioms.• Consider several examples (and non-examples) of incidence geometries.


Next time


In the next lecture we will:• Define an axiomatic system.• Define incidence geometry and consequences of the axioms.

• Consider several examples (and non-examples) of incidence geometries.


Next time


In the next lecture we will:• Define an axiomatic system.• Define incidence geometry and consequences of the axioms.• Consider several examples (and non-examples) of incidence geometries.


Chapter 1: Euclid's Elements - users.miamioh.edu

Documents

Transcript of Chapter 1: Euclid's Elements - users.miamioh.edu