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Euclid's Axiomatic Geometry: Developments & Postulates Video

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  • 0:06 Euclid
  • 0:44 Euclid's Postulates
  • 2:21 The Axiomatic System
  • 3:34 Euclidean Geometry
  • 4:39 Lesson Summary
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Lesson Transcript
Instructor: Yuanxin (Amy) Yang Alcocer

Amy has a master's degree in secondary education and has taught math at a public charter high school.

Learn how the way we do proofs in geometry had its start with Euclid in this video lesson. Learn about his contributions to the geometry we know today. Also learn about the five basic truths that he used as a basis for all other teachings.

Euclid

Euclid was a Greek mathematician who introduced a logical system of proving new theorems that could be trusted. He was the first to prove how five basic truths can be used as the basis for other teachings. He wrote a series of books that, when combined, becomes the textbook called the Elements in which he introduced the geometry you are studying right now. It is in this textbook that he introduced the five basic truths or postulates upon which the whole of geometry at that time was based.

Euclid's Postulates

What are these postulates that he introduced as the basis for geometry? You might be surprised by how simple some of them seem to be. And that is as it should be. These are basic truths that should be self-evident and do not require formal proofs. The five postulates that he introduced are these:

1. A line can be drawn between any two points.

2. Any line segment can be extended to infinity in both directions.

3. A circle can be described with just a center point and radius.

4. A right angle is equal to all other right angles.

5. This last one is also called the parallel postulate. It states that if one line intersects two other lines and forms angles less than 90 degrees on one side, then the two lines will intersect on that side.

Looking at the first four, you can see that these are intuitive and very simple truths. You can see that they are true and do not require a formal proof. The last one may not seem so intuitive, and mathematicians have argued that the parallel postulate requires a proof, but mathematicians have seen over time that it is a basic truth that does not need a formal proof.

These five postulates, or basic truths, are also called axioms, and Euclid used these axioms to introduce his axiomatic system.

The Axiomatic System

The axiomatic system is a system of formal proofs. It provides a system where new theorems are proved using basic truths. Because these theorems are based on basic truths, they can be relied on and used with trust that they will work and will give reliable answers.

How does the axiomatic system work? You can think of it as a lawyer trying to prove a case. A lawyer needs reliable evidence that cannot be refuted. A lawyer needs to review some basic facts that he knows are true, and then he builds on these facts to build new ones that lead to his conclusion of either guilty or not guilty. It is similar with the axiomatic system. A set of basic truths that are self-evident are used to prove new conclusions. Mathematicians use this system to either show a new conclusion or new theorem to be true or false. True theorems are then presented to their students or written in textbooks. All of these theorems that have been proved are now part of what we know as geometry. Many of the theorems you will learn have been proved using the axiomatic system.

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