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### Course: High school geometry>Unit 1

Lesson 1: Intro to Euclidean geometry

# Euclid as the father of geometry

Euclid, often called the father of geometry, changed the way we learn about shapes with his 13-book series, Euclid's Elements. He used basic ideas called axioms or postulates to create solid proofs and figure out new ideas called theorems and propositions. Studying Euclidean geometry helps us think better and solve problems more effectively. Created by Sal Khan.

## Want to join the conversation?

• Is the "Element" textbook used today? Is it still printed for geometry students?
Where can I buy a copy?
• Since theorems in Euclidean geometry are assumed to be true and are the foundation of much of mathematics, wouldn't almost everything we've learned by obsolete if this assumption of truth happened to be false?
• There are other branches of mathematics that do not use Euclid's axioms as their basis, such as spherical geometry and many others. These geometries reject Euclid's axioms and substitute others, and thus the properties of lines and shapes and other things are different from those in Euclid.

But that doesn't mean Euclid is wrong. Euclidean geometry is consistent within itself, meaning the axioms all agree with each other and with all the properties derived from them. That's all you can ask from a branch of mathematics--internal consistency. There is no one universal geometry that satisfies all situations and which contains all possible true statements. So we have to start by defining our terms (axioms), and Euclid was the first one to do that.
• Were any of Euclid's basic geometric assumptions (or axioms/postulates) ever shown to be incorrect?
• There was a big debate for hundreds of years about whether you really needed all 5 of Euclid's basic postulates. Mathematicians kept trying to prove that the 5th postulate (commonly known as the parallel postulate) could be proved from the first four postulates and thus was unnecessary. Some really great proofs were created by mathematicians trying to prove the parallel postulate.

In the late 19th century (approximately 1823), three different mathematicians (Bolyai, Lobachevsky and Gauss) proved independently that there was a different system that could be used that assumed the 5th postulate was incorrect. Later (1868) it was proved that the two systems were equally consistent and as consistent as the real number system. We can see different versions of systems where the parallel postulate is false by assuming that either there are no parallel lines, or that for any line and point not on a line there are an infinite number of parallel lines.

On a side note, in 1890, Charles Dodgson (aka Lewis Carroll author of Alice in Wonderland) published a book with a "proof" of the parallel postulate using the first 4 postulates. Not a great proof and written after it was proven that this could not be proved.
• At Sal talks how postulates are used to deduce theorems. When the postulates were proven does it make them theorems? If not, what is the difference?
• Not quite. The postulates are the things that we assume to be true from the beginning that form the foundation for all of our theorems. There are five in Euclidean geometry: that any two points can be connected by a straight line, that any line segment can be stretched out forever in either direction, that we can always define a circle given a center and a radius, that all right angles are congruent, and that for any line and any point not on that line there is exactly one line parallel to the given line that passes through the given point. None of those postulates can be defined from each other, but with the five of them we can prove everything in geometry.
• I really did not get the video. But I really want a copy of the "Element" textbook so I can learn more. Does anyone know where to get a copy?
• If you search Amazon, you'll find no lack of copies of Euclid's Elements. Alternatively, any geometry textbook that is more than thirty years old follows Euclid's path quite precisely. Also, if you google for Euclid's Elements, the top hit will be a fantastic online companion hosted by Clark University that gives context and commentary to every proposition along the way.
• Someone should have stopped Euclid from creating Geometry
• Geometry allows students to connect mapping objects in the classroom to real-world contexts regarding direction and place. Understanding of spatial relationships is also considered important in the role of problem solving and higher-order thinking skills.
• can somebody please explain euclids 5th postulate cause i dont get it
• Here is what the Fifth Postulate means, though Euclid didn't state it quite this way:

Two lines intersect a third line.
Measure the interior angles of the two lines on the same side of the third line.
Add the two interior angles together.
If the sum of those two interior angles is less than 180°, then those lines will intersect on that side of the third line.
If the sum is greater than 180°, then those lines intersect on the other side of the third line.
If the sum is exactly 180°, then those lines are parallel and will never intersect.

This postulate has never been successfully proven. It only works on flat surfaces (Euclidean geometries). On curved surfaces, the fifth postulate is not true.