# Differences Between Euclidean & Non-Euclidean Geometry

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• 0:04 Who Was Euclid?
• 0:47 Euclidean Geometry
• 1:29 Euclidean vs. Non-Euclidean
• 2:02 Types of Non-Euclidean…
• 3:27 Lesson Summary
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Lesson Transcript
Instructor: Betsy Chesnutt

Betsy teaches college physics, biology, and engineering and has a Ph.D. in Biomedical Engineering

Euclidean geometry is the study of the geometry of flat surfaces, while non-Euclidean geometries deal with curved surfaces. Here, we'll learn about the differences between these mathematical systems and the different types of non-Euclidean geometry.

## Who Was Euclid?

Sometime in the 4th century BCE, a boy was born in Alexandria who would grow up to become one of the most famous mathematicians and thinkers who ever lived. His name was Euclid, which, in Greek, means 'renowned and glorious'.'

Euclid was a famous mathematician in his own time, and he only became more famous and influential in the thousands of years that followed. He wrote a book called The Elements in which he laid out the basic principles of geometry that we still know and use today. His book was the primary textbook used to teach mathematics in the Western world from the time it was written until the beginning of the twentieth century. Even today, much of what is taught in a typical geometry course comes from Euclid.

## Euclidian Geometry

This is called Euclidean geometry and it is the study of the geometry of flat surfaces. In Euclidean geometry, the interior angles of a triangle always add together to make 180 degrees, but as we will see, that is not true in the non-Euclidean geometries.

One of the most important of Euclid's postulates in The Elements was the parallel postulate. In simple terms, the parallel postulate says that if you have a line and a point, there is only one other line that you can draw though the point that will be parallel to the original line. This is definitely true on a flat, two-dimensional surface, but it turns out to not be true in some other situations, including when the surface is curved.

## Euclidean vs. Non-Euclidean

While Euclidean geometry seeks to understand the geometry of flat, two-dimensional spaces, non-Euclidean geometry studies curved, rather than flat, surfaces. Although Euclidean geometry is useful in many fields, in some cases, non-Euclidean geometry may be more useful. For example, suppose you want to measure the shortest distance between points on the Earth. The surface of the Earth is curved, not flat (a fact that Euclid was not aware of). Of course, techniques from non-Euclidean geometry would be much more useful in this case since the Earth is a sphere.

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