In simple terms, the definitions for Euclidean and non-Euclidean geometry?

Question

I missed the first few days of class, the notes I got were not helpful, and neither was the book -

Euclidean geometry: "the collection of propositions about figures which includes or from which can be deduced by those given by the mathematician Eucilid".
non-Euclidean geometry: "geometry in which the postulates are not the same as those in Euclidean geometry."

we've been using these terms before, but they weren't "defined" until page 40. further explanations please?

Answer 1

Euclidean geometry is just another name for the familiar geometry which is typically taught in grade school: the theory of points, lines, angles, etc. on a flat plane. All of Euclidean geometry can be deduced from just a few properties (called "axioms") of points and lines. These properties are all pretty basic and self-evident things like "for every pair of distinct points, there is exactly one line containing both of them".

Although there are multiple postulates included in Euclidean geometry, the most theoretical axiom (which is almost more of a theory), is the Parallel Postulate (sometimes known as the Fifth Postulate). This postulate states that for every line (l) and every point (p) which does not lie on l, there is a unique line, which passes through p and does not intersect l (i.e., which is parallel to l).

Eventually it was discovered that the Parallel Postulate is independent of the other axioms, in the sense that it is logically self-consistent to have some things called "lines" and other things called "points" which satisfy the other axioms but don't satisfy the parallel postulate. Any such collection of things is called a non-Euclidean geometry.

There are many examples. Most concretely, if you do geometry on a curved surface instead of on a flat plane (where now "line" refers to the shortest path between two points, which obviously will not be straight if you are on a curved surface), you typically end up with a non-Euclidean geometry.

Hopefully that's clear enough to help you out. Good luck!

Answer 2

Euclidean Geometry Definition

Answer 3

Euclidean Definition

Answer 4

Non-Euclidian geometry arose from an attempt to prove the parallel postulate from the other postulates. It is the one assumption that Euclid made that seemed like a leap.

Mathematicians eventually found that the parallel postulate was indepenent of the others. You may recall that the postulate says that for any line and a point not on the line -- there is one and only one line that goes through the point that does not intersect the original line.

This led to two other geometries being discovered. The first postulates that there are no lines going through that point that do not intersect the line. This can be acheived on the geometry of a spehere (e.g., Earth). On the surphase of a sphere -- lines are just great circles. All great circles intersect at two points.

The other geometry assumes that there are multiple lines that go through the point that do not intersect. As I recall (and it has been a few decades since I learned this) -- this geometry works on a surface of a manifold that looks something like two horns taped together at the ends.

Answer 5

Euclidean geometry deals with shapes on the xy-plane. That's the stuff you learn about in high school. Non-Euclidean geometry deals with shapes on surfaces that are not flat. The most common real world example is travelling by sea on the earth. Because the earth is round (a sphere almost), a ship may wish to traverse from one island to another (over many thousands of miles). What is the shortest route? Answer: IT IS NOT A STRAIGHT LINE, BECAUSE THE EARTH IS ROUND EVERYWHERE. However, the solution to this problem is to find a curve along the earth's surface that acts like a straight line on the xy-plane. This turns out to be what is called a geodesic.