Geometry: Whose Geometry Is It Anyway?

Whose Geometry Is It Anyway?

Euclidean geometry is sometimes referred to as plane geometry. Non-Euclidean geometry, on the other hand, refers to all of the other geometries. Euclidean and non-Euclidean geometry involve more than proving theorems. The ideas in Euclidean geometry can be used to derive formulas (like the Pythagorean Theorem, for example) or give explicit construction instructions (like how to construct an angle bisector). Using Euclidean geometry, you can estimate the height of a tree or the width of a lake. Unfortunately, these applications were established with the help of Euclid's 5th Postulate, and hold true only in Euclidean geometry. There are useful applications of non-Euclidean geometry as well; for example, navigation is one application of spherical geometry.

You might be wondering how Euclid's 5th Postulate was involved in the development of those applications. Most of the time you probably didn't see it. Euclid's 5th Postulate is sneaky, and one of its favorite places to hide is in the shadows of triangles. One reason that Euclid's 5th Postulate is able sneak around is because it was used to prove that the measures of the interior angles of a triangle add up to 180º. As a result, Theorem 11.1 became contaminated, and any theorem that makes use of a relationship betweenthe interior angles of a triangle is suspect.

That's not to say that any time triangles are involved in a proof, the theorem only holds up in Euclidean geometry. Theorems can often be proven in more than one way. One way might involve triangles, another might involve something completely different. If a theorem can be proven in at least one way that isn't contaminated with the Parallel Postulate, then the theorem will hold in Euclidean and non-Euclidean geometry.