A study of geometry, based on the rules and definitions set forth in Euclid's landmark texts, The Elements. These books included:
Definitions of all terms used throughout the texts and today
Axioms that laid out the essential (and unproveable) rules of his geometry
Theorems and proofs of each that make up the entirety of geometry as we know it today
1. A straight line can be drawn through any two points.
2. A finite line segment can be extended to an infinite line.
3. A circle can be described by a point and a radius.
4. All right angles are equal.
5. If a straight line falling on two straight lines makes the interior angles on the same side less than two right angles (180°), the two straight lines are not parallel.
The fifth axiom can be restated as:
5. Given a line and a point not on that line, there is exactly one line through the point parallel to the original line.
The last axiom is popularly known as the “parallel postulate”.
It’s different from the others, because it’s the only statement that feels like it needs it’s own proof. The rest are self-evident, but the fifth is not.
However, it cannot be proven using the other four axioms. It’s just something we understand to be true, and used as a cornerstone of geometry for over two thousand years.
We also rely on it for many of the theorems presented later in The Elements.
This means that if the axiom is NOT true, many of the facts we know about geometry would be wrong.
The Rise of Non-Euclidean Geometry
By the 1800s, many mathematicians started to wonder what would happen if we ignored the fifth axiom. They developed new geometries that either did not include the fifth axiom at all, or assumed it was actually false.
These new geometries were called Non-Euclidean Geometries. The two principle categories of non-Euclidean geometry include
Spherical geometry: the geometry that applies to shapes drawn on a sphere.
Hyperbolic geometry: the geometry that applies to shapes in a hyperbolic (saddle shaped) plane.
What Can We Lose?
Facts from Euclidean Geometry:
The sum of the interior angles of any triangle will be 180°.
Lines are straight.
The fifth axiom itself: there is only one line through a point parallel to a given line.