The Parallel Postulate

Take a piece of paper and draw a dot anywhere on the page. Let’s call that dot P.

Now, grab a ruler and draw a straight line anywhere else on that page (just make sure it doesn’t go through point P.) Call that line L.

Euclid’s parallel postulate states that there is only one line—just one—that goes through P and does not intersect the line L. In other words, there is only one other line that is parallel to L and goes through P.

As you look at your piece of paper, you may be thinking, “Sure, of course this is true.” It seems like common sense after all. That’s what many mathematicians before you thought for many centuries.

But there’s one thing we haven’t tried…

Grab a ball this time, choose a point and call it P. Then draw another “straight” line on the curved surface of this ball. In this case, just as on the flat paper, we define a straight line as the shortest distance between two points. On a curved surface, this is called a geodesic. If you imagine putting two sticks into the ball and pulling a string tightly between them, the string would be an example of a geodesic.

In this case, there can be many lines that pass through P but do not intersect L.

For a long time, the parallel postulate caused a lot of controversy because it could not be proven from Euclid’s four main axioms, which are:

1. A straight line may be drawn between any two points.
2. Any terminated straight line may be extended indefinitely.
3. A circle may be drawn with any given point as the Center and any given radius.
4. All right angle are equal.

Mathematicians tried to demonstrate that using these axioms, the parallel postulate was a self-evident result… but it turned out not to be so evident.

Then, in the 19th century, a Russian and a Hungarian mathematician, Lobachevsky and Bolyai, discovered non-Euclidean geometry. This involved geometry on spheres, like the example above, and on hyperbolic planes (which are curved inwards). They showed that altering the parallel postulate leads to geometries where the postulate does not hold true.

Nowadays, we tend to think of Euclidean and non-Euclidean geometries as separate frameworks, built on separate sets of axioms. Einstein’s relativity is an example of how we use non-Euclidean geometry, specifically Riemannian geometry, in science. On the other hand, Euclidean geometry is used in more practical aspects of daily life, such as architecture and construction.