Euclid’s Fifth Postulate

Euclid's Fifth Postulate is one of the most famous postulates in the history of math. Let us try and understand it and its implications.

Folding 2 parallel lines in a sheet of paper

First let us do the following paper folding activity. Take a piece of paper which has no straight edge anywhere along its perimeter. Let it be shaped like an amoeba!

Fold a line. Now fold a 2ndline which is parallel to the first line. Remember that visual approximations are not accepted. It has to be geometrically provable that both the lines are parallel! The purpose of having no straight edges is to avoid the temptation of using the straight edge as a reference.

Some introspection would show that the "only way" to fold 2 parallel lines is this; fold the perpendicular to the first line and then fold the perpendicular to the perpendicular! A little more introspection will show that this is another perspective of the Fifth Postulate - a perpendicular to a perpendicular of a line would be parallel to it!

It uses the property that when 2 parallel lines are intercepted by a transversal, the interior angles sum up to a straight angle. Conversely if the interior angles formed when a transversal intersects 2 lines, sum up to a straight angle, then both the lines are parallel.

Philosophers and mathematicians had no difficulty in accepting the first four postulates. But many felt that the fifth postulate is too complex to be fundamental. It was not self-evident. They felt that it can be proved from some other simpler postulate.

The order in which the postulates were listed in the Elements seems to indicate that Euclid included this postulate only when he realised he could not prove it or proceed without it.

For almost 2000 years, several mathematicians, including famous ones tried to prove the fifth postulate from other assumptions. None of them were successful. Many of them were accepted as proofs for long periods until the mistake was found. Invariably the mistake was assuming some ‘obvious’ property which turned out to be equivalent to the fifth postulate

It was finally in the 19thcentury that the problem with the Fifth Postulate was understood.

Mathematicians proved that this postulate is true only if the figures are drawn on a plane surface, like that of a table. It is not true if the figures are drawn on curved surfaces. These gave rise to what are called non-Euclidian geometries!

Non-Euclidian Geometries

Euclid’s Geometry states that only one line can be drawn parallel to a given line and passing through a given point.

By assuming that an infinite number of parallel lines can be drawn, geometers invented what is called “hyperbolic geometry”. Here the plane is not plane but shaped like a horse saddle (concave)

And by assuming that no parallel lines can be drawn, geometers invented what is called “spherical or elliptic geometry”. Here the plane is shaped like a sphere (convex)

We will see more about Spherical Geometry in chapter 32.11.

Power of the Axiomatic Method

The non-Euclidian geometries were invented after the “assumptions” of the axiomatic method was Euclid was fully understood. Understanding gave rise to “flexibility” of thinking about them. The development of non- Euclidian geometries is an example of the logic of the axiomatic system of plane geometry leading to other geometries!

This also showed the criticality of the basic assumptions in any system. There can be no proof without basic assumptions. Hence in any proof, it is important to check the basic assumptions.

It is a pity that the geometry curriculum in today’s schools has reduced the thinking aspects and increased the memorizing aspects.