Saddle Up!

Way back in the years of when, it was speculated that order was needed in math, and geometry was the first victim.

A guy named Euclid decided he would formalise it, so he set out to "axiomise" it....

...and he came up with these, seemingly "self-evident" truths:

Euclid's Five Postulates are:

1. The shortest distance between any two points is a straight line

2. Any straight line can be extended (produced) indefinitely

3. Any point can be surrounded by a circle of any size

4. All right angles are equal to all other right angles

5. Parallel lines never meet

Above you see a translation of them into normal language. The first four don't need much tinkering with, but the fifth was originally stated thus:

"It is true that, if a straight line falling on two straight lines make the interior angles on the same side less than two right angles, the two straight lines, if produced indefinitely, intersect on that side on which are the angles less than the two right angles."

Quite a mouthful, and clearly, in that context, NOT "self-evident".

Was there a way around this seeming contradiction? Apparently there still is not.

Quite a while after "Euclid's Fifth", several different people came forward with different interpretations of it. It turned out there were three. Einstein's general Relativity is based on one of the two others.

The three interpretations are really easy to state, but all of which are mutually exclusive:

Either,

a. parallel lines will always be parallel (Euclid),

or,

b. parallel lines will always meet (Riemann et al),

or,

c. parallel lines will always diverge (Riemann et al)

It was by using these "alternate" geometries, that Einstein came up with General Relativity in 1915.

From this acorn of an idea, there sprouted formalised geometry and a little thing explaining in full the anomaly of the precession of the perihelion of the planet Mercury to the tune of 43 seconds of arc per century.