From the Nile and the Euphrates
Flowed knowledge of an art on which so many other arts are based. From the Fertile Crescent up to Greece
mathematics began to flow
Sometime in the 6th century BCE, the great Geometer, Pythagoras of Samos went to Egypt
He returned even more fascinated with geometrical ideas than he had been when he had left
He knew there was wisdom and possibilities he had to share
He saw geometry as part of a larger whole. Part of a philosophy about the perfection of the Universe
He needed to share this too and he knew how to do it.
He envisioned the study of geometry as one of the disciplines that would lead a human being to be more in touch with the true
perfection of the universe
So he went to Magna Graecia. The Greek colonies in what we would now call
Italy and set up a mystery cult to study philosophy and practice the sacred art of
Geometry and his cult did great
But the thing about mystery cults is well, they like their mysteries
So they're not always great at you know
Writing a bunch of stuff down, thus while the pythagoreans
Taught and shared their knowledge and weren't nearly as secretive as most of these groups
They were more interested in the philosophy of Pythagoras and the ways
Mathematics pointed to a beautiful perfection underlying the universe than they were in providing a unified mathematical system
So enter Euclid, a figure we know surprisingly little about, but whose work had a nearly
indescribable impact on human history
Euclid wrote a book or rather in the parlance of the time thirteen books called the elements for over two
thousand years this work would stand as the height of logical rigor
This book right here is the root of almost all mathematics Modern Geometry, Algebra
Calculus, all of them founded in this work to this day
It is the second most republished work in history after the Bible. In this book
Euclid brought together all of the geometric knowledge of the Ancient World
transcribing the discoveries of the Pythagoreans and others and extending them adding his own proofs and
discoveries to this great catalogue of the known
But what makes this work truly one of the pinnacles of human achievement is how he put it together how it was organised
because the book begins with a small number of definitions,
postulates, and common notions and says that with those everything else, every single thing in Geometry
Follows logically. He then organises his proofs, the various geometric problems
he presents so that they all build off of one another
No proof in the entire book will require knowledge beyond those initial
definitions and the proofs that came before it.
Showing just how far we can go with a few simple ideas
the elements is the
Foundation of mathematical thinking and in a lot of ways the foundation for how we think of logic today. It was a huge achievement
But there was one small issue
That bothered some of those studying this text. An issue that appears to have bothered even Euclid himself
And that was the 5th Postulate. Most of the postulates in the book are fairly simple and straightforward
They say things like you can draw a straight line between any two points or all right angles are equal
But the fifth postulate is not simple in the slightest
It's more complex and it just feels different than any of the rest. How complex is it?
Well,
The 5th Postulate states, quote "If a straight line falling across two straight lines makes internal angles on the same side
less than two right angles
The two straight lines if produced indefinitely meet on that side on which are the angles less than two right angles."
*UGH! That felt gross to say. Feels a lot messier than all right angles are equal to one another right?
So let's just break it down real quick. A straight line falling across two straight lines
Okay.
That's just a lines crossed by two other line somewhere. "Makes internal angles on the same side
less than two right angles." and this is basically saying if the internal or
Interior angles, these angles which face each other right here made by the two lines crossing that third line add up to less than two
Right angles or 180 degrees, "Then the two straight lines if produced indefinitely meet on the side where the angles are less than two
right angles, so, okay.
If that thing I said about the interior angles before is true, then if you extend those two lines forever
They are going to intersect at some points on the side where the interior angles are less than right angles
so
putting all that together, if you draw a line and you have two other lines cross it if their
Interior angles add up to less than 180 degrees
Those lines are eventually going to intersect if you draw them out far enough or put even more simply lines
angled towards each other are going to
Intersect if you draw them out far enough and when you put it that way it actually seems kind of obvious, right?
In fact, we are so used to that concept that it barely even seems worth annunciated
But Euclid was nothing if not thorough and hidden in this concept is another
All-important one because let's look at those two lines crossing the third line again. What are the possibilities here?
Well, if their interior angles on a side are less than 180 degrees
We already know they're going to meet but what if they are greater than 180 degrees?
Well, then the interior angles on the other side are gonna be less than 180 degrees, right?
So they're just gonna intersect on that side
It's basically the same thing just flipped around, but what happens if the two angles add up to exactly
180 degrees? Well
Then by this schema those lines would never meet. What this postulate actually does is define what we today call
Parallel lines, but we know that this postulate was a problem even for Euclid
it's the last postulate he puts in the book and even after he's
Enunciated it
He goes about proving almost every single thing that can be proven in geometry
Without it before at last relying on Postulate 5 to build the rest of what we think of as standard Geometry today
And he wasn't alone in being bothered by Postulate 5's weirdness for over
2,000 years, Postulate 5 would bug people. It feels like it should be a proposition not a postulate
It feels like there should be a logical proof for it
And if we could make such a proof, then all of geometry truly would be consistent
The last lingering question would be answered and we really would have that beautiful system that the Pythagoreans
Desired so much
but if Euclid couldn't find a solution to Postulate 5, who could?
Find out next time as we explore all the ways people built off of Euclid and all the different attempts people made to reconcile this
one last tiny piece of our perfect Geometry.
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