As Rome died,
a man in Athens sat, hard at work.
His name was Proclus,
and he had made perhaps the most monumental discovery of the age:
He had discovered that the solution to Euclid's Postulate 5 offered by Ptolemy,
the other great mathematician of the ancient world,
was false.
More still, he had found a solution himself.
He wrote furiously through the night.
Unfortunately, his solution was false, too.
Let the 2000 years of head-scratching began.
Chaos.
War.
Fire.
And famine.
The fall of empires.
Western Rome collapsed.
And with it, many of its institutions,
including its system of education,
were washed away.
Old ideas,
old books,
vanished from the West.
And one of those books
was the Elements.
For nearly 700 years,
the beautiful geometry of Euclid and the Greeks
would disappear from Europe.
But there was somewhere else,
somewhere further East that safeguarded this knowledge
and kept the light of Rome alive.
This was Eastern Rome,
the Byzantine Empire.
Shield of fledgling Europe while it clawed its way back from its great collapse.
But it's not from the Byzantines that we get Euclid back.
At least, not directly.
You see, while the Eastern Roman Empire was on its long, slow decline,
there was another power on the rise:
Islam.
And during the Islamic Golden Age,
the Abbasid Caliphs sent scholars and traders
up to Constantinople
to acquire and translate as many works of ancient thought
as they could.
And one such work was Euclid.
During the Islamic Golden Age,
massive strides were made in mathematics,
building off the texts of antiquity.
Algebra was systemized for the first time.
Trigonometry was pushed well beyond anything the ancient world knew.
And irrational numbers,
that dark heart of mathematics, which the Pythagoreans felt, ruined its perfection,
were brought into the light.
While dabbling in Euclid,
the great poet mathematician Omar Khayyam
took a huge step forward in number theory by deciding that you could put irrational numbers
on the same number line as rational ones.
This was something that had totally confounded Euclid,
because Euclid had wanted a tool that can measure everything.
And there was no line segment,
no matter how small you made it,
that could be laid out evenly to measure both a rational number and an irrational number,
so he had viewed them as totally seperate things.
Khayyam on the other hand, realize that :
Hey, you can still tell that Pi is bigger than 1,
so they must have some relation.
And this is a big deal,
because it lets us do things with irrational numbers.
This brings us to the famous polymath al-Khwarizmi,
who was not only a strong advocant for the use of the Hindu numbering system,
which would become what we call the Arabic numerals today,
but he expanded on the geometry of Euclid to write a book that gives us the very word "algebra" today.
To show you just how much Euclid was a basis for his algebra,
let's, real quick, do a quadratic equation the way al-Khwarizmi did then.
al-Khwarizmi had this famous equation where he wanted to solve
x^2+10x=39.
Instead of using the quadratic equation like we might today,
he literally made a square of x
and drew the thing.
So, step 1:
let's lay out a line and call it x unit(s) long.
Step 2:
put another x unit(s) long line at a right angle to your first line
and make a square out of the two.
So, now we have x^2.
Step 3:
add 5 units to the right side of your square
and make a rectangle using that extra bit
and the height of the square.
So, you get a rectangle attached to your square,
5 long and x wide.
Step 4:
do the same thing with the bottom side
so you end up with another rectangle,
5 wide and x long,
attached to the bottom of your square.
So, now we have x^2+5x+5x
or, put it another way,
x^2+10x,
which is starting to look a lot like our original equation.
But you'll notice that our square is no longer square.
Now there's a hole in the bottom corner.
But luckily, we know just how big that hole is.
It's exactly 5x5, or 25 units large,
because of how we constructed the rest of the square.
But what do we also know from our original equation?
We know that x^2+10x,
or everything but that little missing square here,
equals 39.
So if we were to add that square in,
we would have an area of 39+25, or 64.
But we also now have a perfect square with each side being x+5 long,
so now we know that (x+5)^2=64,
or put it another way,
√64=x+5.
And √64 is 8,
so x+5=8, which means that x must be 3.
Congratulations, you just did a quadratic equation using nothing but geometry.
And the reason I walk you through all of that was to show you how smart and thorough the guy who figured that out had to be.
al-Khwarizmi was smart.
But you know what bugged him?
Postulate 5.
So he charged mightily at it.
And he did come up with a proof for it,
that unfortunately, was invalid because it relied on things like motion and transformation in a totally static geometrical system.
It was a good try though.
And that poet I mentioned, Omar Khayyam?
Man, did the 5th Postulate get his goat.
He wrote a full book on the thing,
and he came up with a proof,
which unfortunately was really just a re-wording of Euclid's original postulate.
Almost every Islamic mathematician you can name in the Middle Ages took a crack at this.
In one case, a mathematical family even passed it down between generations,
with a son taking up his father's work on perfecting Euclid and eliminating the 5th Postulate.
But in the end, none succeeded.
Still though, they did advance mathematics phenominally,
and built a foundation which later Europeans would use to get to non-Euclidean geometry.
Because, in their attempts to eliminate Postulate 5,
some of the Islamic geometers stumbled on a method that they thought foolproof.
They said:
Well, there are only 2 other possibilities for parallel lines other than Euclid's Postulate 5.
If these supposedly parallel lines don't always maintain the same distance apart,
that means they either have to cross sometime,
or they have to bend away from one another.
If we can just disproof both of those ridiculous cases, then
BAM!
Postulate 5 solved.
Of course, we know now that those other 2 cases aren't actually that ridiculous.
In fact, they're the 2 main forms of non-Euclidean geometry.
But it'll take a few centuries for us to get there.
So in the meantime, how did the Elements and the problem of Postulate 5 make its way back into Europe ?
Well, a 12-century Englishman named Adelard spent a very long time wandering around the Crusader states.
He wasn't really too impressed with how all of this crusade seems to be going, but if there is one thing they succeeded at,
it was providing access to the academic knowledge of the Islamic world.
I mean, why let a little crusade get in the way of a good scholarship ?
So, Adelard took all of these awesome books that he found in Crusader countries back to an abbey in England
and he began to translate them.
And amongst this haul was Euclid's The Elements.
his translation served as basis of all Western versions of this book for 600 years,
until Napoleon accidentally stole an older copy from the Vatican that they just didn't realize they had.
But that's a story for another time.
Join us next week as Adelard's translation began to spread,
and we see the influence of Euclid on the West,
and what Renaissance Europe would do with his ideas.