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. [End Music]

Hi everyone! Sorry for how often we repeated prop 5, but understanding how it's both weirdly simple and yet super complex is important for the rest of the series. There will be math in the rest of these episodes but not as heavy handedly done ; )

-JP

But what is the problem, it simply states that lines will be parallel and will not. Create a geometrical figure, triangle if the angles = 180degrees. Well done mr E, but how can this bother anyone i cannot phatom

Is that intro melody from Act Raiser?

I thought this was a game grumps episode

2:07 A little depressing that the most important book about mathematics ever written loses to a bunch of magical mumbo jumbo.

So… what's the "problem" with Postulate 5? It's never stated outright in the video. Is it that it's just horribly worded? Or because the concept of "parallel" hadn't been invented yet?

All angles of a triangle = 180°

So if two angles combine is less than 180 degree then that means at some point it is forming a triangle hence we can actually calculate the angle at which those two lines will meet.

Please stop showing three flat gears in mesh, you're going to make people believe it's possible to do that

I'll bet you spend a fortune in helium. XD

burn math with fire

So to fitth postulate is: ''Two parallel line must not intersect.''?

Euclidean.

linus has a history channel ? fucken leaf

I thought Pythagorus wasn't a real person, though Pythagoreans were pretty culty. Didn't they send someone out to sea for proposing irrational numbers? Can you imagine what would happen if you tried to show them complex analysis? haha

the 5th postulate is answered by string theory if a flea/spider crawls on those lines the dimensions change and there fore a straight line is changed in terms of perspective so if you changed the angle not just 2 dimensional you change it 3 dimensional the lines actually wouldnt intersect if 2 rockets launch at less than 180 degrees in a 3d sense the still may not hit or…make it outta earths atmosphere

I'm not sure how l got here but l think I've just crossed over in to hell!!….I'm decalectic!!!!

By itself the 5th postulate doesn't imply that parallel lines never meet. For that you also need the 2nd postulate.

At end of my only Geometry class (cir. 1959) teacher told me she was giving me a D so I wouldn't have to take the class again. As I recall her words they were, "you tried hard, but you just can't understand it:".

anyone remember Actraiser for SNES? overworld theme

funny you know that euler is not euler but with euclid it's the same!

If all points on those 2 lines are the same distance from the other line, then they always will be, as long as the lines are straight.

Postulate 5 was my college bands name.

2:13 So what book takes the 3rd place…?

Nice video

I'm obviously not a math person but I really don't see what is befuddling about the 5th Postulate…

Sounds fairly simple to me… Now if proving this means a mathematical formula to prove it I can't help…

If that isn't what the issue is I still don't see what the issue is…

Please, no responses that are nothing but insulting…

In other words, don't be an Intellectual snob…

Break it down for me, please… I really am interested in this…

Postulate Five is a great name for a band

So math starts with Arab and black people and then spreads to Greece, eventually to "the West" who subsequently, despite being among the later adopters, claims all modernity comes from them? Interesting.

Video that teaches something about a topic must definitely have the topic name spelled correctly. It should be "Non-Euclidean".

Where did this perfection of the universe come from?

Modern Science is, in a sense Pythagorean espically his statement that "All is Number", in that quantitative empirical measurement is the fundamental process it uses!

Prop 5 only proves people have issues drawing parallel lines. You all sure like to muddy the waters. I had that one figured out after you babbled the third time about a non problem problem YOU Created. The man had it figured out. YOU FAILED !

Brilliant

Actually, I have to disagree: wonderful achievement though it was, the Elements

dorequire knowledge of something outside the Axioms, Postulates and Definitions: they require a knowledge of theGreeklanguage.Yes, even such a thoroughly mathematical work as The Elements suffers distortion when translated. Most of the time, it does not matter much, so the Elements were still useful when read in Latin or English. But the apparent mistake in he very first proposition, where it is commonly observed he needs another axiom to claim that the circles meet, does not appear so mistaken in the original Greek. That the circles meet at a point follows from the sense of the definition of a circle, that its circle

containsthe interior. Sure, the argument is indirect, and not even mentioned by Euclid, but it is sufficient to prove that the circles really do meet.It is also interesting to notice that for centuries, Euclid was used to teach logic, but no one ever complained about this alleged failure until the 19th century. So why did not one notice before? Did all the previous readers notice the same thing I did about the definition of a circle? Not if they were reading it in Latin or English.

No, there is something more going on: great as Euclid's rigor was, the demand in the 19th century for rigor went much further. It was no longer considered acceptable to rely on knowledge of the language the theorems were written in, it had to have yet a higher level of complete independence. For geometry, this was achieved by the Hilbert Axioms, which include things Euclid never dreamed of, such as topology.

Just wanted to say- the 2 interior angles dont have to be less than right angles. They can be 1 degree and 178 degrees and that postulate will still be true.

it's not imperfect. The criticism is imperfect. classic kill the messenger

some people can't stop themselves from getting bogged down in criticism to the point where they lose sight of the big picture.

What does BCE mean?

Eucilid thought knowing Geometry would make you understand the universe.

Quantum Physics has entered the chatQuantum Physics: This ain't it chief.

No, it's easy to prove: Since the earth is flat we can just use a bubble-level to make 2 parallel boards that extend to the edge.

Postulate 5 is satan's postulate ! ! !

Well, the 5th postulate only works if the universe if perfectly flat. But our universe isn't perfectly flat because of the irregular concentration of mass. So we live in a non-Euclidian world

The more you knowso, parrallell lines are parallell. ok

Why is that such a problem? I don't get it.

6:31. What a big mood for mathematicians

can't stand this vioce, sry

When they do not get to the point, it is because there is no point.

I submitted a translation for the subtitles in Portuguese (Brazil). Please, check my translation so that it can be published and made accessible to Brazilians.

Eu enviei uma tradução para as legendas em Português (Brasil). Por favor, confira minha tradução para que ela possa ser publicada e disponibilizada aos brasileiros

this is so spirit sciency

I think the solution is to first find the problem. Wtf dude?

Damn, I saw this whole video (titled Non-Euclidian Geometry)… to find out it's all about Euclidean geometry. 😀 😀

where can i find an english translation for that book

Oh great, we have Non-Euclidian Geometry and Cults. What's next, raising cthulhu?

but if you have two different interior angles that add up to 180 like 135 and 45 what happens then

A math cult?

NEEEEEEEEEEEEEEEEEEEEEERDS

Eh

Am I the only one who didn't immediately find postulate 5 weird at all and entirely sensible…or am I just really really stupid

1:11 Yes, it was Walpole.

1st Day of high school Geometry.

Math cults

I used your videos on non-euclidean history as inspiration for a presentation at university. I also shamelessly used exclusively pictures from your videos as material. With reference of course.

Needless to say: 1.0 was the result. Thanks a lot guys, you are awesome.

Who actually created the universe

Hint: it's not God

So this is the asshole who invented postulates, theorems and

shudderproofs.1:11 IT WAS WALPOLE

"FIND OUT NEXT TIME ON DRAGON BALL Z!!!"

Euclid power level:

Shaggy

My brain feels dissy

Euclid class SCP

obligitory comment to encourage the youtube recomendation algorithm to share an amazing series!

So that Intro tune… Actraiser city theme. I love it

That feels so algebraic!

I was taught the Fifth Postulate as "If there is a line AB, and a point C that does not lie upon the line AB, then there is exactly one line which includes point C but does not intersect line AB." That explanation fits the alternate name for the Fifth Postulate: The Parallel Postulate. Basically, lines are straight.

Wow, that is so cool, Euclid has doubts regarding his 5th postulate back 2000 years ago, I mean all those physics like Newton were sure in their laws before Einstein's relativity theory and didn't give a f*ck there could possible be some circumstances when that will not work, while Euclid did. Guess that is what differs math from other sciences

i wish i could burn that book

anyone else using subtitles?

Walpole did 9/11

Where did you get the data that says that the elements was the second most copied book in history?

Was it Walpole?

1:11 LOL

"Was it Walpole? YES!!!!!!!!"

I love this running gag in the Extra Credits channel.

So why does The SCP foundation use Euclid as a power/threat lvl

Was it walpole ? (YES)

this is so easy uwu .said an asian

You suck

Postulate 5: If two lines are drawn which intersect a third in such a way that the sum of the inner angles on one side is less than two right angles, then the two lines inevitably must intersect each other on that side if extended far enough.

Postulate 5 Simplified: How to create Anarchy*.

*Just add circle.

We a hear Lovecraft screaming in the distanceDo you know what is wrong with postulate 5?

Ludwig Wittgenstein would be able to tell you philosophical questions are often inappropriate grammar or languages commonly poor descriptive ability and that should be enough to explain why this bothers you people

Well i now know how to pass math exams but i need a time machine and burn that book The Elements

proof is makes sense

i am euclid

incredible, thank you2:10 What's the third book? He looks nice and amiable.

My most reviled subject, oh I DETEST mathematics. Yet I am engaged in learning about geometry.

I SIGNED UP FOR HISTORY NOT MATH! NOBODY SAID THERE'D BE MATH!

Stop plagiarism, encouraging stolen knowledge. Indian mathematicians deserve credit

What is the third most republished book? Harry Potter and the sorcerer's stone?

omg

Pass the Tylenol

Well, not exactly. Pythagorus was in a club in which if you revealed the secrets, you would be KILLED! Also, this has been proven to be NOT logically consistent. But for ordinary purposes, it's OK.

Whoa whoa whoa, Extra Credits. Is that you, Patch Man? You sound so familiar. Huge longtime fan of yours.

Fascinating video by the way.

I really LOVE arts & CRAFTs

Childish