Parabolas and Archimedes - Numberphile
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- เผยแพร่เมื่อ 22 พ.ค. 2021
- This video features Johnny Ball.
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This man just turned a parabola into a multiplication table.
I was amazed. I never thought of it that way, took me pen and paper to understand how that was happening.
If you can describe 2,000 years ago as 'just' then yeah! Pretty neat!
In fact, it always has been a multiplication table, only people like you and me did not know about it. Now we do. I am officially amazed by this. I mean, we did conic sections in school and such, but this here never came up. Amazing, just amazing.
Yeah, it’s a neat trick! You can derive it pretty quickly too. Assume you have points a and b and the parabola y = x^2. Then the line between them is
x (b^2 - a^2)/(b - a) + C
For some C which is the y-intercept. Notice this simplifies to
x (b + a)(b - a) / (b - a) + c = x(b + a) + c = y
Now substitute x=a and y=a^2 and you get
a(b + a) + c = a^2 = ab + a^2 + c
So 0 = ab + c and therefore c = -ab, meaning the intercept is negative the product of a and b.
P.S. Had a sign error that I corrected above.
@@Bodyknock c = -ab
This video felt too short! I could listen to this wise old man tell stories of other wise old men until I myself am a wise old man.
Yes Please Yes
or as i am,a mere confused old man
@@billob1305 ahh but an old man none the less Bill, which is half the battle. If you have managed to survived long enough to be an old man, there is likely to be much more wisdom within you than you realize my friend. Without a doubt you could teach us all something!
if you'd like to do a little more relaxing watching this particular man, he used to do a show when i was a kid, called "Johnny Ball explains it all". You may be able to find videos of it, somewhere.
@@beningram1811 I will have to do just that then wont I! I don't remember such a show although there was a time when not a lot of TV stuff made it to our shores here in NZ.
Thank you very much for this information Ben. I appreciate you taking the time to share, I shall endeavour to make it worth your effort by searching right now.
Cheers.
Something Archimedes never said was "Excuse me." or "I'm sorry." Some might argue that it was because he was rude or stubborn. But I think it was probably because he didn't speak English.
I think you're onto something...
@@kwanarchive Shhhh! Don't tell him that.
He'll become too powerful.
you are the "para" and not the "bolic", which is excelent either way
Kwasia 😂
Pretty sure the words “Excuse me” and “I’m sorry” didn’t exist back then, because English hadn’t even been made when he was alive.
Video: "What is the ratio of the parabola's area to the rectangle's?"
Me: "Well, this would be trivial to compute with an integral, but the Greeks didn't have Calculus, so let's see."
Archimedes: *basically does a Riemann sum
Me: "Well color me impressed."
The Greeks had a lot of calculus. They were obviously missing some of the algebra that was developed later. Newton and Liebniz are credited with summarizing the fundamental theorem, i.e. connecting derivatives and integrals, not with developing derivatives and integrals themselves.
@@stephenbeck7222 But Newton did develop it himself didnt he?
@@leif1075 Newton refined it and leibnitz helped with the notation , but integration and derivation predated those two
Differential calculus was first discovered by Newton. Bhaskara II knew derivative of sine function but he couldn't generalize it to other function. He was calculating speed of stars when time interval was very small.
@@vikraal6974 nobody has generalized it. you can generalize the concept, but there is no generalized algorithm. so ultimately it takes somebody who's not really doing the mathematics, but just bored and poking around at stuff to recognize the connections between different derivatives.
If only Archimedes had algebra to formalise the idea of infinitesimal lines summing to finite weight, he almost surely would have made the connection to what we now know as calculus.
Another related missing piece is the Greek rejection of zero as a number. This caused them to miss concepts like infinitesimals "approaching zero". Interesting to think about the quantum leaps that could have been made in math with the addition of a couple of key concepts!
Interesting! Maybe there would be only something like "approaching to nothing" and still no actual zero.
@@destructfashion the rejection was that math was to solve problems
so if something end up been zero meant it was not possible
and you do not need zero to do things
the quadratic equation instead of making equal to zero take c to the other side and equal it with that
the formula can still work
also greeks had ZERO
when used for LOGISTICS
since when you do inventories of warehouse you may end up not having something
so they had to show that
Am I the only one who finds it interesting that the Greeks or those before weren't clever enough to invent/discover algebra? I mean algebra, at least the basics, is kinda just introducing the concept of variables and then shuffling them around, no?
@@christopherpape4823 you are the only one
since greeks where dealing with different kind of problems
and also 90% of greek text is gone so we do not know how much is lost
Archimedes must have been one of the most brilliant minds of all time
He was an intellectual giant, up there with Gauss and Euler.
Just imagine what his mind would do for humanity if he were alive today
Someone once asked me what you do in Calc 3. I said, "Basically, prove everything that Archimedes figured out 2,500 years ago."
@@anthonyparsotam3611 opens up tiktok.. immediately rolls back into grave
@@frostyusername5011 🤣🤣🤣
I love the manner he explains things, pure tranquility.
Like David Attenborough of mathematics
I honestly find it distracting. He didn't explain a crucial step and hardly anyone noticed. "Let me tell you a story" indeed. Math is not a fairy tale, too bad many treat it like one - the best-told story wins, just like in journalism...
@@shy-watcher
You love listening to yourself speak it seems
@@Andrew-fv4sj It seems you like reading people's minds? How do you know what I do and don't like? I just don't like when people ignore problems with a proof just because the presenter's voice is nice.
the point is the presenters attitude should inspire a hunger for learning and creativity. They might make mistakes, everyone does. this channel isnt about perfection. Nobodys watching this so they can become a mathematician, thats what college is for.
Gauss is probably the only one I can allow to get away with calling Archimedes an idiot :)
And no one can get away with calling Gauss an idiot. Although you can call him an asshole.
And Ramanujan is the only one I can allow to get away calling Gauss an average genius.
Archimedes to Gauss: Isn't this so called calculus thing obvious to most casual mathematician?
@@piratesofphysics4100 look guys we found another delusional indian
WOW Numerphile. It's Johnny Ball, bless his cotton socks.
this comment brings me an endless amount of joy
Huh?
??.
Johnny Ball is a legend to my generation.
More Numberphile videos with Johnny Ball: bit.ly/Johnny_Ball
@@numberphile i reas tommy ball and thought for a moment there was a tommyball reference snuck into the animations
@@Koisheep well, we all know that Johnny Ball was actually one of the first people to ever play tommy ball when it wasn't even invented
Who?
(What why where when)
@@numberphile whenever you get a chance, more of Johnny Ball is a MUST thank you!
If Johnny Ball hadn't been on the TV when I was small it is a certainty that my career track would have been different. He was the first person to show me the emotional payoff you get from understanding a piece of mathematics. He set the stage for the idea that there could be such a thing as mathematical exposition for a mass audience. It's not too much of a leap to say that Numberphile could not really exist without him.
I'm 55 and this gentleman was presenting "playschool" when I was 4 and "think of a number" when I was 11.
Still educating and with the same passion and energy. It's wonderful to see him again ❤
For those asking how Archimedes figured out that the triangle would balance the parabola at 6:43, the proof is skipped in the video.
He used a known property of parabolas that PO/NF = MO/CF, and thus PO x CF = MO x NF but by definition CF = FH and so PO x FH = MO x NF which means that if each PO segment is placed at H, it would balance its corresponding MO segment, and thus the parabola placed at H balances the big triangle.
It's too bad in a Numberphile video that you have to go to the comments to find the actual math.
thank you! your comment completes the video
thanks, I really didn't get where it came from
This comment is underrated
Thanks for the explanation and the identity you provided here, but it also took me time to conclude even after your statements. Do you not think you also forgot to finalize the proof? You did not mention the relation between N and X.
I love how the Greeks figured out mathematics in such a visual, geometric way.
I didn't understand why, at 6:43 , he said "it will only balance at the balance point". Why is that?
The triangle “sits” on the lever at its center of mass
I think this is a severe issue with the video - that is one large leap of reason that I can't imagine archimedes making but from the video it is not clear what the reasoning would be
@@epicsmashman6806 Still didn't got it. Why will the parabola balance it at 3 times the distance?
"the proof is left for the reader as an exercise"
This is explained in coordinates on Wikipedia with a straight parabola and triangle under "Method of Mechanical Theorems", which is the Archimedes work where this argument and diagram come from. The quadratic relations are difficult to see intuitively on a skewed parabola.
You think this is amazing? Archimedes’ calculation of the volume of a sphere is even more incredible.
Please ask prof. Ball to do do another video on Archimedes calculation of the volume of the sphere using levers/balancing! Pleaseeee
So, how you liking their new video?
Heyyyy)
Numberphile released a travesty of a video on the volume of sphere. Ball makes it sound like Archimedes dunked shapes in water to "prove" their volumes add up. Completely skips the actual law of levers method Archimedes wrote in "Method" (the palimpsest).
His voice is so soothing
johnny ball is a legend
That's as British accent as it can get.
Reminded me of the Geico gecko a bit.
I really love this accent.
I just read the wikipedia article on this, and boy can I say this video did not at all do justice to the proof. Trying to represent this proof in two dimensions makes it practically impossible to tell what’s going on. I didn’t even notice him *attempting* to prove the most important part of the demonstration. Basically, for anyone confused, imagine on a lever, the triangle is placed starting at the fulcrum, following along the lever to the left, increasing in height. On the other side, the parabola will be placed *sideways* (extremely important part that was not clearly explained due to the lack of 3D) meaning it rests on a single point of the ruler.
One can then pair up, line by line, pieces of the triangle and that of the parabola, and using the property of levers that states force=mass*(distance to fulcrum) and prove that they apply the same force. The triangle has mass x and distance x, and the parabola has mass x^2, but a constant distance of 1. Since these both multiply to the same thing, they apply an equal force, and balance.
Notice how this *only works* in three dimensions where the parabola can lie sideways, unlike the triangle.
With every line being equal, the entire shapes must be equal. Archimedes didn’t just say, like this video suggests, “I bet it will balance”. He actually demonstrated it in a really beautiful way that this video just does not offer.
Oh, that's a great and an elegant approach! I still have questions, though, but I can't find the article you're talking about. Can you tell me its name?
Upd: Never mind, I found one: The Method of Mechanical Theorems
This man was quite a big part of my childhood, he's an amazing story teller.
at first i was think "how he is going to figure it out without integrals at that time" and than he uses integral mindblown.
Except he didn't. He only almost did. He knew the answer approximately (no doubt from measurement) and assumed the exact solution. Here he "almost" invented calculus which would have allowed him to prove it. I put "almost" in quotes, because this is more like the first step to inventing calculus -- there's three or four more big insights or ideas needed to get to calculus.
@@GreylanderTV Well, if there was only one person intellectually capable of these insights, that would be Archimedes. But he was a few centuries too early, and lacked the mathematical tools.
@@jasondoe2596 Yes, no knock on Archimedes. See my main comment for context.
@@GreylanderTV Hey Scott, genuine question, what are the 4 additional big insights to get to calculus?
@@iancheung3587 using equations for one, they used shapes. Then I'd guess limits, then perhaps derivatives, also just more knowledge on graphs (rectangular, not graph theory). It'd be possible without our ways of equations and variables, but much harder
It is so difficult for me to imagine mathematics without the technology commonly called graphing.
Or paper for that matter
Thanks Descartes!
??.
Crazy to imagine how brilliant Arhimedes must've been at a time when he had nothing to start from. He had to think of this all by himself, with no previous knowledge from the past. We are truly blessed to have all the information we have today. He had nothing, but still managed to create the base of mathematics. Astonishing.
He had a lot of previous knowledge as stated by the video i.e levers and the center of mass of a triangle. 3:03 Still very brilliant, thought of the world and mathematics in a way that wouldn’t be grasped fully for another 2000 or so years
5:57 I absolutely love the mathematical discoveries where it's like "By definition I cannot" but then the genius goes "But what if I did anyways?"
I’ve never been lost in a numberphile video before but I have no idea what just happened
Yeah, same here. I love Numberphile because it takes complicated math things and makes them accessible, but this video didn't achieve that at all.
@@maxxie8058 same
@@maxxie8058 Indeed, it took a fairly simple geometrical concept and dressed it up so it became incomprehensible
I feel the reasoning behind Archemides' intuition as to why the parabola's "weight" balanced that of the triangle at the "fulcrum" was not explained at all. We got all this beautiful setup for a punchline that said Archimedes' guessed this was true and it turns out he was right. Unfortunately the video does not clearly explain WHY Archimedes was correct. It left me feeling like there is an itch I can't reach to scratch.
@@e2DAiPIE because he was doing an integral without knowing it
Among other things, back in the day on the BBC, this chap did a couple of series for kids called 'Think of a Number' and 'Think Again'. They were inspiring and engrossing. As far as I know there has never been anything else quite like them. I doubt if they would be made now with the BBC the way it has become.
"Look Around You".
Honestly, sounds like the Richard Attenborough of Mathematics.
As somebody whose brain is just not made for arithmetic, it is amazing how fond my memories of Think of a Number and its follow up shows are. He's one of the people who showed me that mathematics is a kind of wonderful magic and despite my lack of arithmetic skills, I have been in love with maths ever since.
Thank you, Johnny!
This man needs a math series on Netflix, I am always enthralled by his videos, the way he weaves stories with math along with his David Attenborough like voice and enthusiasm gets me everytime.
Archimedes was just about to invent calculus... and then realized he needed a bath.
If only he didn't born at war at the time....
More Johnny please! I occasionally teach middle school children and the way he teaches is very inspiring.
What an amazing video. Men like Archimedes, Gauss, Euler and Pascal were so important for our evolution as a species.
6:38 - there's a massive, unexplained leap of logic around here. Why would putting all the POs at that location balance? Why that distance? What's special about parabola vs random squiggly shape that is also inscribed by the original triangle?
If Archimedes determined it with a physical piece of cardboard/wood/marble/etc, that's worth mentioning, but it's not. If it was determined through some insight, that was not mentioned either.
Read Archimedes' "Method" Proposition 1, and his "Quadrature of the Parabola" Proposition 5. Key idea is he can prove MO/OP = CF/FN = HF/FN. So arbitrary triangle line MO placed distance FN from fulcrum balances with parabolic section line PO placed distance FH from fulcrum.
The genius of Archimedes is astonishing, not just an incredible matematician but probably the first scientist before science itself existed as we now know it: as far as I now his Principle of buoyancy is the oldest principle of physics which stands exactly as it was formulated more than 2200 years ago
Archimedes papers of mathematical physics, the law of the lever and the floating bodies are the first true papers in physics.
Johnny Ball!! My hero as a kid. MORE JOHNNY BALL PLEASE!!!!!!!
I wouldn't be surprised if the Library of Alexandria had contained bits and pieces of Calculus invented by mathematicians all over the ancient world before it burned down.
Nice to see a childhood hero still knocking about :)
More Numberphile videos with Johnny Ball: bit.ly/Johnny_Ball
OMG Johnny Ball!!!!! Johnny is the only reason I can count!
How do you know that the "weight" of the parabola would balance out the "weight" of the triangle? There's no basis (in this video) for that conclusion.
Read Archimedes' "Method" Proposition 1, and his "Quadrature of the Parabola" Proposition 5. Key idea is he can prove MO/OP = CF/FN = HF/FN. So arbitrary triangle line MO placed distance FN from fulcrum balances with parabolic section line PO placed distance FH from fulcrum.
@@willjohnston2959 thanks for the explanation. Not sure why this key part of the proof is overlooked in the video.
It’s so impressive what the Greeks were doing, well done those guys
Its mostly Archimedes by himself tbh
@@steliostoulis1875 Eratosthenes was pretty great.
@@steliostoulis1875 Euclid
@@steliostoulis1875 Euclid also played a major role in mathematics
@@steliostoulis1875 and ofcourse the father of algebra , Al - Khwarizmi is very important too ( he wasn’t Greek tho but still )
Fantastic! the beauty of mathematics lies in the fact that it describes the beauty and harmony of the world
Lovely to see you again, Johnny! You were a feature of my adolescent education!
The legend Johnny Ball...his calming voice and amazing way of explaining things, any other 40 somethings here remembering their youth and watching him on tv after school or a saturday morning?
So happy to find this channel, I missed Mr Ball, so much positivity made my childhood bareable. We lack such sincere educational presenters.
I grew up with stories about Archimedes and other coll mathematician and science stories., My grandfather past away last year. I'll never forget all those amazing stories he told me about him, some probably hyperbole. He would've loved this video.
AMAZING video. Thank you for producing and sharing this.
this type of meaning of calculus was around for maybe 1000s years, the story goes like this, the governor wanted to punish a certain farmer and told him to count every rice grain in a metric ton without a single loss of grain or would forfeit his life, the farmer back and said next day he counted all and it was 10^7 grains, farmer asked the governor to verify if he wanted, but he couldnt so he asked how? the farmer replied, a spoonful of rices had 100 grains, 1 dipper had 1000 spoonfuls of them, 1 ton had 100 dippers, after that the governor never bothered the farmers again
More Johnny Ball please, my childhood just came back to me. Still one of the best teachers ever!
I wish I could sit and converse with this man. Many stories, experiences, lessons and things to learn from him.👏
Still listening to Johnny Ball all these years later. What a legend.
Archimedes knew what he was doing when he decided to not publish calculus, he is a hero.
Imagine how far human civilization could have gone further if calculus was invented back then
Yes dudd, newton, gauss, euler and etc could've then make a whole new discovery/invention more brilliant than calculus
I wouldn't be surprised in this scenario if it was lost during the middle ages and not rediscovered again until the time of Newton, or something like that
Genghis Khan could've had a nuke.
@@lock_ray yeah, in the "middle ages" when universities first started
It's really wild how much of modern mathematics existed thousands of years ago, but all scattered around the world in isolated disconnected pieces. If only we got along better and had fewer language barriers...
Yay! The great Johnny Ball! More, please!
Grew up with Johnny Ball explaining stuff to me as a kid on British TV 30 years ago. He's maths' answer to David Attenborough: an absolute legend.
So pleased to see him still educating: and educating *me* no less(!) all these years later!
What a joy to see Johnny again!!
Love you Johnny Ball! Loved watching you as a kid.
Started watching without reading the caption and a few seconds in the voice made me say “That’s Johnny Ball!” To people my age (mid forties) in the UK this guy is an iconic part of our childhood!
The moment I saw this episode was about parabolas, my mind flashed back to Johnny Ball teaching me about them as a kid on the brilliant Think of a Number.
Couldn’t believe it when this one turned out to be him again. Fantastic stuff!
The nice mixture of the history of math and the lovely geometrical interpretation of things!
This is so calming, I love it
Absolutely wonderful mathematician, wonderful explanation, and the reveal at the end!
Love the sound effects
Johnny Ball! Love him!! Thanks Numberphile!
this is *_absolutely_* wonderful
Johnny Ball!
One of my childhood heroes.
What a great demonstration
Holy crud, that is an amazing mathematical intuition, right there. As soon as they showed the sums of all those slices, I literally got a chill. How is this story not more well-known?!
Great video. And always amazing to see what conclusions were made back then with the limited tools they had.
Woah, the great Johnny Ball, TV legend and enthusiastic educator extraordinaire!
Man, Sunday morning science and maths are the best! Coffee, apple fritter, and Johnny and Brady to make my day! Cheers
Johnny Ball did more for my maths ability than school ever did. Some of the best afternoon children’s TV in the UK ever. 👍
What a stunning explanation... Fantastic
Man the MO and PO part is just pure thinking excellence.
"there's only one true parabola"
The Parker Parabola
I miss numberphile Matt.
Gloria in x-squaris!
(sees parabolas measured with triangles)
Parabolati confirmed!
So much fun, thank you!
5:22 Why did he double that line? And why put the fulcrum there?
Why not triple, or quadruple? I don't understand that part.
Think back to the ruler, 6 inches on one side of the fulcrum and 6 inches on the other. F being the fulcrum then FH must be the same length as CF to "balance" without any PO or MO (the dominoes)
Yeah, it does seem like there's a bit that's glossed over.
"I bet this balances at that point." Why? Sure, it balances somewhere, but why does it have to balance there?
The video really skipped over this, and I had to pause at 7:00 and stare at it for 5 minutes to figure out that each PO balances each MO individually, so he could check a few points and after seeing each one that he checked balance, he assumed that they all balance. For example the one at the center of the rectangle is half the distance from F and P is half the height of M.
Love any vids on ancient greek geometry. Please do more!
Wonderful video. Wonderful Johnny Ball. Wonderful story about Archimedes and integral calculus.
Wow. Great explanation! And what a great discovery!
I always thought Zeno with his arrow and hare paradoxes was on the cusp of inventing calculus as well. But got shut down, and probably body slammed instead.
Zeno inspired Eudoxus to define real numbers as limits (although they didn't quite use this terminology), and Archimedes came right after Eudoxus.
What an amazing voice!
Thank you for your charming video. It looks like Cavalieri’s picked up on Archimedes work and extended it.
this was such an exciting video!
He is actual time travel, I went from 42 to 7 in a heartbeat
More of Johnny Ball, please!
oh what an amazing story! thank you all
Happy birthday Johnny!
what proof did archemides use to prove that it should balance?
idk about a mathematical proof but he could have just cut shapes out a material and weighed them on a lever.
What do you mean?
Method of exhaustion, you can find them online. It's basically old calculus.
@@Nnm26 ok, i see. thanks. I know the method, I will check out the implementation of Archimedes later. thanks again.
Probably exist a more rigorous proof in one of his books
For anyone confused about the "Why should they balance", it took me a while to be convinced, but the reasoning is as follows. Use the same point names as in the video.
Consider a single particular MO. Based on the properties of a parabola and triangle similarities (This is where I had to put it on coordinate axes, but there are certainly classical theorems I don't know about chords of parabolas), you can show that MO:OP=AC:AO.
Next, by triangle similarity, you can show that AC:AO=FC:FN.
Finally, you have that FC:FN=FH:FN simply because FC=FH.
This forms a chain of equalities, therefore
MO:OP=FH:FN, which is the lever equation. Thus, Line MO at point N and line OP at point H balance on F.
So, at point H, we have the stack of lines corresponding to the parabolic section, and spread out on the other half of the lever, we have the lines of the triangle. Levers obey superposition, so if all their individual parts balance, the total also balances. However, any shape can be treated just as a weight at the center of mass, which for a triangle is easy to calculate as shown, point X. Thus, the weight of the triangle all spread about the lever is the same as the same weight at point X. This gives us a lever with 2 point masses, and we can easily deduce the ratio of the weights based on the ratio of the lengths, which we know by construction.
Johnny Ball! Love you mate. You're one of the reasons for my love of maths :D
Loved it!
Best numberphile video I’ve seen in a while
Absolute brilliance!
What I keep saying in defence of seeming "idiocy" is: It is very difficult (if not impossible outright) to choose what to seriously think about in depth, mainly, before you have done it, it is very difficult (if not outright impossible) to know if the problem or the results might be interesting. If for Archimedes Calculus was a fire-and-forget kind of tool, that's because he did not need it for any other problem (likely).
We need more johnny videos! Ball and Sins!
Getting Johnny Ball on is amazing. 80s kids rejoice!
Wow! Great Job on this video!
GREAT AS ALWAYS.
Wow! This was such a great video! I never would have thought that Archimedes thought about calculus 2000 years ago, way way before Leibniz... with such creativity in his methods too! ♥️
Thx I was having so much trouble understanding. This wise man just saved my life😅😅
Beautiful explanation
This video made me incredibly happy