How Archimedes Almost Broke Math with Circles
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- čas přidán 26. 06. 2024
- Archimedes proved the area formula for a circle by dividing the shape into infinitesimally small pieces. The concept was behind some of his greatest mathematical achievements, but rested on a paradox that wasn't addressed rigorously for thousands of years. Even Isaac Newton left some key questions unresolved when he wrestled with this paradox 1800 years later.
I skipped the detailed proof in this video - you can find an excellent full explanation of Archimedes' proof of the area formula of the circle by @DanielRubin1 here: • Area of a Circle, the ...
*A few clarifications and comments, based on helpful viewer feedback*
•While Archimedes is often credited with the invention of the screw style water pump, and it bears his name, evidence shows that this invention existed in Egypt before Archimedes, and he was likely the first to "demonstrate and fully explain its properties." books.google.com/books?id=5cP...
•Another source, cited in Wikipedia, says that the screw may have been an improvement on a similar pump that existed in the hanging gardens of Babylon: muse.jhu.edu/issue/2473 (I have not independently vetted this article since it comprises such a small detail in my video)
•Some have pointed out that Leibniz was given short shrift in my video. Had I known that this video would go viral, I would have held it back for a week or two to fix that problem! As it is, I hope to cover the Newton/Leibniz controversy in a future video.
•It's worth clarifying that parts of the story of Archimedes' death are likely apocryphal, in particular his last words. In the video, I referred to it as a "legend" because there's no question that it's a good story. Some 300 years after Archimedes, Plutarch reported multiple versions of Archimedes' death and his last words. There is probably a version of the truth in there somewhere, but storytelling and myth making surely played a role as well.
Time Stamps:
00:00 - Introduction
00:51 - The challenge of curves
01:44 - The area of a circle
03:27 - The paradox of infinitesimals
04:42 - History after Archimedes
06:45 - Calculus in the modern world
07:11 - Archimedes' life and death
_____
The inspiration for this video comes from:
Infinite Powers: How Calculus Reveals the Secrets of the Universe, by Steven Strogatz amzn.to/3RagI0v
I highly recommend this book to anybody interested in reading more. It's a great read, even if you're not a "math person."
______
Additional Resources:
Calculus Reordered: A History of the Big Ideas, by David M. Bressoud
amzn.to/3G9o9Pj
Infinitesimal: How a Dangerous Mathematical Theory Shaped the Modern World, by Amir Alexander
amzn.to/46s3uR8
Isaac Newton, by James Gleick
amzn.to/49QPNhG
_____
The Works of Archimedes: archive.org/details/worksofar...
Résumé des leçons données à l'École royale polytechnique sur le calcul infinitésimal, par M. Augustin-Louis Cauchy: gallica.bnf.fr/ark:/12148/bpt...
Newton's Fluxions: cudl.lib.cam.ac.uk/view/MS-AD...
Archimedes' proof from The Measurement of a Circle:
media.bloomsbury.com/rep/files...
Annotation of the proof from the American Mathematical Society:
www.ams.org/publicoutreach/fe...
More information about Archimedes' inventions and myths:
www.nytimes.com/2013/06/25/sc...
Circular Reasoning: Who First Proved That C Divided by d Is a Constant? by David Richeson
Discussion of who gets credit for which ideas:
www.jstor.org/stable/10.4169/...
Additional information ("all of Europe knew less in 1500 than Archimedes did on the day he died"):
inters.org/Whitehead-Western-...
_____
Thank you to @JonathanWhitmore for feedback on the script and providing the Manim animation at 00:12. Thank you to @CreateSmarter for technical help and valuable editing feedback on this video.
Note: Amazon links are affiliate links which help support the channel at no additional cost to you.
Thanks for watching! I am looking forward to making more videos like this, so drop a comment if there's anything you'd like to see.
@CUJ-CZcams Ooh that's interesting I haven't heard that before. I'll have to look into it.
@@bensyversenyeah please make an intuitive video about it for us 🙏
Great video, love this! A video about pythagoras and his gang would be interesting. Also, maybe complex numbers.
Please make a video on integration 😊😊
An obvious one is the correspondence between Pascal and Fermat that birthed probability theory. The suggestion about Descartes is also a good one.
Centuries later, Cicero visited Syracuse in search of Archimedes' tomb, which had been described having a giant cylinder and a sphere marking it. He finally discovered it hidden among the brambles, and had to tell the Syracusians with him the significance of it, because they didn't know who Archimedes was.
Fuckin Sicilians lol
interesting
Do we still have it?
Could you provide some links where can I read that in detail?
@@mandala-YIN.YANG-either in Plutarch's life of Cicero, one of his Parallel Lives, or in Cicero's own Tusculan Disputations, he mentions finding it.
His last words "μη μου τους κύκλους τάραττε" meaning "do not disturb my circles" are used to this day in Greece when we want to get rid of someone who's annoying us or disrupting our work
Interesting. I first thought it is in honour or to remember Aristoteles. But could it not rather be that the saying is older, and so the point of Aristotele's last words is this double meaning? Think it's not so easy to tell what was first ?
It is also used in Hungarian, in the meaning of leave me alone.
The Mughals disturbed him apparently already back then.
Obviously his actual last words are unknown, and we have a text from 30 AD that mentions this legendary utterance in Latin.
Thousands of years later, in the 19th century, it was translated into an archaizing form of Greek, what it might have been like.
This legendary saying was not passed on over 2000 years. It was merely rediscovered and popularized in the 19th century.
The genius of Archimedes is mind boggling and his demise was a monumental loss to science.
He was absolutely extremely ahead of his time. I forget whether I said this in the video, but it’s worth noting that while his death was dramatic, he did live to the age of ~75.
That and the destruction of the library of Alexandra.
Yeah this is something I’d like to research more for a possible future video
@@sphakamisozondi I was thinking of including that in my OC , but didn't want to digress too much.
@@bensyversen That would be an excellent choice.
General Marcellus was very upset at Archimedes death. He took only 2 things from Syracuse , 2 machines made by Archimedes which are said to accurately show the positions of all the planets. Until recently it was considered an exaggerated legend but with the discovery of the Antikythera mechanism we now know that it was certainly true.
I feel like I recently saw a documentary on this...wasn't there a dragon that people jumped out of?
The Antikythera mechanism is actually a moon calendar...
@@burnstick1380 No it is a planetarium that accurately tracks all the known planets as well as the moon.
@@billshiff2060 whats your source?
Mine is clickspring on youtube, he literally rebuild the antikythra (He's a watchmaker) and studied it and found that it is a lunar calender (well it does predict the moon phases too but not entire planets)
@@burnstick1380 Dr Tony Freeth and company which is where clickspring gets HIS information.
Michael Wright which is where Freeth got a lot of HIS information.
Look up "A Model of the Cosmos in the ancient Greek Antikythera Mechanism"
Click spring could not complete his model because Freeth came to a different conclusion about the display and it has to be re designed.
8:26 "do not disturn by circles." would be an awesome last words, like he defended something to his death, truly a great mathmatician and inventor
Do not disturb my circles. *
"Get off my spheres"
@@JO-ch3el"get off my balls"
@@JO-ch3el😂😂
"Dont touch my balls"
Well done! The time that you spend on the visuals really makes the concepts clear. Looking forward to seeing more!
Thanks Jonathan! And thank you for the help!
This is unironically one of the highest quality math videoes i have seen, and the visuals really help!
Thank you! Planning to make more
Math usually scares me, it was always my worst subject. But as someone who is so passionate about astrophysics and other subjects that require math, this video was fun and comprehendable for me. You have a great talent, I wish you luck on your CZcams endeavors :)
Thank you very much!
maths is more often than not taught by people who, while passionate, can't for the life of them empathize with different mental approaches to things; and of course sometimes just by idiots to begin with.
combine that with a society that vibes with the idea that maths should only ever be understood by nerds and it's easy to get scared by the subject.
and yes, there are also people with dyscalculia, and they deserve help with their disability. however, the majority or people who are bad at maths are so bc of bad and discompassionate teaching methods, which is extremely sad.
Great video, Ben! Love the concept of teaching math through the lens of history, dramatic music, and visuals. I'd like to see more videos like this.
Thank you Henrik! I’m looking forward to making more.
The quality of this video does not match your subscription or view numbers! So under rated!
Thank you so much for this! This is my first time making this kind of video but I’m looking forward to making more.
actually no, this was mid
ayo wtf just checked the sub count and was shocked
im in!
@@qwertyuiophmid compared to what?
i just checked the view count, i thought this would be in the millions. This is very high quality work. both mathematically and storytelling ability
Thank you, I appreciate that!
Don't confuse quality with popularity. Most people just want quick amusement. Mathematics is a niche passtime.
We always called those proofs "epsilon delta stuff" in first-year calculus. We all dreaded it. I had no idea it wasn't invented until the nineteenth century! No wonder it stood out so much from what was otherwise a fun and relatively easy course.
Yeah I really believe that giving people a sense of history with calculus (and math in general) would go a long way towards helping people appreciate it instead of thinking of it as something to dread.
Ben! I'm so happy to see this wonderful video of yours getting the recognition it deserves! Keep it up, man!
Thank you Joel! You really helped me bring up the quality quite a bit with your feedback.
Fantastic video - well edited and well explained!
Thanks Elias!
I just love seeing great high-level content coming from small channels. Great work! Expecting more videos from you 😀
Thank you! I’m looking forward to making more
The creativity, pacing and visual energy of this video were all incredibly excellent! However I did feel underwhelmed by the video’s denouement, wherein Cauchy’s epsilon solution turned out to be the same as Archimedes. Felt like there is a lot more to be explored in the concepts of infinity - hope to see more in future videos!
Thank you, I appreciate this comment. I'll be the first to tell you that I've greatly simplified the history here in order to give a quick throughline. I did try to be careful to say that Cauchy's "strategy was remarkably similar" to exhaustion -- similar type of strategy, but not the same. I'd love to explore it further in another video.
man!! you deserve more, really good video
Thank you so much!
Congrats.... A well done video. Good luck and success to you and your future content.
Thank you very much!
Man, where have you been all these years?
The city needs you!
Keep uploading to help me and my generation winning the mathzilla fight.
Please don't ever let anyone delete these videos, these are life saviours
Thank you!
Nice video. This is your first big banger. The pressure is on to follow it up with another one. You're fortunate that math history is full of cool stories. I'll give you a shot. +1 sub
Thank you! Yes, math history is full of great stories and I will do my best to do them justice.
Amazing! Good narration and the video quality looks like it was made by a channel with millions of subs 👍👍👍
Thank you very much!
I just looked at subs and assumed something about 500k not 40 you deserve so much more
Thank you very much! This is my first video of this type that I’ve made and I just posted it yesterday. I’m looking forward to making many more!
Wonderful! Best video on the subject!
Thank you very much!
my top 10 maths greats list:
10) Alhazen (optics)
9) Leibniz (binary code)
8) Descartes (coordinates)
7) Ramanujan (fractal proofs)
6) Russell (math philosophy)
5) Euclid (geometry)
4) Newton (calculus)
3) Euler (e^[iπ] + 1 = 0)
2) Archimedes (π)
1) Gauss (non-Euclidean geometry, FFTs, Normal curve, etc.)
noooo euclid what did the tier list do to him 😢
@@larevolution13 i think all of these guys are very close to each other in terms of importance. I also realize now that there are names I may have forgotten to mention that have also contributed greatly to the expansion of mathematical knowledge.
Bernoulli (fluid dynamics), Listing (knot theory), Hamilton (multidimensional complex numbers), Al-Khwarizmi (algebra), Riemann (integral calculus), and Galileo (astronomical measurements) are a few of the names that definitely should have muddied this list
@@kitcutting Leibniz has done so much more than binary lol. Same for Euler, he probably didn't care that much about e^ipi+1 because he knew it was a special case.
@@darthmath1071 Obviously all of these mathematicians have done more than what I have put down. I am just listing examples of some of the fields of maths that they have contributed to. I have a tremendous level of respect for all of these guys but after careful deliberation, I would personally rate them as you see them.
“calculus: Leibniz vs Newton,” “did Archimedes really come up with pi and prove that it was transcendental,” “did Euler even come up with that equation or was it just an extrapolation of one of his proofs,” there are so many things up for debate here that I have no time to talk about lol. Just take the original comment for what it is
i failed calc 2, but as a classical musician i appreciate how you fit the background music to the time period. that chopin prelude is one of my favs
Thank you!
Excellent video, very interesting. Thank you 👍😊
Thank you
Very underrated video, keep going!
Thanks!
Beautiful and educational video! I hope you grow in the future, seriously quality content.
Thank you very much!
imagine if that roman didn't stab Archimedes
Would not make that much of a difference.
Archimedes made much more discoveries than we know anyways. Most of his discoveries were lost by the time the Church got their hands on his manuscripts and decided to wash them and write hymns to God on 'em instead.
So, I'll forgive the soldier. The priest, however...
This is an excellent combination of history, mathematics, story-telling, and visual presentation of information. If there was an Oscar 🎥🏆 for CZcams videos, you would have won by a wide margin!
Thank you so much!
bro dropped the hardest maths video and thought we wouldn't notice
Wow!! Very interesting!!
Thank you!
I was wholly disheartened to see that you do not have 50 more videos for me to binge watch. Marveling content and presentation, thank you for sharing this with us!
Thanks! Working on it!
Those last words are something else “do not disturb my circles” absolute legend
Great video, love this kind of content. Thank you!
Thanks!
Bare in mind the two soldier's sent to capture Archimedes were explicitly told not to kill or even harm him.
Wow.. Rarely I get the flyback feeling after 7 years of uni studies. You did that perfectly and reminded me why I did that to begin with randomly stumbling over your clip, solid thanks.
Thank you!
Awesome video and very informative. Thank you
Thanks!
Great video, you have gained a subscriber
Thank you!
Kudos on this episode!
More please!
Thank you!
Incredible animations! What are you using to make them? A video of this quality like this looks like it would take a long time
Thank you! Believe it or not, I made most with Keynote, the Apple version of Powerpoint. There's one little animation in the intro that someone made for me using Manim, the math animation programming language. Yes, it was kind of time consuming! Hoping to get better at that with practice though.
Very impressive! Keep it up man
THank you!
I just found an excellent full explanation of Archimedes' proof by exhaustion of the area formula of the circle here: czcams.com/video/R1HUtt2oo7A/video.htmlsi=r-FQ-OfaES7XVmUM
This is a masterpiece. More videos like this please.
Thank you! Hoping to do more
cool! very fascinating!
Thank you!
😊 very informative 👏 and useful video and your visuals are fabulous 👌
Thank you!
I think my biggest disappointment in mathematics is that points cannot be adjacent to each other. Points either overlap, or they're separated by infinitely divisible space, and it drives me nuts.
When they overlap can you even say there's two points? Or do you have just one?
If you wanted an adjacent point, you'd basically have to invent new math I guess. And the neighbour points would have to be undefinable / described by limits essentially. I'm not sure what use they'd have but ye that does conceptually sound annoying 😅
There actually are particular topologies where points can be adjacent to eachother. You basically have to rethink the notion of distance, or even more fundamentally, of "separation"
@@lorenzoputignano8829
Where can I read about this
You can have adjacent points if you have a non-continuous space.
Circle and pie is something where we can see and feel infinity. This makes mathematics beautiful.
Loved the video
Thank you!
This youtube channel is so underrated
I remember in Calc I that my professor talked about the implications of pi. It didn't mean a lack of sides or just 1 side, but an infinite number of sides.
Or no sides at all.
Make more great videos like this one please
Thank you, I'm planning on it!
Very cool Video, quite interesting! However, the "Calculus in the modern World" part with the stockfootage felt like a bit much xd
Great video.
Thank you!
Good video! This Ben Syversen guy seems smart.
What a coincidence, you have the same last name as me!
Your video is so good that let me think your subscribers are 2M instead of 2k for a second there
Thank you!
That is because the people did not understand fully the idea of limit. The paradox like when the arrow is thrown, it should stay in equilibrium and should not move because it is stationary in very very small amout of time, and it should be stationary in another small time interval, and it keeps going. When you add up all the infinitesimally small pieces, you add up to get the whole thing. Seems Archimed had understood this and that is why he is GREAT.
Brings back a good memory from freshmen year of highschool where I was stumped over the circle equation lol I think we had a math masters sub in man even though I had that guy for a day I miss his enthusiasm for math lol really made me feel less crazy in that class lol
This is top tier quality. You should have millions of subs
Thank you! Someday I hope. I’m working on more as we speak.
not even 3 k subs? damn man keep making videos like this and you'll be 300k soon. best of luck
Thank you!
You have 5k subs right now. Remember me when u hit a mil ❤
Haha thank you. Hoping to get there! Gotta finish my next video first…
I hate how school really undersale how powerful Archimedes and Neton are, they just do a "ohh they invent this" and did not and can not eleborate on how truly amazing the details are
I strongly agree with this comment. I hope to tell more of those stories here in the future!
Very cool. Thanks for sharing
Thanks!
Bro imagine society if Archimedes lived longer
This channel is underrated
Thank you!
Okay, this is silly, but I love how much his cadence seems to line up with the background music. It's like he's rapping Archimedes' praises. And he even got killed by a cop-a true OG
I wasn't thinking of that but I do love hip hop!
You got your 403th subscriber, still late but fine. Wish I could know about this channel earlier then now
Before Wednesday I had only been posting boring explanation videos, so you haven't missed much! Hoping to do plenty more.
Great video. I would like to ask you something. What exactly do you use for those drawings?
Thank you! I made the circle sectors in Inkscape and the animations in Keynote. Some of the background images come from Midjourney.
@@bensyversen Great. Thank you very much for your answer. You really did amazing job...
Thank you!
damn, onlt 350 subs
the quality of this video is actually impressive, i'm happy that at the least it's getting traction with nearly 19k views
Thank you very much!
Excellent video!
Thank you!
This is something that helped me to understand different sized infinities.
Imagine you have a pair of totally fair dice. You throw each pair and you ger a number of pips between 2 and 12. However, you are more likely to get some numbers (like 6) than others (like 2 or 12). Because some numbers result from more combinations.
If you threw the dice an infinite number of times you will get an infinite number of each result. However, the dice will follow the same statistic. You will have more 6's than you have 2's or 12's, even though you have an infinite number of both.
This is possible became infinity is not a number.
"You will have more 6's than you have 2's or 12's, even though you have an infinite number of both." No you won't, you will have a countable infinity of each of them.
I really appreciate both of these comments. That idea of “countable infinity” vs “uncountable infinity” is one that could he interesting for me to explore in a future video.
Subscribed 😊
Thank you!
“Do not disturb my circles.” 😵💫
Question for Ben Syversen. Near the end you mention Archimedes said the volumn and area of a sphere is 2/3 of a closely enclosing cylinder. This reminded me of what I think is a similar fact. The volumn of the intersecting of two cylinders is 2/3 that of a closely enclosing cube. What would the surface area of this shape be? Thanks for interesting video.
Hmm I'm not if I can picture what you're describing, or if I'd be the right person to answer your question to be honest. I think you'd be able to calculate it with multivariable calculus, but mine is quite rusty I'm afraid.
An area is cut into infinitely small pieces: a / inf = 0
An individual piece is zero.
All the small pieces together add up to the original area
small piece + small piece = big piece
So if a the small pieces are 0 it‘s like saying 0 + 0 + 0… = a
That is why a / inf is not 0. the individual pieces approach zero.
Onw of my favourite things to do in high school was to prove where the formulae we were using came from. But I couldnt figure out the area of a circle formula.
great vid keep em coming
Thanks, will do!
made an equation to calculate pi to 10 decimal places using the idea that a circle is a polygon with infinite sides. what I found is that how you approach zero (degrees for each tringle in the polygon) is really important. I had to approach zero with 2Pi/x while 1/x didn't work.
This video is going to pop off and receive millions of views
Thank you I hope so!
I loved your video thank you very much.
Thank you!
What program did you use to make the math animations?
I used Inkscape to draw the circle sectors and I made the animations using Keynote
On Finnish we have a common saying, "to mess up ones shapes." Shapes on this sense refers to plans, but I would imagine it's origin is in Archimedes.
You could make a video on calculating the sun's angle at a given time and one's latitude.
Thank you!
Great video about an interesting topic
Glad you enjoyed it
Loved the video. Just one thought, you could let go of the background music during your explanation! Looking forward to more
Thank you! Yeah I was back and forth about the music in that section
Seems like you found a winning video format bro. =]
Thank you!
bro got da capo in the background in the beginning lmao
The good old case of a soldier "just following orders" 😊
Very good video, the opening music is a little too prominent but that’s about it.
Thank you
I, too, am a fan of circles. And I, too, dislike them being disturbed.
E.T.Bell ex president of American Mathematical association and author of "Mens of mathematics" ; says in his introduction to the chapter on Archimedes " that Archimedes genius is so great that from the number of math breakthrough that he anticipated ,if he rose up today (1965) and took an undergrad course in physics,he would understand Bohr, Einstein,Dirac better than they understand their own theories. "
I mean damn that's some high praise more than 2 millenium after your death🙆🏽♂️🙆🏽♂️.
I love this. I think that Archimedes, like Newton, are actually still underrated to this day even with all that is said about them.
@@bensyversenTrue
Very interesting video!
Thank you!
@@bensyversen I thank you for your effort!
I friggen love that guy
Great video .. encouraged me to study for my maths test...😂😂
Thank you and good luck with the test!
Mate, i loved it. I guess this is somewhat like the book by steven strogatz: Infinite powers-the story of calculus. Have you read that? In any case, looking forward for more videos like this. 3b1b, numberphile, mathologer are some other greats. A well rested dog made jist one video in its lifetime and I loved that one, it was #Some2. Lastly, if you could prove that the ratios of sides of a right triangle remain the same as long as the angles are same regardless of the lengths of the sides. Many commit a fallacy in doing so, they unwittingly assume that it must be so in proving it. I guess you are understanding what I say, english after all is not my native tongue
Thank you very much! I'm glad you enjoyed the video. Yes, this was definitely inspired by Infinite Powers. I'd like to tell more stories based on the ideas from that book. I think Strogatz has a great way of talking about this stuff and deserves to be shared with more people. I tried to point people to Infinite Powers in the video description but maybe next time I'll find a way to put something onscreen that doesn't interrupt the storytelling.
I love the channels you mentioned! Also, Veritasium is a big inspiration.
I'll look into that proof -- I've got some ideas for next videos, and I'm trying to find ideas from math that specifically have an element of history or personal drama involved that makes for an exciting story as well as being educational.
@bensyversen For one thing, you made my day, or rather I should say night! Only new creators bother to answer all comments but it is also understandable as they get well known the amount of comments gets grandeous. definitely veritasium must be a inspiration. Though he makes few strictly mathematical videos, the ones he makes are excellent (most thanks should be given to his math friend).
@bensyversen not any professional myself, but I would really like to recommend these science/Math/history books-
Surely you are joking Mr feynman
Homo sapiens by yuval noah Harare
Higher Algebra by Barnard and Child
Higher algebra by Hall and Knight
I have all the books mentioned and I definitely liked them! Hall and Knight algebra is concise and most suitable for the first course of reading. The Barnard and child version is a big tome and suitable for those who have mastered the hall and knight. Again, Heartfelt thanks for the video and your reply.
@bensyversen curious about any other way we could stay in contact.
Well I am definitely a new creator and I am very happy to see people writing in the comments!
Thank you for those book recommendations. I read Surely You're Joking Mr. Feynman many years ago and was thinking of returning to it now that I'm starting to make videos like this. The others I'm not familiar with and I'll be excited to dig in.
Are you on Twitter/X? I have an account there: @ben-syversen. Also, you can find my email in the about me section of my CZcams page...I hesitate to post it in the comments to avoid any spam bots.
this needs more views
Thank you!
Amazing video
Honestly blew me away. Archimedes, can I marry your brain? 🤯😍😂
If you approximate a circle to a rectangle by doing small slices, having half the slices in one direction and the other half in the opposite one, you end up with a rectangle with height equal to the circle radius, and length equal to half the circunference. The area of that square would be
r * (pi*d/2) = r * pi*r = pi * r^2
And thats the area of a circle.
No shit bro he just explained that in the video
@@rogumann838 yep, the video cover is a clickbait, that's why I said that before watching it XD
Alessi showed with ruler and compass:
pi = 4*sqr[1/phi] = 4*sqr[phi-1] = 4*sqr[2/(1+sqr[5])]