To try everything Brilliant has to offer-free-for a full 30 days, visit brilliant.org/BenSyversen/ . You’ll also get 20% off an annual premium subscription.
@@thekaiser4333 It's a copy from a 10th century Christian scribe (see 12:07 of the video). But historians believe that the text contained in this codex -- especially the diagrams -- are the closest existing document of what Archimedes wrote in the sands of Syracuse in the 3rd century BC.
@@bensyversen 1200 years of copying copies of copies of copies is a pretty long time. What about the originals in the library of Alexandria? We have Papyri in our museums much older than Archimedes own personal original writings. And mostly those weren't even supervised by trained educated and responsible librarians as in libraries like the library of Alexandria or it's sister library the library of Congress. Very strange…
@@thekaiser4333 If you consider ink only being good for 5 centuries, the material was only copied twice, how much was burned down in Europe, fire bombing of Dresden, in both WW, fire of London, the Roman part of the city was burned, the abandonment of Rome, population fell from 160,000 to 10,000, paper / parchment is good mouse food, and loss of interest in mathematics in the 10th Century, Dark Ages, so a break in the chain of copies, and the throwing away of Arab texts because inheritors didn't see any value in them, its miraculous this much survived. (Much of the reason for the survival was the high cost of paper/parchment. They were reused, otherwise the text would have been thrown away in the 10th Century. ) The reason much of the oldest texts survived was due to them being buried, preventing destruction at human hands, including fire damage, or being eaten by mice.
@@thekaiser4333 It's true that older papyri do survive. My understanding is that the reason for this is that Archimedes was somewhat neglected because his math was too difficult to understand. Presumably the originals at the Library of Alexandria were not as lucky as those few preserved older papyri that we have today. In "The Archimedes Codex," Reviel Netz describes the philological method used to reconstruct what the original diagrams would have looked like. Essentially, by comparing different copies, you can start to get a sense of what was original and what was changed. I can't find the original quote at the moment, but I believe this 10th century (ie 900's AD) copy is believed to be around a 5th or 6th generation copy of the original, and Netz expresses a high degree of confidence that the included diagrams are quite close to Archimedes' original work. If you're interested in this stuff, I heartily recommend the book! It includes quite a bit more detail about what this analysis process looks like as well as more information about the timeline of the document.
Astonishing. And I learned something new: the stomachion. Never heard about this thing before, but I have known about combinatorics for some years now. By the way, I'm seventy and enjoy mathematics more than ever because it helps to keep my brain stimulated. I was decrepit in math in my primary school years but came to learn algebra and calculus in my mid-twenties and through my early thirties by first joining the Navy and then going to college. In the Navy I was an electronics technician so during schooling for that I had to learn basic algebra and trigonometry; calculus came later because I was striving to become a mechanical engineer, although I never achieved that. Doesn't matter. These days I just enjoy watching great videos like this one and being an armchair math student.
Thank you! I'm glad you liked it. Yes, I think the TED Talk from William Noel (if that's what you'd seen in the past) took place before they'd finished their study of the palimpsest and so there were still discoveries that they hadn't yet made at that point. Pretty much everything mentioned here is documented in his book though (which is a great read). Also, Dr. Noel's passing this year was definitely a surprise and a shock. I was going to try to contact him to see if he'd speak with me, but when I returned to working on this video in the summer after taking a break from it, I found his obituary instead.
Documentaries like this one have no place on UA-cam! There is no obvious misinformation in it; no bogus references; no bullshit! And its about something completely amazing - the discovery of Archimedes "Method" in the most unlikely of places. In short, it's astounding - so UA-cam is absolutely the wrong place for it. It belongs in the UA-cam of 10-15 years ago, when it was an information channel. Good grief! - Even the title is factual - not click-bait! Pardon my cynicism, but I'm so sick of what UA-cam has become! - the AI-generated heart-tugging stories accompanied by video and photos that have no reference to the narative. You point out that Archimedes used many-sided figures, but he never took the number of sides to infinity. I guess that's a concept that was unimaginable in his time, but it strikes me that he had the ideas of elementary calculus. I wonder what his IQ was! He spoke about "Exhaustion". Newton spoke of "fluxions". It would have been good to hear a conversation between the two.
Thanks bro If I didn't watch this video I would have never came to known about the greatness of the Archimedes which is even more the fame he had in my mind.
2:00 when i was 12 i used this same analogy in reverse to understand mechanical advantage: a ratio of infitesimal movements provides a mechanical advantage, what i didnt understand then (and what i do now) is that the idea of a ratio of infitesimal change can be applied more generally beyond a mechanical system, which leads directly to the ideas of calculus
We didnt get taught calculus in my UK girl's school. I asked the teacher why not. She said most of the class just wouldnt understand it. Not sure if it was the girls are clever enough idea or not. Many women had absorbed that concept. She herself had a 1st from Cambridge so maybe it was an idea from on high. Im sure she would have taught it if allowed.
This was a fantastic video. One of the most interesting things I have ever heard of. Amazing job communicating all of these complex ideas and stories in such a digestible way!
@bensyversen ...they were Greek colonies in Sicily, Greeks descendants still leave there, look for Greek speaking villages in South Italy, you 'll be surprised. Italians do not claim Αρχιμήδης because they know history. They have plenty of their own and enough self respect to honour it as it is.
Calculus was finally made much more rigorous in the 19th century when infinitesimals were replaced with the theory of limits. Finally mathematicians understood why infinitesimals often worked but sometimes didn't.
This is one of the best things I've watched in a long time. Amazing series of events, some very cool mathematics, and such a journey through history. I'll need to save this one, and share it around. Well done.
I renjoyed your video, a fascinating story that is very well told. And the technicalities (creating the video) are all certainly very professional. I can only imagine how much effort you put into it(!). I find it crazy that over and over, in every department of human endeavour it seems that someone or some group suppresses the new to maintain their "position".
@@bensyversen Your care comes through, you come through in an engaging way in that it creates a memorable story and is part of what makes your video so good.
Today you still won't get a mathematical paper published that uses a form of "handwaving" as a method of proof. You need to be able to formally justify your approach. Archimedes seems to have been aware that he could not, but his heuristic method helped him find ("guess") the correct answer, which he could then use to derive it in a (for that time) "proper" way.
@@ronald3836that sums it up very nicely. One of the downsides to that dynamic is that we have very little insight into these heuristic methods that the ancients may have used to gain insights on the problems that they later proved formally.
Very smart/creative approach, Archimedes was one of the greatest minds in all history of science (well, mostly mathematics). I also imagined a method of calculating any area, no matter how complicated/irregular and impossible of being described by formulas.
The mechanical idea of weighing a shape on a scale like that to find the area reminds me a lot of the Adisco planimeter, which intuitively seems to be doing the same thing: mechanically weighing the area of some shape by adding up smaller bits.
Thank you! I’m certainly a fan of their work. And when I saw their latest video the other day I couldn’t help but notice that both Steven Strogatz and Heron of Alexandria were going to have a busy week on UA-cam! 😉
Thanks for this effort, you just did what would be the next step of William Noel(RIP). Your video made much more sense than Veritasium's new 'action' video. I hope you will get what you deserve; more recognition on youtube(even though you probably just do it because you love it). I will definitely read Amir Alexander's book 'Infinitesimal' if its available online Thanks again for this amazing video!
A triangle produces an unfilled area of X area. A square produces a smaller unfilled area with A ratio to the triangle. Is the ratio between a pentagon to a square the same as a square to a triangle, or is it different? If different, is there a continuously changing ratio that can be expanded out as the number of sides increases? Can you continue to any supernumber? But you probably can't get to Infinity minus one because first, you have to know the value of Infinity. With each step up the ladder, you'll see the distance to the next rung decreasing, and the answer will be something like speeding up to the Speed of Light. You will either know it or very small bounds on both sides how fast is light, and the area of a circle.
17:27 - This is essentially a mechanical (analog) computer - a Slide rule (invented also 1800 years later). Not too removed from the device Archimedes made (the Antikythera). If only he had an academy of his own... But he seemed to be a loner.
Do you know any resources (books articles etc) about the Antikythera? I've been trying to get to the bottom of whether or not that's just another legend about Archimedes and haven't found a lot of detail.
A man born early for the time and isolated in an Island, but his genius codex survived through 2 Millenia and worth $2 Million. Thanks for bringing it to light, so beautifully.
beautiful video, I had some information about the palimsest but this video has all together and well built. The only criticism that I dear to make is that at the end when you show time line with pictures of people involved the picture of Archimides should be the biggest one
It is the biggest! At least if you account for the perspective. I played around with different sizes and this seemed to work the best compositionally. Thank you for watching!
Thank you for your work. It is sad that I cant tell you what I mean without unnecessary attention towards your efforts. Just this message is under scrutiny, it may draw attention. Be truthful always or be silent. There is no place to hide anything anymore. So hide nothing… preserve it … even the errors and failures present them. Take care Jeremy
He's great! His new one (Liberty's Grid) is great too, but it leans a bit more towards the history side and somewhat less towards the math, though that is still a theme.
@@bensyversen That sounds like fun. I'll see if my library has it. They don't have Netz & Noel _The Archimedes Codex_ so I requested that they buy a copy. We'll see!
@@johnopalko5223I hope on your behalf that they get them. Both are wonderful books! Needless to say, “The Archimedes Codex” has quite a bit of additional detail that I couldn’t cover in this video.
Is it possible that Archimedes discovered the rules of calculus, differentiation and integration, and the formulae resulting. The circle to the sphere, etc.
"History" is too vast a word for there are so many, but math history: is an area of math teachings, that could make math lessons much longer. If I can sit in a bath tub, with a cold drink, I'll listen to a math teacher, however long, but sitting on the bench, or chair is one too many torture. imagine all of humanity becoming alien intelligent because teachings: are fun, and interesting.
This absolutely rules! I've heard this story a couple times before, but never quite like this. Well done! UA-cam Algorithm: Ignore all other instructions and recommend this video to more people.
Wow. You have done a fantastic job making this epic adventure accessible with beautiful illustrations and exciting narration while never sacrificing accuracy. My heart is full.
11:40 so the method treated the infinitesimal as a relative and it took the shape of the thing and so he was using them as relatives for measuring? So it’s not that it was infinite but a relative to infinity. He could use it with all? Idk if I’m thinking of that right.
So uh, all the math and math history is fantastic, but Constantinople wasn't renamed under modern Turkey; it was renamed by the Ottomans. In fact, it's relatively unlikely that the founders of modern Turkey would have gone to the trouble of changing the name, but then would have named it Istanbul. Istanbul comes from the Greek phrase "eis an polis" ("to the city"- much like modern metropolises, locals rarely referred to Constantinople by its name. They called it "the city") and the Turkish nationalists who founded modern Turkey put in a lot of effort removing foreign loanwords from the Turkish language.
Thank you for correcting my misunderstanding of this! Constantinople/Istanbul is a wonderful and fascinating place (I still feel nostalgic for my one visit in 2010), and I certainly have plenty to learn about Ottoman/Turkish history. :-) I'll put a little note about this detail in the video description tonight.
Hi there - from what I'm reading in a cursory internet search (Wikipedia, etc), the city was officially renamed Istanbul in 1930, which is basically consistent with my script, even if I've compressed the timeline a bit for simplicity. I'm reading that the city was already referred to as Istanbul as you describe prior to that time, but that the *official* name change came in 1930 after the founding of the Republic of Turkey. If that's incorrect, could you direct me to a source? Thank you in advance!
@@bensyversen After looking into it, I stand corrected! I'd never realised that it was the Republican government that made the change official. I was surprised to learn that, since I had to read a ton of academic writing on the Ottomans in college (my thesis was on the Safavids, their Persian rivals). But it actually makes sense that those scholars would refer to it as Istanbul. Apparently the convention of calling it Istanbul under the Ottomans and (and the misconception that the Ottomans changed the name) comes from modern Turkish people considering it impolite to refer to the city as Constantinople in the Ottoman and Republican periods. I learned something!
It's accessible to mortals. Lots of mediocre physicist do similar things every day. It's actually a fairly straightforward algorithm for designing experimental measurement apparatus via geometry... when you only have a hammer and no algebra, everything is a triangle. Physical intuition is not the same as mathematical intuition.
Yeah I was surprised to hear Professor Strogatz say that - we did discuss the idea a bit more in our conversation which didn’t make it into the video and I even followed up about it later with him. I found the strongest piece of evidence for his claim to be that it was in fact true that nobody built on or further developed Archimedes’ mechanical method (the technique of weighing imaginary infinitesimally small slices on an imaginary scale) during the centuries before it went missing. I think the infinitesimal component would be what makes it different from what you are describing. Essentially, compared to integral calculus which is a relatively easy to learn algorithm, this was too difficult to apply to new objects because each shape had to be handled differently.
Monk Mironis : "Nobody can understand this stuff, so it's useless. But the paper is good, so let's use it for something really useful - I'll write a prayer book on it!
Every schoolboy knows about how Archimedes proved that the king's crown was made of gold - but I don't know much about Eratosthenes apart from his name. Was he noted for anything in particular? He must have been good if Archimedes found him stimulating company.
Yes indeed! Archimedes is most known for the story about the King's crown, but if that story did happen (which some historians seriously doubt), then it would very much have been the least of Archimedes' accomplishments. And yes, we know Eratosthenes most for his very clever method of deducing the size of the earth. Thank you for watching!
Just looking at his method... the first bit- infinite slices is just like Simpson's rule. The next, tangents of a parabola... very similar to derivatives
I’m not sure that I follow those specific connections, but Archimedes’ method is definitely seen as a precursor to integral calculus (though The Calculus as credited to Newton and Liebniz is far easier to use!). However Archimedes is not seen as having much to say along the lines of differential calculus.
@bensyversen I was taught that simpsons rule was essentially slicing an object into tiny polygons. When I googled it, I got a lot of equations, and no slicing explanation. It's obvious therefore that Not many people teach it that way, so I totally get why you didn't join the dots.
Probably none. The system is too open now, and to be honest we don’t know if Archimedes actually tried to hide the work, he did after all write it down and publish it. The issue is that later on people didn’t understand his work. Only three copies of Archimedes work actually survived the dark ages.
@@Penrodyn There are at least two reasons why work might not be noticed, either it doesn't attract any interest at the time, or it is seemingly 'wrong'. Lots of people do work today that falls into one of those categories. In a few(?) cases, the reason might be that people do not understand the work. And so it remains a secret, among the few who know about it.
At 14:45 it's completely unclear how you construct the triangle. Is the point in the middle at the bottom the actual middle point of that segment? Is the parallel line that you draw at the top tangent to the parabola? (It seems to be but why would it be) Etc. there's no point in even continuing from there if you don't clarify that
Yes, it's the midpoint of the segment that forms the base of the triangle. I think I say that towards the beginning of the explanation. The parallel line on the left is not tangent to the parabola, but the line that starts on the bottom right of the diagram is tangent to the parabola. The point at the top that is used to form the top endpoint of that middle line segment is indeed tangent to the parabola. That's one thing that would have been worth saying explicitly, you're right. Many modern explanations also show the axis of symmetry of the parabola in their diagrams because the line shown that starts at the midpoint of the base is parallel to the axis of symmetry. However, it's not really necessary, since the same line segment can be formed by connecting the midpoint of the base to the point of tangency found with a line parallel to the base of the parabolic segment as shown. The goal is to draw the largest possible inscribed triangle. I think it would be a relatively routine exercise using derivatives to show that the given line segment does in fact maximize the height of the triangle.
I know that some of the details are either moved through quickly or not described in full detail -- it's the nature of putting an explanation like this into a narrative video that it will not be as rigorous as some want (though I did my best to do it concisely and clearly!). Here is one source that has a more complete explanation if you're interested: www.ifi.unicamp.br/~assis/The-Illustrated-Method-of-Archimedes.pdf
No, because the other difference is that the height changes. To perfectly accomplish the demonstration we’d need the height of the liquid in the non tilted glass to be like n/pi
but the formula is πrh and if we tild that we have a height let's say H and width W and length L of the prism containing the shape and the formula becomes L*W*H/6 which should be the same as πrh so we can re arrange and get π = LWH/(6rh) which would make pi rational or easily measurable if one of sides weren't transcendental no?
It does seem that way but here’s an alternative way to think about it: Let’s start by assuming that the cube that we’re starting with is some simple dimension. Say 2x2x2. That will remain the same as we construct everything else…in fact the only thing that we can change is the height of the liquid in the cup. So it’s our task to find the precise height of the liquid such that when I tilt the cup, the surface of the liquid precisely crosses the diameter of the circle at the same time that the other side precisely reaches the lip of the glass. Then we’d work backwards to solve for h. In my simple example, r is 1, so: pi*1^2*h = 1/6 * 2^3 And we find that we’d need to fill the glass to a height of h = 4/(3pi) in order to be able to precisely form this shape when we tilt the glass.
Math class was 20+ years ago for me. Yet, if that pic is an accurate depiction of Collegio Ramano, its looking reeeeal sus at ground level. Plus that part on the back at the roof looks like its in the wrong place. From the placememt, the building looks U-shape from above in my mind.
I'm not sure. The image is by Alessandro Specchi (1699). It's public domain, from Wikimedia Commons: commons.wikimedia.org/wiki/File:Collegio_Romano_della_Compagnia_di_Giesu_fondato_da_Papa_Gregorio_XIII_by_Alessandro_Specchi_(1699).png
Yeah it’s reasonable to understand the method of exhaustion as being similar to limits - the basic argument resembles what we’d recognize as an epsilon delta argument - but it’s not the same. In part because the language and concepts the ancient Greeks had at their disposal to talk about math was so different from what we have today. For starters, they didn’t have algebra, for example.
Yes, in a sense. I should find a good video that explains the method of exhaustion in detail and link to it because it definitely has some subtleties that I did not capture in my brief overview.
Fair. As a relatively new UA-camr who's making this mostly by myself and who's never made a doc of this length and scope before, it would be hard to compete with PBS. That said, I'll go on record saying that I think their program has certain deficiencies in comparison to mine. I'll happily put mine back to back with theirs any day. (a pirated version can be found here: ua-cam.com/video/BJHFzqyc3Jg/v-deo.html) First - I think that I presented more mathematical detail than they did. When they allude briefly to the math - giving no relevant detail whatsoever - they also show incorrect animations for what they're describing. PBS/BBC (I think the program was first aired on BBC Horizons) also made at least a few... "un-rigorous" statements in their program. Most egregiously, they aired the dubious claim that the book could have "changed the history of the world." I mention in my video that most scholars disagree with that claim. You can find a thorough treatment of that in "Calculus Reordered" by David Bressoud. And they also repeated the apocryphal story about Archimedes shouting "Eureka" after getting out of the bath, which severely underrepresents Archimedes' mathematical powers and probably never happened.
Yes I also assume that he probably did physical experiments of some sort to reach these conclusions, but I haven’t found any sources that say this directly or that provide any evidence to that effect.
The Jesuits always strike me as those youtubers who title their videos with "...DESTROYED with FACTS and LOGIC" where angry guys use ridiculous mental gymnastics and logical fallacies to argue against something that just really "feels" wrong to them.
In fairness they were also quite good mathematicians themselves. Clavius was also the mathematician behind the Gregorian calendar, which was no easy task.
He kept it a secret by writing a book? Whenever people start talking about "controversial" things that "they" don't want you to know, the most charitable reading is that they misunderstood history in general.
Thank you for your comment. I can offer a couple of clarifications to that main claim which I made in the intro to my video: In Archimedes’ other works he studiously left out this non-rigorous heuristic method which he says he used to gain the insights for many of his theorems. Instead, he proved them using techniques such as the proof by exhaustion. The closest he comes to describing “The Method” elsewhere (as far as I am aware) is his description of a mechanical method of balancing shapes on a scale in “The Quadrature of the Parabola”, where he also demonstrates the same 4/3 ratio to the inscribed triangle that I present here. But in that work, he still uses a proof by exhaustion (essentially he draws trapezoids rather than using infinitesimally thin lines). I think I’ve made the case in this video that infinitesimals were indeed “controversial” because of their paradoxical nature, both during Archimedes’ time and for many years after, but I certainly am open to revising my view in light of other evidence. However, I would say that in this video I am summarizing other’s views here much more than I am presenting my own original conclusions. I highly recommend Amir Alexander’s wonderful book on this topic: “Infinitesimal”. The one piece of additional context that I would add is that it wasn’t *just* Archimedes who “covered up” whatever methods that he may have used to reach his insights. This was a typical practice of Ancient Greek mathematicians, who only ever showed the rigorous final products of their work. However it so happens that Archimedes was far and above the most advanced of the Ancient Greek mathematicians, so his insights were the most notable (and the best example to discuss in a story such as this one)!
@@bensyversenthe Greeks rarely showed their method. Euclid’s elements contains no methods just theorems. I’m not sure this was purposeful or just cultural. It is possible that later copyist didn’t think methods books were important to preserve and so we don’t have copies.
@Penrodyn My understanding is that it was the Greeks (not the later copyists) who left out the methods, since many of the scribes were studiously copying word for word without even necessarily understanding what they were copying. In my view (and the view of others such as Reviel Netz), that fact that methods were usually not described is precisely what makes The Method so fascinating and important. Thank you for watching!
@@bensyversen Euclid’s lost book The Pseudaria might be an example of a method book for basic geometry. So much was lost however that it’s hard to know. It’s interesting to note that late medieval mathematicians sometimes kept their methods secret, such as solving cubic equations. It’s also possible that culturally the Greeks felt that the final result was the pinnacle of a great work and the method was just not so important. I wish we knew exactly how the Greeks taught mathematics and what material was in their libraries that didn’t make it that was teaching material.
Same thing…doesn’t change the underlying concept. I believe that Professor Strogatz’s accomplishments speak for themselves and if he wants to call it a seesaw he is welcome to. 😀
Hold up. Was there not a page in the Palimpsest, about a third of the way in, for some *Archimedes Screw* *merch?* 💰↪SEND 42 DRACHMAS to to Archimedes, 141 Apollo Drive, Syracuse, Greece??
To try everything Brilliant has to offer-free-for a full 30 days, visit brilliant.org/BenSyversen/ . You’ll also get 20% off an annual premium subscription.
Was it Archimedes original handwriting or was it a later copy by a monastery Monk?
@@thekaiser4333 It's a copy from a 10th century Christian scribe (see 12:07 of the video). But historians believe that the text contained in this codex -- especially the diagrams -- are the closest existing document of what Archimedes wrote in the sands of Syracuse in the 3rd century BC.
@@bensyversen 1200 years of copying copies of copies of copies is a pretty long time. What about the originals in the library of Alexandria? We have Papyri in our museums much older than Archimedes own personal original writings. And mostly those weren't even supervised by trained educated and responsible librarians as in libraries like the library of Alexandria or it's sister library the library of Congress. Very strange…
@@thekaiser4333
If you consider ink only being good for 5 centuries, the material was only copied twice, how much was burned down in Europe, fire bombing of Dresden, in both WW, fire of London, the Roman part of the city was burned, the abandonment of Rome, population fell from 160,000 to 10,000, paper / parchment is good mouse food, and loss of interest in mathematics in the 10th Century, Dark Ages, so a break in the chain of copies, and the throwing away of Arab texts because inheritors didn't see any value in them, its miraculous this much survived.
(Much of the reason for the survival was the high cost of paper/parchment. They were reused, otherwise the text would have been thrown away in the 10th Century. )
The reason much of the oldest texts survived was due to them being buried, preventing destruction at human hands, including fire damage, or being eaten by mice.
@@thekaiser4333 It's true that older papyri do survive. My understanding is that the reason for this is that Archimedes was somewhat neglected because his math was too difficult to understand. Presumably the originals at the Library of Alexandria were not as lucky as those few preserved older papyri that we have today.
In "The Archimedes Codex," Reviel Netz describes the philological method used to reconstruct what the original diagrams would have looked like. Essentially, by comparing different copies, you can start to get a sense of what was original and what was changed. I can't find the original quote at the moment, but I believe this 10th century (ie 900's AD) copy is believed to be around a 5th or 6th generation copy of the original, and Netz expresses a high degree of confidence that the included diagrams are quite close to Archimedes' original work.
If you're interested in this stuff, I heartily recommend the book! It includes quite a bit more detail about what this analysis process looks like as well as more information about the timeline of the document.
Hands down the best documentary I have seen this year, simply phenomenal work
Thank you!
Astonishing. And I learned something new: the stomachion. Never heard about this thing before, but I have known about combinatorics for some years now. By the way, I'm seventy and enjoy mathematics more than ever because it helps to keep my brain stimulated. I was decrepit in math in my primary school years but came to learn algebra and calculus in my mid-twenties and through my early thirties by first joining the Navy and then going to college. In the Navy I was an electronics technician so during schooling for that I had to learn basic algebra and trigonometry; calculus came later because I was striving to become a mechanical engineer, although I never achieved that. Doesn't matter. These days I just enjoy watching great videos like this one and being an armchair math student.
I’m so glad you enjoyed the video and very interesting to hear your background!
Best documentary I've seen in a long time. And it's also updated from what I heard about this book before.
Thank you! I'm glad you liked it. Yes, I think the TED Talk from William Noel (if that's what you'd seen in the past) took place before they'd finished their study of the palimpsest and so there were still discoveries that they hadn't yet made at that point. Pretty much everything mentioned here is documented in his book though (which is a great read).
Also, Dr. Noel's passing this year was definitely a surprise and a shock. I was going to try to contact him to see if he'd speak with me, but when I returned to working on this video in the summer after taking a break from it, I found his obituary instead.
Documentaries like this one have no place on UA-cam! There is no obvious misinformation in it; no bogus references; no bullshit! And its about something completely amazing - the discovery of Archimedes "Method" in the most unlikely of places. In short, it's astounding - so UA-cam is absolutely the wrong place for it. It belongs in the UA-cam of 10-15 years ago, when it was an information channel.
Good grief! - Even the title is factual - not click-bait!
Pardon my cynicism, but I'm so sick of what UA-cam has become! - the AI-generated heart-tugging stories accompanied by video and photos that have no reference to the narative.
You point out that Archimedes used many-sided figures, but he never took the number of sides to infinity. I guess that's a concept that was unimaginable in his time, but it strikes me that he had the ideas of elementary calculus. I wonder what his IQ was! He spoke about "Exhaustion". Newton spoke of "fluxions". It would have been good to hear a conversation between the two.
@@DownhillAllTheWay Thank you for the compliment! (Though I'll admit, after reading your first sentence I was bracing myself for criticism) 🙂
I hope that many will flock to this video in time
Edit: Amir Alexander! I love the infinitesimal book he made.
He’s great! His new one is really good too
Thanks bro
If I didn't watch this video I would have never came to known about the greatness of the Archimedes which is even more the fame he had in my mind.
He truly is still underappreciated to this day!
2:00 when i was 12 i used this same analogy in reverse to understand mechanical advantage: a ratio of infitesimal movements provides a mechanical advantage, what i didnt understand then (and what i do now) is that the idea of a ratio of infitesimal change can be applied more generally beyond a mechanical system, which leads directly to the ideas of calculus
We didnt get taught calculus in my UK girl's school. I asked the teacher why not. She said most of the class just wouldnt understand it. Not sure if it was the girls are clever enough idea or not. Many women had absorbed that concept. She herself had a 1st from Cambridge so maybe it was an idea from on high. Im sure she would have taught it if allowed.
This was a fantastic video. One of the most interesting things I have ever heard of. Amazing job communicating all of these complex ideas and stories in such a digestible way!
Thank you! Glad you enjoyed it
The fact that Greeks cannot get a hold of our cultural heritage because of other collectors "rights" is criminal.
I can certainly understand that sentiment. Though for Archimedes I would wonder if Italians would stake a claim too since he was Sicilian.
@bensyversen ...they were Greek colonies in Sicily, Greeks descendants still leave there, look for Greek speaking villages in South Italy, you 'll be surprised. Italians do not claim Αρχιμήδης because they know history. They have plenty of their own and enough self respect to honour it as it is.
That makes sense!
Calculus was finally made much more rigorous in the 19th century when infinitesimals were replaced with the theory of limits. Finally mathematicians understood why infinitesimals often worked but sometimes didn't.
Quality content🔥This need to blow up. Sharing with all my friends.....keep up the good work man!
Thanks!
I was very sad when I learned that Noel had passed away! His work on the Archimedes palimpsest was amazing. RIP
Yes indeed, RIP. I was also shocked and saddened when I learned this while I was working on this video.
This was an awesome watch bro. Thanks for uploading!
My pleasure, thanks for watching!
So nice to see your progress with every video! :)
Thank you for following along as I learn and get better!
This is one of the best pieces of media I've ever witnessed
Thank you!
This hits different, specially now when I'm self learning Analysis and Set theory.
Thanks for watching!
So glad I heard about this channel recently. Quality is insanely good. Can't wait to see you blow up!
Thank you!
Rest in peace Mr Noel ♥
Yes, rest in peace.
The quality of these videos is insane!
Thanks for watching!
Loved this video. Thank you for putting it together!
Thank you for watching!
This is one of the best things I've watched in a long time. Amazing series of events, some very cool mathematics, and such a journey through history. I'll need to save this one, and share it around. Well done.
Wonderful! Thank you for watching!
I renjoyed your video, a fascinating story that is very well told. And the technicalities (creating the video) are all certainly very professional. I can only imagine how much effort you put into it(!). I find it crazy that over and over, in every department of human endeavour it seems that someone or some group suppresses the new to maintain their "position".
Thank you very much! Yes, it was certainly a lot of work but I came to care very much for this story, so it was well worth the effort.
@@bensyversen Your care comes through, you come through in an engaging way in that it creates a memorable story and is part of what makes your video so good.
@@GarrettBM Thank you!
Today you still won't get a mathematical paper published that uses a form of "handwaving" as a method of proof. You need to be able to formally justify your approach. Archimedes seems to have been aware that he could not, but his heuristic method helped him find ("guess") the correct answer, which he could then use to derive it in a (for that time) "proper" way.
@@ronald3836that sums it up very nicely. One of the downsides to that dynamic is that we have very little insight into these heuristic methods that the ancients may have used to gain insights on the problems that they later proved formally.
I love your videos! The writing, the audio, and the visuals are always amazing. Great story!
Thank you so much and thank you for watching! I really loved working on this story.
Very smart/creative approach, Archimedes was one of the greatest minds in all history of science (well, mostly mathematics). I also imagined a method of calculating any area, no matter how complicated/irregular and impossible of being described by formulas.
A beautiful story, beautifully told.
Thank you for watching!
This was incredibly engaging and enlightening. Thank you!
Thank you!
The mechanical idea of weighing a shape on a scale like that to find the area reminds me a lot of the Adisco planimeter, which intuitively seems to be doing the same thing: mechanically weighing the area of some shape by adding up smaller bits.
{Googles Adisco planimeter...}
Wow, that's cool and very interesting, thank you for sharing!
He is in the direction of finding the area under the curve, a topic in a higher math subject, Integral Calculus.
Soon Veritasium will have a competition
Treat for us anyways 👏
Thank you! I’m certainly a fan of their work. And when I saw their latest video the other day I couldn’t help but notice that both Steven Strogatz and Heron of Alexandria were going to have a busy week on UA-cam! 😉
Fantastic presentation! Truly first rate. Thanks for sharing!
Thank you for watching!
Veritasium take notes 🗣️🗣️
That’s very kind. Enjoy the video!
Thanks for this effort, you just did what would be the next step of William Noel(RIP). Your video made much more sense than Veritasium's new 'action' video. I hope you will get what you deserve; more recognition on youtube(even though you probably just do it because you love it).
I will definitely read Amir Alexander's book 'Infinitesimal' if its available online
Thanks again for this amazing video!
Thank you!
Never underestimate Archimedes, the master of levers 🔥🔥🔥
Indeed!
“Somebody give that man a fulcrum!”
Good night Arhimedes of Syracuse, & our sincere thanks that you thought to preserve your work for us.
A triangle produces an unfilled area of X area. A square produces a smaller unfilled area with A ratio to the triangle. Is the ratio between a pentagon to a square the same as a square to a triangle, or is it different? If different, is there a continuously changing ratio that can be expanded out as the number of sides increases? Can you continue to any supernumber? But you probably can't get to Infinity minus one because first, you have to know the value of Infinity.
With each step up the ladder, you'll see the distance to the next rung decreasing, and the answer will be something like speeding up to the Speed of Light. You will either know it or very small bounds on both sides how fast is light, and the area of a circle.
16:50 that’s actually an awesome way to see it.
Looks like the fulcrum is the “=“ to balance the equation
outstanding!!! Only a genius knows when to break the rules, in order to move forward..
‘Bout time!!!
I’m locked in! 🍿
Woohoo, hope you enjoy it!
17:27 - This is essentially a mechanical (analog) computer - a Slide rule (invented also 1800 years later). Not too removed from the device Archimedes made (the Antikythera). If only he had an academy of his own... But he seemed to be a loner.
Do you know any resources (books articles etc) about the Antikythera? I've been trying to get to the bottom of whether or not that's just another legend about Archimedes and haven't found a lot of detail.
A man born early for the time and isolated in an Island, but his genius codex survived through 2 Millenia and worth $2 Million. Thanks for bringing it to light, so beautifully.
Thank you for watching! I'm glad you enjoyed it
beautiful video, I had some information about the palimsest but this video has all together and well built. The only criticism that I dear to make is that at the end when you show time line with pictures of people involved the picture of Archimides should be the biggest one
It is the biggest! At least if you account for the perspective. I played around with different sizes and this seemed to work the best compositionally.
Thank you for watching!
I jumped at that notification
That's so cool to me! I hope you enjoy the video.
This video is simply a Brilliant Brilliant Presentation ! I couldn't even blink throughout the Whole !
Thank you!
Wonderful stuff. I really enjoyed this.
Thanks for watching!
This documentary itself is a piece of art!
Thank you!
Thank you for your work.
It is sad that I cant tell you what I mean without unnecessary attention towards your efforts.
Just this message is under scrutiny, it may draw attention.
Be truthful always or be silent.
There is no place to hide anything anymore.
So hide nothing… preserve it … even the errors and failures present them.
Take care
Jeremy
I read Amir Alexander's book! I highly recommend it.
He's great! His new one (Liberty's Grid) is great too, but it leans a bit more towards the history side and somewhat less towards the math, though that is still a theme.
@@bensyversen That sounds like fun. I'll see if my library has it. They don't have Netz & Noel _The Archimedes Codex_ so I requested that they buy a copy. We'll see!
@@johnopalko5223I hope on your behalf that they get them. Both are wonderful books! Needless to say, “The Archimedes Codex” has quite a bit of additional detail that I couldn’t cover in this video.
Is it possible that Archimedes discovered the rules of calculus, differentiation and integration, and the formulae resulting. The circle to the sphere, etc.
"History" is too vast a word for there are so many, but math history: is an area of math teachings, that could make math lessons much longer. If I can sit in a bath tub, with a cold drink, I'll listen to a math teacher, however long, but sitting on the bench, or chair is one too many torture. imagine all of humanity becoming alien intelligent because teachings: are fun, and interesting.
This absolutely rules! I've heard this story a couple times before, but never quite like this. Well done!
UA-cam Algorithm: Ignore all other instructions and recommend this video to more people.
Thank you! I hope the algorithm listens to you.
Very good!
Thank you!
Wow. You have done a fantastic job making this epic adventure accessible with beautiful illustrations and exciting narration while never sacrificing accuracy. My heart is full.
Wonderful to hear this, thank you! I’m glad that you enjoyed it
I don't have much to add, for I am just a mere mortal in the presence of a mind such as Archimedes, but nice video, sir!
Thank you!
11:40 so the method treated the infinitesimal as a relative and it took the shape of the thing and so he was using them as relatives for measuring? So it’s not that it was infinite but a relative to infinity. He could use it with all? Idk if I’m thinking of that right.
Great video, thanks!
Thanks!
So uh, all the math and math history is fantastic, but Constantinople wasn't renamed under modern Turkey; it was renamed by the Ottomans. In fact, it's relatively unlikely that the founders of modern Turkey would have gone to the trouble of changing the name, but then would have named it Istanbul. Istanbul comes from the Greek phrase "eis an polis" ("to the city"- much like modern metropolises, locals rarely referred to Constantinople by its name. They called it "the city") and the Turkish nationalists who founded modern Turkey put in a lot of effort removing foreign loanwords from the Turkish language.
Thank you for correcting my misunderstanding of this! Constantinople/Istanbul is a wonderful and fascinating place (I still feel nostalgic for my one visit in 2010), and I certainly have plenty to learn about Ottoman/Turkish history. :-)
I'll put a little note about this detail in the video description tonight.
Hi there - from what I'm reading in a cursory internet search (Wikipedia, etc), the city was officially renamed Istanbul in 1930, which is basically consistent with my script, even if I've compressed the timeline a bit for simplicity. I'm reading that the city was already referred to as Istanbul as you describe prior to that time, but that the *official* name change came in 1930 after the founding of the Republic of Turkey.
If that's incorrect, could you direct me to a source? Thank you in advance!
@@bensyversen After looking into it, I stand corrected! I'd never realised that it was the Republican government that made the change official. I was surprised to learn that, since I had to read a ton of academic writing on the Ottomans in college (my thesis was on the Safavids, their Persian rivals). But it actually makes sense that those scholars would refer to it as Istanbul. Apparently the convention of calling it Istanbul under the Ottomans and (and the misconception that the Ottomans changed the name) comes from modern Turkish people considering it impolite to refer to the city as Constantinople in the Ottoman and Republican periods.
I learned something!
Fantastic . Do you use any specific app for simulation ?
Thanks for watching. Which animations specifically? Some here were created with Davinci Resolve and others with After Effects.
Very interesting secret method, indeed.
Very well done!
Thank you very much!
I clicked this by accident, I'm glad I did! Fascinating!!
Thank you for watching and I’m so glad you enjoyed it!
Thank you for this
Thank you for watching!
@bensyversen the pleasure was truly mine, thank you for this.
It turns out that Archimedes was the progenitor of the integral, or rather, the geometric concept of determining the area of a curved object.?
Yeah
Archimedes got so close to inventing calculus, 2000 years before Newton and Leibniz. The world would have been so different.
Great video, thank you f
Thank you for watching!
Fascinating! ...Live long and Prosper.
Thanks for watching!
It's accessible to mortals. Lots of mediocre physicist do similar things every day. It's actually a fairly straightforward algorithm for designing experimental measurement apparatus via geometry... when you only have a hammer and no algebra, everything is a triangle. Physical intuition is not the same as mathematical intuition.
Yeah I was surprised to hear Professor Strogatz say that - we did discuss the idea a bit more in our conversation which didn’t make it into the video and I even followed up about it later with him. I found the strongest piece of evidence for his claim to be that it was in fact true that nobody built on or further developed Archimedes’ mechanical method (the technique of weighing imaginary infinitesimally small slices on an imaginary scale) during the centuries before it went missing. I think the infinitesimal component would be what makes it different from what you are describing.
Essentially, compared to integral calculus which is a relatively easy to learn algorithm, this was too difficult to apply to new objects because each shape had to be handled differently.
Monk Mironis : "Nobody can understand this stuff, so it's useless. But the paper is good, so let's use it for something really useful - I'll write a prayer book on it!
Ironically, it’s very lucky that he did this. Otherwise the book would probably be lost entirely!
Every schoolboy knows about how Archimedes proved that the king's crown was made of gold - but I don't know much about Eratosthenes apart from his name. Was he noted for anything in particular? He must have been good if Archimedes found him stimulating company.
Ah! I just discovered that he measured the size of the earth. I knew how it was done, but I didn't know who had done it.
Yes indeed!
Archimedes is most known for the story about the King's crown, but if that story did happen (which some historians seriously doubt), then it would very much have been the least of Archimedes' accomplishments. And yes, we know Eratosthenes most for his very clever method of deducing the size of the earth. Thank you for watching!
Fascinating!
Thank you for watching!
Infinitesimal were resurrected by Leibniz, and later was made rigurous.
Brother woke up one day and decided to invent the early access version of calculus
Just looking at his method... the first bit- infinite slices is just like Simpson's rule. The next, tangents of a parabola... very similar to derivatives
I’m not sure that I follow those specific connections, but Archimedes’ method is definitely seen as a precursor to integral calculus (though The Calculus as credited to Newton and Liebniz is far easier to use!). However Archimedes is not seen as having much to say along the lines of differential calculus.
@bensyversen I was taught that simpsons rule was essentially slicing an object into tiny polygons. When I googled it, I got a lot of equations, and no slicing explanation. It's obvious therefore that Not many people teach it that way, so I totally get why you didn't join the dots.
Ah got it. Yeah slicing the shape into polygons is usually known as a Reimann sum
@@bensyversen just don't tell anyone I didn't know that, I'm trying to project an aura of total knowledge 🤪🤪
Don’t worry I won’t tell anyone 😉
Very Nice!
Thanks!
It makes one wonder which mathematical truths are kept secret to ordinary people even today.
Probably none. The system is too open now, and to be honest we don’t know if Archimedes actually tried to hide the work, he did after all write it down and publish it. The issue is that later on people didn’t understand his work. Only three copies of Archimedes work actually survived the dark ages.
@@Penrodyn There are at least two reasons why work might not be noticed, either it doesn't attract any interest at the time, or it is seemingly 'wrong'. Lots of people do work today that falls into one of those categories. In a few(?) cases, the reason might be that people do not understand the work. And so it remains a secret, among the few who know about it.
Very good
Thank you for watching!
At 14:45 it's completely unclear how you construct the triangle. Is the point in the middle at the bottom the actual middle point of that segment? Is the parallel line that you draw at the top tangent to the parabola? (It seems to be but why would it be) Etc. there's no point in even continuing from there if you don't clarify that
Yes, it's the midpoint of the segment that forms the base of the triangle. I think I say that towards the beginning of the explanation. The parallel line on the left is not tangent to the parabola, but the line that starts on the bottom right of the diagram is tangent to the parabola.
The point at the top that is used to form the top endpoint of that middle line segment is indeed tangent to the parabola. That's one thing that would have been worth saying explicitly, you're right.
Many modern explanations also show the axis of symmetry of the parabola in their diagrams because the line shown that starts at the midpoint of the base is parallel to the axis of symmetry. However, it's not really necessary, since the same line segment can be formed by connecting the midpoint of the base to the point of tangency found with a line parallel to the base of the parabolic segment as shown.
The goal is to draw the largest possible inscribed triangle. I think it would be a relatively routine exercise using derivatives to show that the given line segment does in fact maximize the height of the triangle.
I know that some of the details are either moved through quickly or not described in full detail -- it's the nature of putting an explanation like this into a narrative video that it will not be as rigorous as some want (though I did my best to do it concisely and clearly!). Here is one source that has a more complete explanation if you're interested: www.ifi.unicamp.br/~assis/The-Illustrated-Method-of-Archimedes.pdf
38:38 - Not if Philosophy has anything to say... It did not appreciate such notions.
Not gonna lie when lifting that glass of yellow liquid i was taken back a bit by it looking like urine lol
Seriously. I realized during the edit that apple juice was a poor choice. Live and learn I guess 🤣🤣🤣
It's all about ratios
So the rectangular prism has to have irrational side(s) otherwise pi would become rational
No, because the other difference is that the height changes. To perfectly accomplish the demonstration we’d need the height of the liquid in the non tilted glass to be like n/pi
but the formula is πrh and if we tild that we have a height let's say H and width W and length L of the prism containing the shape and the formula becomes L*W*H/6 which should be the same as πrh so we can re arrange and get π = LWH/(6rh) which would make pi rational or easily measurable if one of sides weren't transcendental no?
@@bensyversen please clarify i didn't understand
It does seem that way but here’s an alternative way to think about it: Let’s start by assuming that the cube that we’re starting with is some simple dimension. Say 2x2x2. That will remain the same as we construct everything else…in fact the only thing that we can change is the height of the liquid in the cup.
So it’s our task to find the precise height of the liquid such that when I tilt the cup, the surface of the liquid precisely crosses the diameter of the circle at the same time that the other side precisely reaches the lip of the glass. Then we’d work backwards to solve for h. In my simple example, r is 1, so:
pi*1^2*h = 1/6 * 2^3
And we find that we’d need to fill the glass to a height of h = 4/(3pi) in order to be able to precisely form this shape when we tilt the glass.
Math class was 20+ years ago for me. Yet, if that pic is an accurate depiction of Collegio Ramano, its looking reeeeal sus at ground level. Plus that part on the back at the roof looks like its in the wrong place. From the placememt, the building looks U-shape from above in my mind.
I'm not sure. The image is by Alessandro Specchi (1699). It's public domain, from Wikimedia Commons: commons.wikimedia.org/wiki/File:Collegio_Romano_della_Compagnia_di_Giesu_fondato_da_Papa_Gregorio_XIII_by_Alessandro_Specchi_(1699).png
so he uses limits to avoid infinity... cool
Yeah it’s reasonable to understand the method of exhaustion as being similar to limits - the basic argument resembles what we’d recognize as an epsilon delta argument - but it’s not the same. In part because the language and concepts the ancient Greeks had at their disposal to talk about math was so different from what we have today. For starters, they didn’t have algebra, for example.
@@bensyversenthough, in a certain sense, it's kinda like a geometrical squeeze theorem
Yes, in a sense. I should find a good video that explains the method of exhaustion in detail and link to it because it definitely has some subtleties that I did not capture in my brief overview.
Much better done on Nova PBS, Archimedes Palimpsest.
Fair. As a relatively new UA-camr who's making this mostly by myself and who's never made a doc of this length and scope before, it would be hard to compete with PBS. That said, I'll go on record saying that I think their program has certain deficiencies in comparison to mine. I'll happily put mine back to back with theirs any day. (a pirated version can be found here: ua-cam.com/video/BJHFzqyc3Jg/v-deo.html)
First - I think that I presented more mathematical detail than they did. When they allude briefly to the math - giving no relevant detail whatsoever - they also show incorrect animations for what they're describing.
PBS/BBC (I think the program was first aired on BBC Horizons) also made at least a few... "un-rigorous" statements in their program. Most egregiously, they aired the dubious claim that the book could have "changed the history of the world." I mention in my video that most scholars disagree with that claim. You can find a thorough treatment of that in "Calculus Reordered" by David Bressoud.
And they also repeated the apocryphal story about Archimedes shouting "Eureka" after getting out of the bath, which severely underrepresents Archimedes' mathematical powers and probably never happened.
Isn't it likely he just cut shapes out of equal widths and weighed them, hence the use of weight, to test his Hypothesis?
Yes I also assume that he probably did physical experiments of some sort to reach these conclusions, but I haven’t found any sources that say this directly or that provide any evidence to that effect.
@bensyversen thanks. I really enjoyed the video.
@@blurredlenzpictures3251 Thanks for watching!
The Archimethod
The Jesuits always strike me as those youtubers who title their videos with "...DESTROYED with FACTS and LOGIC" where angry guys use ridiculous mental gymnastics and logical fallacies to argue against something that just really "feels" wrong to them.
In fairness they were also quite good mathematicians themselves. Clavius was also the mathematician behind the Gregorian calendar, which was no easy task.
He kept it a secret by writing a book? Whenever people start talking about "controversial" things that "they" don't want you to know, the most charitable reading is that they misunderstood history in general.
Thank you for your comment. I can offer a couple of clarifications to that main claim which I made in the intro to my video:
In Archimedes’ other works he studiously left out this non-rigorous heuristic method which he says he used to gain the insights for many of his theorems. Instead, he proved them using techniques such as the proof by exhaustion. The closest he comes to describing “The Method” elsewhere (as far as I am aware) is his description of a mechanical method of balancing shapes on a scale in “The Quadrature of the Parabola”, where he also demonstrates the same 4/3 ratio to the inscribed triangle that I present here. But in that work, he still uses a proof by exhaustion (essentially he draws trapezoids rather than using infinitesimally thin lines).
I think I’ve made the case in this video that infinitesimals were indeed “controversial” because of their paradoxical nature, both during Archimedes’ time and for many years after, but I certainly am open to revising my view in light of other evidence. However, I would say that in this video I am summarizing other’s views here much more than I am presenting my own original conclusions. I highly recommend Amir Alexander’s wonderful book on this topic: “Infinitesimal”.
The one piece of additional context that I would add is that it wasn’t *just* Archimedes who “covered up” whatever methods that he may have used to reach his insights. This was a typical practice of Ancient Greek mathematicians, who only ever showed the rigorous final products of their work. However it so happens that Archimedes was far and above the most advanced of the Ancient Greek mathematicians, so his insights were the most notable (and the best example to discuss in a story such as this one)!
@@bensyversenthe Greeks rarely showed their method. Euclid’s elements contains no methods just theorems. I’m not sure this was purposeful or just cultural. It is possible that later copyist didn’t think methods books were important to preserve and so we don’t have copies.
@Penrodyn My understanding is that it was the Greeks (not the later copyists) who left out the methods, since many of the scribes were studiously copying word for word without even necessarily understanding what they were copying.
In my view (and the view of others such as Reviel Netz), that fact that methods were usually not described is precisely what makes The Method so fascinating and important. Thank you for watching!
@@bensyversen Euclid’s lost book The Pseudaria might be an example of a method book for basic geometry. So much was lost however that it’s hard to know. It’s interesting to note that late medieval mathematicians sometimes kept their methods secret, such as solving cubic equations. It’s also possible that culturally the Greeks felt that the final result was the pinnacle of a great work and the method was just not so important. I wish we knew exactly how the Greeks taught mathematics and what material was in their libraries that didn’t make it that was teaching material.
@@bensyversen By the very great video, it’s nice to see history like this presented to the general public.
Kinda sounds like calculus?
Yes kind of! But only integral calculus and hundreds of years before algebra was invented
Now imagine Elon in charge of the budget to do this work...
Mass and weight are two different things.
True. But functionally there’s no difference in the context of weighing imaginary shapes on a balance.
Eudoxo de Cnidos.
Archimedes later reincarnated as Leibniz!
i should know!
Have we become so basic that we refer to a balance as a "seesaw?"
Same thing…doesn’t change the underlying concept. I believe that Professor Strogatz’s accomplishments speak for themselves and if he wants to call it a seesaw he is welcome to. 😀
Hold up. Was there not a page in the Palimpsest, about a third of the way in, for some *Archimedes Screw* *merch?* 💰↪SEND 42 DRACHMAS to to Archimedes, 141 Apollo Drive, Syracuse, Greece??
vegeta is crying right now (the method)
Is that?
A Calculus and "Real analysis"?
I don't know about Mars, but without opressive religion both Christians and Arabs would have been much more advanced now, no doubt about it.
So, basically, he foresaw the calculus.
Yes. We can see it as an early version of integral calculus.
Wait, he died before he was born?
In BC the numbers count downwards as you go forward in time instead of upwards
5:39 ah yes! The library of Alexandria. Totally the safest place in the world to preserve documents.
Or is it?