This video series will end after 2 more videos, and these 2 videos will be released simultaneously. There will also be a very brief summary at the end of the last video. Stay tuned and subscribe to Mathemaniac for more videos!
Right now I cannot think of any other ideas (other than the 2 videos promised) that fall into the umbrella of the video series, so I will extend it only if I can think of other methods to do integrals without integration. I will still continue producing videos in the meantime, probably opening up a new video series! (Just probably...)
So, the method of exhaustion is based on the Archimedean Property which correctly stated says: Given any magnitude x (whether commensurable or incommensurable with any other magnitude), there are commensurable magnitudes (or rational numbers) m and n, such that m < x < n. Archimedes did not recognise those imaginary objects called "real numbers". The only numbers he knew about were the rational numbers. Any magnitude that could not be measured was called alogos in Greek, meaning irrational, but not "irrational number"! To the Ancient Greeks: A number is a name given to a measure that describes a magnitude or size. I could write much more, but why waste my time - especially when my comment might not even be published...
Me too! While a lot of things can be dealt with much more efficiently with algebra and calculus, Ancient Greeks did prove to us that if we think hard enough, they can be, surprisingly, dealt with by geometry only!
Ancient Indian texts are much knowledgeable and richer than anything in the world..... But sad part is that due to ill effects of british rule we still follow a crap education system..... And foreigners gain knowledge , steal ideas from our texts and patent it on their name
@@maxwellsequation4887 To be fair It was after a worldwide civilisation reset 1500 years ago (in real history, Dr Anatoly Fomenko, Sir Isaac Newton, (both mathematicians!), etc). Which one of the equations, not 1 of the 4 Maxwell Heaviside equations?.
@@warrior3357 The knowledge in the texts are preserved relics of a previous worldwide civilisation, it belongs to all of humanity. Due to British rule Britain also has a crap education system.
When Newton went to enroll in Trinity College, one of this tutors Isaac Barrow tested his knowledge and was disappointed by his lack of skills in proof writing. He recommended Euclid's Elements and so Newton took his word and studied the volumes in 2 years. He not only finished those books but he studied other books including "Conics" by Apollonius that was under the rugs at the time. By the time he finished college, he was a mathematical giant. None of his peers had the tools to treat Curves the way he did. He wrote Principia in the language of geometry to respect his predecessors.
The name "Principia" was not random. Newton name it after Euclid's Elements or "Principles" in Latin. Many important thinkers wrote their own versions of Elements in their field like Elements of Philosophy by Thomas Hobbes , Spinoza Metaphysics, Whitehead and Bertrand Russel Elements of Mathematics
Thank you for the perfect explanation. As a viewer I would recommend you including the keyword "quadrature" into the description, cause I was looking for it and stumbled upon your video, and watched it out of curiosity although it did not say anything about quadrature in the title.
Thanks for the suggestion! I just thought nobody is going to search for quadrature and so didn't add it in the description. I am going to add it in the description now.
Cool video! I‘d really like to hear more about different ways mathematicians were thinking about modern mathematics back in the day! One little thing: I think the music is slightly too loud/too distracting :)
How wonderful! The Algorithm suggested this after I commented that if Archie lived longer, he would have developed another flavor of Calculus. (I'm commenting before actually seeing the video; looking forward to do so now)
Looks like Ancient Greece has lost its Glory.... No one could ve surpassed Greece in mathematics during the era of Euclid and Archimedes *Field medal(Nobel prize of mathamatics) has a depiction of Archimedes but ironically no Greek citizen ever won this prize*
In the words of Arthur C. Clarke "If Christianity and Turkish occupation hadn't stopped the Greek civilization from its advanced progress in mathematics , chemistry and physics , the Greeks could have been to space 600 years before the Americans". A little perspective and history knowledge might answer your questions...And greece never was 1.5 billion people like india...
@@Lalakis Keep in mind that That population of India at the time was 30 million (Agrarian civilization in India had disappeared during bronze age collapse i.e Indus valley civilization had collapsed some time ago due to climate change and rivers drying and the nomadic less advanced Indo-Aryan tribes had just now began to settle and urbanize) and population of ancient Greece is 20 million ( including Greek settlements in Anatolia, the Greek archipelagos were so overpopulated that greeks would go settle as far as western India ) at the time. More importantly Indo-Aryans had caste system where institutions of Knowledge, teaching and learning was exclusively reserved for brahmins top 5%-10% of society others where prohibited to do the same occupation as them. Rest of the population were supposed to be warriors, merchants, peasants, they were not encouraged to think for themselves or philosophize and where dependent on the liturgical class which was more interested in theology . Whereas Greek society encouraged even soldiers to be scholars and also their scholars to be soldiers. "The society that separates its scholars from its warriors will have its thinking done by cowards and its fighting by fools" - Thucydides
@@clovebeans713 ancient Indian society was not based on cast. Everyone used to read and educate themselves in ancient India. Some Upanishads(ancient hindu texts) are composed by the sons of dasas (servant)
Isn't calculus a development over time and not just one person and also what about infinite series for pi, sine, cosine, greeks were still using chords and the chord tables came after archimedes
The logic of this video contains a loop: derivation of the formula x = at^2/2 itself requires integration. I suppose the real Archimedes calculated the areas geometrically. P.S. Sorry for the nitpick, I've finished the video and saw your remark on simplification. Still, the loophole is there.
no you know what the real nitpick here is? The fact that the kinematic equations that were used in this example could not have been used by Archimedes because they were invented by Newton in the 17th century.
Not necessarily, S=ut+1/2at^2 can easily be derived even without calculus. We had the derivation of the equations of motion back in 9th grade. One can also look at the area under the v-t curve (It's a trapezium).
@@daigakunobaku273 But when one finds the area of the trapezium, they're just doing what archimedes does in this video, that is, it was somewhat possible for him to do this in his time (though he didn't and Newton probably came up with it later on, that guy does everything).
@@thelosttomato4020 no, they are not. The problem is not finding the area of the trapezium itself, but realizing the connection of it with the distance travelled by an accelerated body. Even the physical concept of acceleration, as far as I know, didn't exist in Archimede's times, let alone the Cavalieri's principle and the fundamental theorem of calculus. Besides, at the time a belief was widely held (at least among Greek philosophers, heavily influenced by Aristotle) that a slung body first ascends diagonally, until the "force" that acts on it is not "exhausted", and then likewise descends. Also, Archimedes himself filled the area under the parabola with clverly chosen triangles of decreasing sizes, and this approach was problem-specific.
Most likely the ancient Greeks did in fact discover calculus, but the documents didn't survive to the modern time, since the libraries of Alexandria, Constantinople, and others didn't survive to the present time. So many inventions of the ancient world have been lost to history including Greek fire, the Pharos lighthouse, ancient Egyptian and Roman concrete, Aqueducts and plumbing, Antikythera mechanism, Saqqara planes, Lycurgus cup, Archimedes heat ray, Aeolipile, automatically opening doors etc. Eratosthenes calculated the distance from the Earth to the Sun. The Romans had almost modern level chemistry, mass production of food additives such as Lead(II) Acetate and Sulfur Dioxide. I think that Leonardo Da Vinci and other Renaissance men didn't really invent anything, they just copied ancient Greek and Roman manuscripts. So yeah, I wouldn't be surprised if ancient Greeks actually had working knowledge of calculus. We just don't have the books, or they are locked up within the Vatican's archives. I think that if you go back in time and take an ancient Greek scientist with you, they would be able to understand 20th century science within a year or two of concentrated study.
Again you don't have to believe in infinity. You can just take it as the definition of limit. But ultimately it is meant to be a way to calculate the sum of infinite terms.
This is a good video, but I think the most interesting thing I saw was in the description. You said he might be credited with calculus "if ancient Greeks were not so afraid of the concept of infinitesimals". I would LOVE to hear more about the reaction to his work from mathematicians of his era
I made two videos on Fibonacci sequence already! One is about how combinatorics is related to Fibonacci numbers, and the other is about the different ways of computing the general term of Fibonacci numbers.
So for the largest red square, to its top and right there are two identical yellow squares of the same size as the red one. Then for the second largest red square, you also have two yellow squares to its top and right, and so on. So there are three copies of the same set of red squares in the entire figure, hence red squares occupy a third of its area.
No... because x^2 represents a parabola, and we have used some physics for this argument as parabolas are just trajectories of objects under gravity. However, there are other ways to derive the integral of x^n without the power rule; see my "Power rule of integrals not using integration; definition of e | DIw/oI #7" for that method.
Even if he did not, contemplated an infinite series, he have still eluded to calculus, almost like HERO making his steam apparatus have fail utilize the steam power, but have also eluted to the potentials of steam energy. Pythagoras invented the square of two sides yet did not invented the sq. Root of 2. Just remarkable.
...Exodus 26:13 ≈ Pi... Moses was first to record Pi in 1440 BC. Josephus the Historian's description of the Tabernacle in 94 AD was inaccurate. This oversight was discovered in 2015 AD. 330 Exodus 26:8 eleven curtains each 30 cubits long 15 Exodus 26:12 one of the curtains is folded in half to 15 cubits long - 1 Exodus 26:13 makes Pi each curtain overlap/seam adds up to 1 cubit = 314 3.14 = 314 circumference/100 diameter ≈ π ratio (100 cubit court per Exodus 27:9-18) ................. This Wilderness Tabernacle archeological discovery is similar to the Dead Sea Scrolls and Martin Luther's 95 Theses. ................. History of finding π: -(1900-1680 BC) One Babylonian tablet indicates a value of 3.125 for π -(1650 BC) The Rhind Papyrus the Egyptians gave the approximate value of π 3.1605 -(1440 BC) Moses recorded Pi in the Exodus blueprints rediscovered in (2015 AD) 3.141592653... or Exodus 26:13 ≈ Pi -(500 BC) India's Aryabhata approximation was 62,832/20,000, or 3.141 -(429-501 BC) Zu Chongzhi a Chinese mathematician 3.1415926 - 3.1415927 -(250 BC) Archimedes from Syracuse showed between 3.1408 and 3.1429 ................. The knowledge of Pi was lost from Exodus near 1000 BC. Josephus the Historian in 94 AD did NOT know about Exodus 26:13 makes Pi to properly explain the Tabernacle blueprints. He deferred to the Temple's structure and not Exodus 25-26-27. Pi is found in the spiral of the double helix in your DNA. This knowledge has been lost for 3000 years. Consider King Josiah & the Prophetess Huldah rediscovering the scriptures, right? In short, this monumental oversight was corrected in 2015. What's next? ...Exodus 26:13 ≈ π...
Archimedes didn't use the method that I described obviously, and the method he used was way more geometric - but this is a way simpler and more engaging way, like I have said in the end.
@@mathemaniac Yeah the method is cool and all but the physics approach is dependent on calculus which wasn't discovered back then so it would have been better if you used other method instead.
The physical intuition failed me completely, I had to stop the video and pull up the algebraic formulas to prove using derivatives, which alas is calculus again. Though since vectors were invented way later than calculus, and is way less intuitive to me, I always find people who use that as the go to explanation perplexing.
Perhaps it is because different people have different ways of understanding mathematical concepts, and for a lot of people, graphical / physical concepts are more intuitive, which is probably why a lot of people including me use those concepts as the go-to.
@@mathemaniac It was never explained though why or how that analogy to gravity would be justifable. Why would vectors be related to that line segment at all? It produced the results we wanted, but it seemed arbitrarily chosen and not a consequence of algebraic graphs. I'm missing the connecting link between them
@@anonymouscoward2491 Archimedes used a tedious geometric method to do something that the average high schooler could do in half a minute using calculus. The geometric methods would be very complicated for other curves, if it is possible at all. Why would you still think calculus is inferior?
I think I was in middle school when I wanted to calculate the area of a circle without using pi. I think I accidentally stumbled upon the method of exhaustion (More luck than smarts lol) and used something like that. I only learned about the method later in life though.
No, but I don't want to reveal the video making method just yet because I fear people will judge me - it's far from professional. Will make a video about how I made these videos sometime in the future.
He didn't discover calculus actually... It's like to say that James Clark Maxwell discovered E=mc² Same work was done by Chinese mathematicians liu hui.. Indian mathematician aryabhatt used infinitesmall to study the rate of change . he did it to study the orbit of moon. Much of the key notions ideas of differential calculus was found in the book of bhaskarcharya II.. He introduced differentiations and derivatives. He was perhaps the first to conceive the differential coefficient and differential calculus... 15th century mathematiciam madhava of south east discovered power series infinite series and expansion of trigonometric functions. Power series forms the basis of complex analysis.... Many pples see madhava as a founder of mathematical analysis
You never mentioned why he or somebody else did not finished his work. It was because of Judeo-Christian destruction of ancient knowledge. Shortly, after Judeo-Christianity gained political power in Rome, they started a persecution of other religions and thinkers. Archimedes works were mostly destroyed by them. The surviving papyrus where he almost discovers calculus has a script on Greek or Hebrew, a prayer to protect against "demons"...
One of the big steps they needed was the concept of the derivative. It's hard to accept the claim that Archimedes was close to developing calculus if he didn't have a concept of derivatives. No derivatives. No fundamental theorem of calculus. No rules of integration like the power rule or substitution.
The concept of calculus does not only involve derivatives and integrals; it's fundamentally about the study of infinitesimals. Archimedes developed the concept of limits already, and it is just one step away from infinitesimals. In fact, Archimedes also was a pioneer of the idea of differential calculus. He found the tangent to a curve other than the circle as well (of course not using the modern methods - you can search for it if you wish). What I am trying to say is that, even if Archimedes method does not look anything like modern calculus, it still captures a very quintessential part of it. The calculus methods now might be like the Archimedes' ones if history took a different path.
@@mathemaniac if archimedes wasn't killed prematurely by the romans, he might have had changed history propelling advancement of civlization a millenium earlier. Think about that for a moment.
@@quelorepario In the words of arthur c.clarke "If Christianity and Turkish occupation hadn't stopped the Greek civilization from its advanced progress in mathematics , chemistry and physics , the Greeks could have been to space 600 years before the Americans".
@@mathemaniac he could have formulated various principles of integrations using derivatives. We doesn't use exhaustion method to compute the area of a curved figure anymore Exhaustion method was infact still a famous method known to many peoples before Newton and leibniz. *It's precursor to calculus same as electromagnetism is precursor to theory of relativity.* We cannot say Maxwell discovered e=mc2
Greek pure geometry rejected neusis because it leads via infinitesimals to the empirical absurdity of Zeno paradoxes. Greeks were right, look at what attempt to make infinitesimal calculus consistent part of pure math lead to: post-truth post-modern language games of Formalism posing as foundational mathematics. No thanks, Analysts and your coordinate system neusis. To maintain sanity I'll stay with Greeks, Berkeley and Intuitionism.
I am sorry for hurting your feelings. But he could have discovered calculus if he spent more time stealing and understanding ancient Indian mathematician..
Lol came 9 yr old kid claiming that everything was stolen from india there was no evidence on any concept of calculus until 14th century in Kerala but they invented power series of trigonometry not integration and differentiation
@@mauryanempire7503 well I am not claiming that I think you probably know something called real analysis which is the base of calculus... If you study that you will understand that how much important power series and their inventions in Kerala school is ... And also I don't need to give you any source if you are interested you will search for the truth .. I am not here to change your belief system... Thank you
@@Pavan_Gaonkar_abcwe known that Kerala school invented power series but it was during 14th century and it is considered as a precursor to calculus.The real calculus which is integration and differentiation is invented by Europe by Leibniz and newton and secondly Madhava power series expansion is only applicable to trigonometry functions like sine, cosine and arctant and secondly newton discovered these series after his publication of his work on calculus
@@Pavan_Gaonkar_abcEven method of exhaustion which is the precursor to integration is also a base of calculus as it talks about finding areas of objects.
This video series will end after 2 more videos, and these 2 videos will be released simultaneously. There will also be a very brief summary at the end of the last video. Stay tuned and subscribe to Mathemaniac for more videos!
That's a shame, I was looking forward to a lot more videos in this series. Will you consider extending it someday?
Right now I cannot think of any other ideas (other than the 2 videos promised) that fall into the umbrella of the video series, so I will extend it only if I can think of other methods to do integrals without integration.
I will still continue producing videos in the meantime, probably opening up a new video series! (Just probably...)
So, the method of exhaustion is based on the Archimedean Property which correctly stated says:
Given any magnitude x (whether commensurable or incommensurable with any other magnitude), there are commensurable magnitudes (or rational numbers) m and n, such that m < x < n.
Archimedes did not recognise those imaginary objects called "real numbers". The only numbers he knew about were the rational numbers. Any magnitude that could not be measured was called alogos in Greek, meaning irrational, but not "irrational number"!
To the Ancient Greeks:
A number is a name given to a measure that describes a magnitude or size.
I could write much more, but why waste my time - especially when my comment might not even be published...
I'm in awe of how much the ancient greeks did with geometry only
Me too! While a lot of things can be dealt with much more efficiently with algebra and calculus, Ancient Greeks did prove to us that if we think hard enough, they can be, surprisingly, dealt with by geometry only!
They were absolute geniuses, in a time when humans were hunting for food.
Ancient Indian texts are much knowledgeable and richer than anything in the world.....
But sad part is that due to ill effects of british rule we still follow a crap education system.....
And foreigners gain knowledge , steal ideas from our texts and patent it on their name
@@maxwellsequation4887 To be fair It was after a worldwide civilisation reset 1500 years ago (in real history, Dr Anatoly Fomenko, Sir Isaac Newton, (both mathematicians!), etc).
Which one of the equations, not 1 of the 4 Maxwell Heaviside equations?.
@@warrior3357
The knowledge in the texts are preserved relics of a previous worldwide civilisation, it belongs to all of humanity.
Due to British rule Britain also has a crap education system.
When Newton went to enroll in Trinity College, one of this tutors Isaac Barrow tested his knowledge and was disappointed by his lack of skills in proof writing. He recommended Euclid's Elements and so Newton took his word and studied the volumes in 2 years. He not only finished those books but he studied other books including "Conics" by Apollonius that was under the rugs at the time. By the time he finished college, he was a mathematical giant. None of his peers had the tools to treat Curves the way he did. He wrote Principia in the language of geometry to respect his predecessors.
The name "Principia" was not random. Newton name it after Euclid's Elements or "Principles" in Latin. Many important thinkers wrote their own versions of Elements in their field like Elements of Philosophy by Thomas Hobbes , Spinoza Metaphysics, Whitehead and Bertrand Russel Elements of Mathematics
great video ..... Archimedes is one of the greatest genius in all history
Couldn't agree more!
I wanted to learn some history and got a full math lesson. I loved it, it was beyond me but very fun learning the concept. Great video!
The more I study, more I am astonished by how close classical antiquity was from an Industrial Revolution.
I've just discovered your channel right now, and man ... you are AWESOME! gonna marathonate all of it lol!
Cheers from Brazil
Thank you for the perfect explanation. As a viewer I would recommend you including the keyword "quadrature" into the description, cause I was looking for it and stumbled upon your video, and watched it out of curiosity although it did not say anything about quadrature in the title.
Thanks for the suggestion! I just thought nobody is going to search for quadrature and so didn't add it in the description. I am going to add it in the description now.
Cool video! I‘d really like to hear more about different ways mathematicians were thinking about modern mathematics back in the day!
One little thing: I think the music is slightly too loud/too distracting :)
How wonderful! The Algorithm suggested this after I commented that if Archie lived longer, he would have developed another flavor of Calculus.
(I'm commenting before actually seeing the video; looking forward to do so now)
Explained beautifully. Good deployment of Euclidean geometry.
The video was pretty good and pretty well explained but the background music doesnt let me inmerse in the video
I subscribed before I watched this video, I can tell you make really good videos
Aw thanks!
Man, this channel is underrated!
incredibly good video.
Thanks so much!
Looks like Ancient Greece has lost its Glory....
No one could ve surpassed Greece in mathematics during the era of Euclid and Archimedes
*Field medal(Nobel prize of mathamatics) has a depiction of Archimedes but ironically no Greek citizen ever won this prize*
In the words of Arthur C. Clarke
"If Christianity and Turkish occupation hadn't stopped the Greek
civilization from its
advanced progress in mathematics , chemistry and physics , the Greeks could
have been
to space 600 years before the Americans".
A little perspective and history knowledge might answer your questions...And greece never was 1.5 billion people like india...
@@Lalakis Maybe even more than 600 years.
@@Lalakis Keep in mind that That population of India at the time was 30 million (Agrarian civilization in India had disappeared during bronze age collapse i.e Indus valley civilization had collapsed some time ago due to climate change and rivers drying and the nomadic less advanced Indo-Aryan tribes had just now began to settle and urbanize) and population of ancient Greece is 20 million ( including Greek settlements in Anatolia, the Greek archipelagos were so overpopulated that greeks would go settle as far as western India ) at the time. More importantly Indo-Aryans had caste system where institutions of Knowledge, teaching and learning was exclusively reserved for brahmins top 5%-10% of society others where prohibited to do the same occupation as them. Rest of the population were supposed to be warriors, merchants, peasants, they were not encouraged to think for themselves or philosophize and where dependent on the liturgical class which was more interested in theology . Whereas Greek society encouraged even soldiers to be scholars and also their scholars to be soldiers.
"The society that separates its scholars from its warriors will have its thinking done by cowards and its fighting by fools"
- Thucydides
@@clovebeans713 ancient Indian society was not based on cast.
Everyone used to read and educate themselves in ancient India.
Some Upanishads(ancient hindu texts) are composed by the sons of dasas (servant)
You are a great teacher
Isn't calculus a development over time and not just one person and also what about infinite series for pi, sine, cosine, greeks were still using chords and the chord tables came after archimedes
The word "quadrature" sounds much more elegant than just "area".
Hi, amazing video. What software was used for making it?
Archimedes has to be one of the smartest people of all time!
The application of gravitational acceleration is ingenious.
The logic of this video contains a loop: derivation of the formula x = at^2/2 itself requires integration. I suppose the real Archimedes calculated the areas geometrically.
P.S. Sorry for the nitpick, I've finished the video and saw your remark on simplification. Still, the loophole is there.
no you know what the real nitpick here is? The fact that the kinematic equations that were used in this example could not have been used by Archimedes because they were invented by Newton in the 17th century.
Not necessarily, S=ut+1/2at^2 can easily be derived even without calculus. We had the derivation of the equations of motion back in 9th grade. One can also look at the area under the v-t curve (It's a trapezium).
@@thelosttomato4020 which is literally integration, just without saying the scary word out loud.
@@daigakunobaku273 But when one finds the area of the trapezium, they're just doing what archimedes does in this video, that is, it was somewhat possible for him to do this in his time (though he didn't and Newton probably came up with it later on, that guy does everything).
@@thelosttomato4020 no, they are not. The problem is not finding the area of the trapezium itself, but realizing the connection of it with the distance travelled by an accelerated body. Even the physical concept of acceleration, as far as I know, didn't exist in Archimede's times, let alone the Cavalieri's principle and the fundamental theorem of calculus. Besides, at the time a belief was widely held (at least among Greek philosophers, heavily influenced by Aristotle) that a slung body first ascends diagonally, until the "force" that acts on it is not "exhausted", and then likewise descends. Also, Archimedes himself filled the area under the parabola with clverly chosen triangles of decreasing sizes, and this approach was problem-specific.
Most likely the ancient Greeks did in fact discover calculus, but the documents didn't survive to the modern time, since the libraries of Alexandria, Constantinople, and others didn't survive to the present time. So many inventions of the ancient world have been lost to history including Greek fire, the Pharos lighthouse, ancient Egyptian and Roman concrete, Aqueducts and plumbing, Antikythera mechanism, Saqqara planes, Lycurgus cup, Archimedes heat ray, Aeolipile, automatically opening doors etc. Eratosthenes calculated the distance from the Earth to the Sun. The Romans had almost modern level chemistry, mass production of food additives such as Lead(II) Acetate and Sulfur Dioxide. I think that Leonardo Da Vinci and other Renaissance men didn't really invent anything, they just copied ancient Greek and Roman manuscripts.
So yeah, I wouldn't be surprised if ancient Greeks actually had working knowledge of calculus. We just don't have the books, or they are locked up within the Vatican's archives. I think that if you go back in time and take an ancient Greek scientist with you, they would be able to understand 20th century science within a year or two of concentrated study.
Derivatives were unknown to them. They were only aware of the concept of tangents
Didn't archimedes calculate the integral of x^2 from -1 to 1 or something like that
Excellent explanation. Thank you. Now I'm looking for the story of the palimpsest...
How can any limit of any series equal its sum if there is always error by definition.
There is always an error as long as you only count finite terms. The limit is not a sum of finite terms.
Again you don't have to believe in infinity. You can just take it as the definition of limit. But ultimately it is meant to be a way to calculate the sum of infinite terms.
This is a good video, but I think the most interesting thing I saw was in the description. You said he might be credited with calculus "if ancient Greeks were not so afraid of the concept of infinitesimals". I would LOVE to hear more about the reaction to his work from mathematicians of his era
That is enlightening. 👍
Links don t work😥
Which links?
This subject is awesome, but the background music pisses me off.
Yep, I do know that the background music is not appropriate here - I did try to improve in videos after this one though :)
Make some video on fibonachi sequence
I made two videos on Fibonacci sequence already! One is about how combinatorics is related to Fibonacci numbers, and the other is about the different ways of computing the general term of Fibonacci numbers.
@@mathemaniac both are really amazing.! Excellent explanation.! 😍
This ... was not said properly.
Why the red squares' area is one-third of all?
So for the largest red square, to its top and right there are two identical yellow squares of the same size as the red one. Then for the second largest red square, you also have two yellow squares to its top and right, and so on.
So there are three copies of the same set of red squares in the entire figure, hence red squares occupy a third of its area.
@@mathemaniac Oh, I understand. Thank you. By the way, can we apply for x^n rather than only x^2 as you have described?
No... because x^2 represents a parabola, and we have used some physics for this argument as parabolas are just trajectories of objects under gravity. However, there are other ways to derive the integral of x^n without the power rule; see my "Power rule of integrals not using integration; definition of e | DIw/oI #7" for that method.
Even if he did not, contemplated an infinite series, he have still eluded to calculus, almost like HERO making his steam apparatus have fail utilize the steam power, but have also eluted to the potentials of steam energy. Pythagoras invented the square of two sides yet did not invented the sq. Root of 2. Just remarkable.
archemedes was the first western mathematician to discover calculus pretty much I mean what you are describing is the reiman integral
...Exodus 26:13 ≈ Pi...
Moses was first to record Pi in 1440 BC. Josephus the Historian's description of the Tabernacle in 94 AD was inaccurate. This oversight was discovered in 2015 AD.
330 Exodus 26:8 eleven curtains each 30 cubits long
15 Exodus 26:12 one of the curtains is folded in half to 15 cubits long
- 1 Exodus 26:13 makes Pi each curtain overlap/seam adds up to 1 cubit
= 314
3.14 = 314 circumference/100 diameter ≈ π ratio (100 cubit court per Exodus 27:9-18)
.................
This Wilderness Tabernacle archeological discovery is similar to the Dead Sea Scrolls and Martin Luther's 95 Theses.
.................
History of finding π:
-(1900-1680 BC) One Babylonian tablet indicates a value of 3.125 for π
-(1650 BC) The Rhind Papyrus the Egyptians gave the approximate value of π 3.1605
-(1440 BC) Moses recorded Pi in the Exodus blueprints rediscovered in (2015 AD) 3.141592653... or Exodus 26:13 ≈ Pi
-(500 BC) India's Aryabhata approximation was 62,832/20,000, or 3.141
-(429-501 BC) Zu Chongzhi a Chinese mathematician 3.1415926 - 3.1415927
-(250 BC) Archimedes from Syracuse showed between 3.1408 and 3.1429
.................
The knowledge of Pi was lost from Exodus near 1000 BC. Josephus the Historian in 94 AD did NOT know about Exodus 26:13 makes Pi to properly explain the Tabernacle blueprints. He deferred to the Temple's structure and not Exodus 25-26-27.
Pi is found in the spiral of the double helix in your DNA. This knowledge has been lost for 3000 years.
Consider King Josiah & the Prophetess Huldah rediscovering the scriptures, right? In short, this monumental oversight was corrected in 2015. What's next?
...Exodus 26:13 ≈ π...
But the fact that distance is proportional to time squared comes directly from calculus......
Archimedes didn't use the method that I described obviously, and the method he used was way more geometric - but this is a way simpler and more engaging way, like I have said in the end.
@@mathemaniac Yeah the method is cool and all but the physics approach is dependent on calculus which wasn't discovered back then so it would have been better if you used other method instead.
The physical intuition failed me completely, I had to stop the video and pull up the algebraic formulas to prove using derivatives, which alas is calculus again. Though since vectors were invented way later than calculus, and is way less intuitive to me, I always find people who use that as the go to explanation perplexing.
Perhaps it is because different people have different ways of understanding mathematical concepts, and for a lot of people, graphical / physical concepts are more intuitive, which is probably why a lot of people including me use those concepts as the go-to.
@@mathemaniac It was never explained though why or how that analogy to gravity would be justifable. Why would vectors be related to that line segment at all? It produced the results we wanted, but it seemed arbitrarily chosen and not a consequence of algebraic graphs. I'm missing the connecting link between them
@@anonymouscoward2491 Archimedes used a tedious geometric method to do something that the average high schooler could do in half a minute using calculus. The geometric methods would be very complicated for other curves, if it is possible at all. Why would you still think calculus is inferior?
There was no ancient Greece.
Dr Anatoly Fomenko, "History: Fiction or Science?".
Oh Archimedes you were so close.
If the Romans hadn't killed him..... now we will never know.
Heck we would be colonizing the solar system by now
I think I was in middle school when I wanted to calculate the area of a circle without using pi. I think I accidentally stumbled upon the method of exhaustion (More luck than smarts lol) and used something like that. I only learned about the method later in life though.
Is it manim?
No, but I don't want to reveal the video making method just yet because I fear people will judge me - it's far from professional. Will make a video about how I made these videos sometime in the future.
For reference, manim is a Python library 3b1b has made for his videos.
Mathemaniac who cares if you’re using MS paint? The real crime is the background music.
Nice
He didn't discover calculus actually...
It's like to say that James Clark Maxwell discovered E=mc²
Same work was done by Chinese mathematicians liu hui..
Indian mathematician aryabhatt used infinitesmall to study the rate of change . he did it to study the orbit of moon.
Much of the key notions ideas of differential calculus was found in the book of bhaskarcharya II.. He introduced differentiations and derivatives. He was perhaps the first to conceive the differential coefficient and differential calculus...
15th century mathematiciam madhava of south east discovered power series infinite series and expansion of trigonometric functions.
Power series forms the basis of complex analysis....
Many pples see madhava as a founder of mathematical analysis
Almost, keyword "almost"
I don't think bhaskara ii discovered derivative only some initial concept of differentiation
Any proof that he conceived diferential calculus nor derivative
"things will be getting more complicated"
Me thinking :things are already complicated 😅 should I just accept defeat and leave it here?
The Ancient Greeks were geniuses of our world.
Yes indeed!
mindblowing
You never mentioned why he or somebody else did not finished his work.
It was because of Judeo-Christian destruction of ancient knowledge. Shortly, after Judeo-Christianity gained political power in Rome, they started a persecution of other religions and thinkers.
Archimedes works were mostly destroyed by them. The surviving papyrus where he almost discovers calculus has a script on Greek or Hebrew, a prayer to protect against "demons"...
The world should have stayed pagan.
Use Tau, not Pi!. Great video though
Great…
And Newton took the credit.🤷🤷
Shut the fuck up you stole it from Babylon
yeah if i was there i def would have invented it lol
ARCHIMEDES! NO! IT'S FILTHY IN ZERE...
One of the big steps they needed was the concept of the derivative. It's hard to accept the claim that Archimedes was close to developing calculus if he didn't have a concept of derivatives. No derivatives. No fundamental theorem of calculus. No rules of integration like the power rule or substitution.
The concept of calculus does not only involve derivatives and integrals; it's fundamentally about the study of infinitesimals. Archimedes developed the concept of limits already, and it is just one step away from infinitesimals.
In fact, Archimedes also was a pioneer of the idea of differential calculus. He found the tangent to a curve other than the circle as well (of course not using the modern methods - you can search for it if you wish).
What I am trying to say is that, even if Archimedes method does not look anything like modern calculus, it still captures a very quintessential part of it. The calculus methods now might be like the Archimedes' ones if history took a different path.
@@mathemaniac if archimedes wasn't killed prematurely by the romans, he might have had changed history propelling advancement of civlization a millenium earlier. Think about that for a moment.
@@quelorepario In the words of arthur c.clarke "If Christianity and Turkish occupation hadn't stopped the Greek civilization from its
advanced progress in mathematics , chemistry and physics , the Greeks could have been to space 600 years before the Americans".
@@mathemaniac he could have formulated various principles of integrations using derivatives. We doesn't use exhaustion method to compute the area of a curved figure anymore
Exhaustion method was infact still a famous method known to many peoples before Newton and leibniz.
*It's precursor to calculus same as electromagnetism is precursor to theory of relativity.*
We cannot say Maxwell discovered e=mc2
@@mathemaniac he just used tangents. He was not aware of differential calculus
Greek pure geometry rejected neusis because it leads via infinitesimals to the empirical absurdity of Zeno paradoxes.
Greeks were right, look at what attempt to make infinitesimal calculus consistent part of pure math lead to: post-truth post-modern language games of Formalism posing as foundational mathematics.
No thanks, Analysts and your coordinate system neusis. To maintain sanity I'll stay with Greeks, Berkeley and Intuitionism.
math sucks
I am sorry for hurting your feelings. But he could have discovered calculus if he spent more time stealing and understanding ancient Indian mathematician..
Source -trust me bro 😂😂😂
Lol came 9 yr old kid claiming that everything was stolen from india there was no evidence on any concept of calculus until 14th century in Kerala but they invented power series of trigonometry not integration and differentiation
@@mauryanempire7503 well I am not claiming that I think you probably know something called real analysis which is the base of calculus... If you study that you will understand that how much important power series and their inventions in Kerala school is ... And also I don't need to give you any source if you are interested you will search for the truth .. I am not here to change your belief system... Thank you
@@Pavan_Gaonkar_abcwe known that Kerala school invented power series but it was during 14th century and it is considered as a precursor to calculus.The real calculus which is integration and differentiation is invented by Europe by Leibniz and newton and secondly Madhava power series expansion is only applicable to trigonometry functions like sine, cosine and arctant and secondly newton discovered these series after his publication of his work on calculus
@@Pavan_Gaonkar_abcEven method of exhaustion which is the precursor to integration is also a base of calculus as it talks about finding areas of objects.