When Descartes Challenged Fermat (and Lost)

⬣ LINKS ⬣
⬡ PATREON: / anotherroof
⬡ CHANNEL: / anotherroof
⬡ WEBSITE: anotherroof.top
⬡ SUBREDDIT: / anotherroof
How were tangents derived before calculus? And why did Descartes and Fermat, two of history’s more renowned mathematicians, hate each other? Watch to explore their methods to solve the tangent line problem and the origins of their bitter rivalry.
Check out my series on building numbers from the ground up:
• Mathematics from the G...
⬣ TIMESTAMPS ⬣
Intro - 00:00
Descartes and his Circle Method - 03:34
Fermat and his Adequality Method - 15:13
Rivalry - 29:09
Folium of Descartes - 31:26
Conclusion - 42:18
Outro - 46:08
⬣ CORRECTIONS ⬣
In some sources, “Discourse on Method” is described as a work of three essays. Elsewhere, it is described as the introductory work to those three essays.
⬣ INVESTIGATORS ⬣
Are you still out there? Check my logic video for an update!
⬣ REFERENCES ⬣
[1] R. Descartes, La Geometrie, Trans. David Eugene Smith and Marcia L. Latham. Open Court Publishing Company: La Salle. (1952), pp. 95-112.
[2] Selected Correspondence of Descartes, Jonathan Bennett 2017:
www.earlymoderntexts.com/asse...
[2a] to Mersenne, end of xii.1637, paraphrased.
[2b] Fermat to Mersenne, iv or v 1637.
[2c] to Morin, 13,vii.1638.
[2d] to Mersenne 27.vii.1638
[2e] to Mersenne, ix.1641.
[2f] against Fermat, 1.iii.1638.
[2g] to Mersenne, 27.v.1638.
[2h] to Mersenne, xii.1638.
Huge thank you to Hal Hellman and his excellent “Great Feuds in Mathematics” for compiling many of the sources used in this video.
[3] Hal Hellman, Great Feuds in Mathematics, 2006.
[4] D. E. Smith, A Source Book in Mathematics, McGraw-Hill, 1929, pp. 389-96.
[5] Fermat to Mersenne, December 1637; Fermat Oeuvres, vol. 2, p. 116. Translated by Daniel Curtin.
[6] Fermat to Mersenne, February 1638; Fermat Oeuvres, vol. 2, pp. 132-33. Translated by Daniel Curtin.
[7] G. F. Simmons, Calculus Gems, 1992 p. 101.
[8] Fermat to Clerselier, March 10, 1658. Translated by elucidation by James Nicholson.
[9] J. D. Nicholson, as detailed in Hal Hellman, "Great Feuds in Mathematics" 2006.
[10] L. T. More, Isaac Newton, Scribner’s 1934 p. 185.
⬣ CREDITS ⬣
All music by Danjel Zambo.
Images:
Explosion: [www.vecteezy.com/free-png/fir...] Fire Explosion PNGs by Vecteezy.
Laser beams: [www.pngkit.com/view/u2t4i1w7e...]
Sunglasses: [www.vecteezy.com/free-vector/...] Meme Glasses Vectors by Vecteezy.

[@DrDeuteron]

Descartes reasoning was circular, but no more than Fermat's was triangular.

[@pedroisern9838]

i would like your comment but it has 69 likes xd

[@theobserver314]

@@pedroisern9838 Not anymore.

[@WithinEpsilon]

In the end, Fermat used far more rational thinking than did Descartes (the ratio of two sides of similar triangles).

[@herbie_the_hillbillie_goat]

Clever comment. 😉

[@Nothingtonnobodson]

420 blaze it

[@Zveebo]

Crazy to see how close Fermat got to calculus with his method - what a pity he didn’t quite put everything together at the time.

[@nikitakipriyanov7260]

I bet he had actually invented the calculus, but it was too long to give it in the margin.

[@honourabledoctoredwinmoria3126]

Both methods can actually invent calculus, because you can define a general derivative equation from Descartes's method and use that to prove the chain rule and define implicit differentiation, and then you have the tool to solve for anything that can be written as an analytic function in two variables. I think the reason neither of them pushed any farther is partly because Descartes went onto other things and then died before coming back to this, but more importantly, both of them did not fully appreciate that in going from a geometry problem to an algebra problem, you have created new tools that can be used in ways that neither algebra or geometry could before. Both of them were too fixated on giving the original constructions. Neither went to wondering what the properties of the function that gives you the gradient at a point. It's vey easy for us to make the jump that the derivative is itself a function, in fact one of a class of functions, and we can examine properties of this class of functions. In doing so, we can find that there are such properties that do not depend on the properties of any particular point or any particular derivative function PROVIDED IT EXISTS THERE. That's why Newton and Leibnitz almost immediately proved the chain rule followed by the fundamental theorem of calculus and Newton "proved" the expanded binomial theorem within a year of creating methods of finding the derivative. (He didn't conceive of a function that would not be continuous or analytic) And Fermat never did, because ultimately Fermat was still in the space of constructing individual slopes. I think Newton suspected that Leibnitz had made such a leap to think of problems in the general, although he didn't think Leibnitz succeeded in making the conclusion of defining derivative and integral without stealing his method.

[@DrDeuteron]

@@honourabledoctoredwinmoria3126 when you say "analytic function" I think "complex analysis"...is that too much? with all due respect to another doctor ;-)

[@RobBCactive]

Fermat seems to have had too much fun triggering Descartes, sending indirect letters so they couldn't simply be ignored. The invent a to make extra equations then divide by a and immediately set a=0 looks like trolling. That it turned out to be more general was piquant.

[@parthsavyasachi9348]

Many hundreds of years before them indian mathematicians used integration type method to calculate value of pi.

[@firefox7857]

I know this video is about how Fermat's method eventually won out, but I can't help but feel impressed with how Descartes came up with a method that didn't involve limits! Even if it was limited.

[@felipe1st]

no no, it wasn't limited, it was limitless!! cause, you know, no limits... sorry, the exit is.. that way...? k bye

[@ToriKo_]

@@felipe1stthe exit is perpendicular to that bend over there, but tbh I’m not sure you could work it out...

[@atzuras]

Jokes of the early XVII century must not use limits, unless they are in latin.

[@mitchwyatt9230]

The limits are hidden in the details - specifically in setting a=0 after dividing through by a.

[@typo691]

@@mitchwyatt9230 That's for Fermat's method. OP is talking about Descartes' method with the circle, which doesn't involve limits

[@APaleDot]

Descartes: "You moron, you absolute buffoon" Fermat: "I eagerly await your criticisms, for you are truly one of the greatest mathematicians of our time" Another Roof: "As you can see from the previous exchange, Fermat was a sarcastic asshole" I thought it was going to be a one-off bit, but it just kept happening. 🤣

[@NoneOne_]

read 'great' as 'fat' and he was a sarcastic asshole

[@DrDeuteron]

and we thought trolling was invented with the internet, no, it is part of the one constant of history: human nature,

[@pyropulseIXXI]

That isn't sarcasm. Decartes was one of the greatest. So, if you call one of the greatest "one of the greatest" and then claim sarcasm.... that literally makes no sense So you are being sarcastic about something that is true? That is called gen z humor, where you are being ironically unironic in an insincere way

[@APaleDot]

@@pyropulseIXXI Yeah, it never read like sarcasm to me. But then Another Roof just kept calling him a sarcastic asshole.

[@AnotherRoof]

@@APaleDot Hmm, going to step in to defend myself here! I never called Fermat a "sarcastic asshole" but I said that his words can be interpreted as sarcastic. As @pyropulse7932 says, there is the literal, sincere interpretation. I was only offering an alternate, more sarcastic, interpretation. And I also say at 19:34 that the correct interpretation is an open question and then say: "listen to this quote from Fermat and decide for yourself" because I don't know which is correct and it's up to the reader to decide. Since Fermat is no longer around we can't ask what he really meant, but I would add that the sarcastic interpretation isn't my invention. As I state in the video, JD Nicholson (who translated many of the letters between the two) and Hal Helman offer both interpretations as Fermat wrote with "politesse," which as Helman writes is "a term in French meaning right-thinking attention to manners but with less than noble purposes." I hope this clears up this point!

[@kilianklaiber6367]

I admire Fermat's courage to publish a mathematical method that was not quite right but still worked for some unkown reason. A rigorous understanding of calculus really requires an understanding of infinite limits. This was introduced in 1821 by Cauchy and Weierstrass. Thus, we must credit a whole bunch of ingenious mathematicians with the development of calculus.

[@MrMiddleWick]

He would be an amazing software developer

[@renanmaas3502]

The relation between Fermat and Descartes looks like it comes straight out of a pixar movie lol, the shy, discredit and polite Fermat being challenged by an arrogant and famous mathmathician who cant accept defeat after seeing he just got outplayed at his own game

[@buschtoens]

This has to be the best video in a long while the algorithm has fed me. I love your style of presentation and enthusiasm. The history trivia made it so much more entertaining. Kudos!

[@cadekachelmeier7251]

"Okay, now we're going to divide by 'a'." "Okay" "And /now/ we're going to assume 'a' is 0." "-_-"

[@rmsgrey]

A better phrasing for that second step (in terms of mathematical validity, if not historical accuracy) would be "assume that 'a' is close enough to 0 that we can ignore anything with an 'a' in"

[@thewhitefalcon8539]

​@@rmsgreybetter yet, take the limit as a goes to zero

[@rmsgrey]

@@thewhitefalcon8539 Correct mathematical jargon, sure, but "better"? At best, it would be anachronistic to invoke the nineteenth century concept of limits to resolve an argument in the seventeenth century...

[@CliffSedge-nu5fv]

Reminds me of when my algebra students divide both sides of an equation by x instead of factoring it out.

[@DoggARithm]

Fermat's skill combined with his disdain for proofs set the perfect counterexample for Euler

[@GreatCollapsingHrung]

If you had asked me yesterday to tell you about Fermat, the only thing I would have remembered off the top of my head would be the "Last Theorem." This video has made me even more aware of the genius of the man. It also made me aware of the douchebaggery of Descartes.

[@AnotherRoof]

I didn't know quite so much either until I started writing this video! I stumbled upon the Folium in a Dictionary of Mathematics (because that's the sort of book I flick through). That led me down the rabbit hole of this rivalry and I knew it was a story I had to tell!

[@theflaggeddragon9472]

After that marginal scribal, Fermat actually proved his "last theorem" for the case of n = 4! So he certainly did not have a general proof.

[@blableu4519]

@@theflaggeddragon9472 Wow! But why did he figure it out for n = 24 specifically?

[@theflaggeddragon9472]

@@blableu4519 Just n = 4 haha (! was an exclamation, not factorial). The reason was that pythagorean triples, i.e. integer solutions to x^2 + y^2 = z^2 were well understood at the time of Fermat. Given two integers r,s, the triple (r^2-s^2, 2rs, r^2+s^2) is a Pythagorean triple (easy to check), and all Pythagorean triples are of this form (a bit trickier to verify). Using this fact, he showed that the equation x^4 + y^4 = z^2 (stronger than FLT for n=4 !) is not solvable; given a minimal solution to the equation, and a pair (r,s) as above, one can show that you can construct a smaller pair (r',s') which gives a contradiction. Do it as an exercise!

[@honourabledoctoredwinmoria3126]

​@@theflaggeddragon9472 I actually think what he meant was that he had a proof for n =3 and n =4 and he was postulating the general, because he singles out those two cases. It is true that the Latin grammar means that the too big to fit in the margin is about the general case and not the n =3 and n =4, but that could be an error on his part. I don't think there is any way he could have thought he had a proof for every natural number over 2.

[@JeffreyLWhitledge]

I really like your use of props. They make the story much easier to follow.

[@antonbashkin6706]

This is a brilliant and priceless math lesson, I don’t know where else besides KZbin I could find this kind of content. The algorithm cannot reward this video enough.

[@Its2Reel4U]

Great video! As a hobbyist, Fermat might not have had all the tools and the formal rigor, but damn did he have intuition! I can't help but wonder if he might have actually had a "marvelous proof" for his last theorem. One that, in a similar way to his approach to tangents, was perhaps not formally sound, but could have provided some stunning insights that were well ahead of their time.

[@gauravbharwan6377]

I don't think fermat had a proof because do you think he was that stupid to leave the proof of something which is as general as last theorem, couldn't he bring more paper Another reason why Fermat won is that he was a lawyer so he knew how to handle people and controversy

[@caiodavi9829]

​@@gauravbharwan6377 he didnt publish his version of analytics geometry for years. perhaps he did not publish his proof oh his last theorem for similar reasons. well never know, i guess

[@gauravbharwan6377]

@@caiodavi9829 hmm, the only thing that favors him in this matter is that he was in ranks of strong mathematicians.

[@Galahad54]

Fermat probably had a method that worked in the general case, but it turns out that particular primes need special care. Now, if the 'little theorem' could somehow drastically reduce the special primes cases, then yes, but 350 years of search had to wait for the invention of 4-6 whole new fields of maths to yield the special primes to the proof. Such fun things as =- let's call it analytic geometry of convex functions, Riemann functions, ... Plus the invention of electronic computers with enough memory to hold the calculations involved.

[@FloydMaxwell]

Fermat invented calculus, but didn't have room on the page to spell it all out.

[@dizwell]

What a great video. Thank you. I'm an historian before a mathematician, but you set the context beautifully and even I was able to spot the significance of setting a=0. Goosebumps stuff!

[@stephenhicks826]

What a wonderful 47 minutes of Mathematics History. I was spellbound from beginning to end. I never knew Fermat was so close to unravelling the secret of differential calculus, he was just about there. Wow!

[@dimitrisfotinakis4076]

It's 2:20 in the morning, I am really to fall asleep, and this video pops up in my feed, 40 minutes later, after I saw the whole video, I was firstly amazed by the genius of both Decartes and Fermat, their methods, even before the existence of calculus, are amazing. Secondly I fell in love with the way of exhibition of the their methods, very good channels with very good quality of content, continue strong

[@moonshine7753]

This channel is so underrated right now, I hope you blow up sometime because you deserve it. (Also, Fermat got so close to calculus without actually reaching it, you might call that "Parker Calculus")

[@gauravbharwan6377]

😀😀😀😀😀 parker calculus Nailed it

[@krozjr5009]

I’d be happy to credit Fermat as the inventor of the derivative. Is it informal, rough round the edges, and very much *not quite right* and wonky? Yeah. Is it blindingly recognisable in hindsight what it is and turned out to be? Also yes.

[@DrDeuteron]

as a physicist who shun formal mathematics (in the office), [thumbs up emoji].

[@isi2973]

@@DrDeuteron Pesky mathematitians, cannot even accept 0^0 = 1. :p

[@pyropulseIXXI]

It isn't the derivative though, so you'd be wrong

[@pyropulseIXXI]

@@DrDeuteron I double major in physics and mathematics with the explicit purpose of getting a PhD in theoretical physics, and I cannot stand my id**t physics peers that shun formal mathematics

[@DrDeuteron]

@@pyropulseIXXI I can 't stand the neighborhood of the disjunction of the sets of all people who curse or censor "idiot"

[@josephblattert6311]

I love when my passions for history and math can combine. I hope this channel gets more attention!

[@DrDeuteron]

omg, please read The Mathematical Experience by David and Hersh...a brilliant book of vignettes about everything we care about.

[@pyropulseIXXI]

So you just haphazardly go through life all willy nilly, just waiting for your 'passions' to combine? How ridiculous

[@andrewzhang8512]

@@pyropulseIXXI thats the dumbest interpretation ever

[@bradsword5822]

I am hoping you stick with this type of content. It’s great! You’re still building a base of watchers, so I hope you don’t get discouraged, given the effort you put in.

[@AnotherRoof]

Thanks! Luckily, I really enjoy making these videos, so even if the view count is low I'm still proud of the result. Here's hoping the algorithm looks kindly on this one!

[@gnashr4366]

@@AnotherRoof this video is now one of my favourites on KZbin. I really appreciate how much effort you put in to your research in order to be able to explain this in a captivating method to a wider audience.

[@gumbo64]

@@AnotherRoof didn't realise it was a long-form video until i was like 25 minutes in lol

[@HighKingTurgon]

Delightful that Fermat's method breaks down at the one point on the folium where the curve is not differentiable.

[@spham_99]

“Like all the best problems in mathematics, it’s simple! Except it isn’t!” Lmaooo 1:34

[@caladbolg8666]

I heard about their methods in my history of math class, but seeing the details and examples is really great. The production and charm is top notch as always!

[@lossen1984]

You deserve a subscription for that oustanding performance and for your story-telling capabilities! Incredibly exciting to learn how calculus was in some way even formulated before the presence of Leibniz and Newton!

[@AleXander-eo3iz]

I thoroughly enjoyed this. Although Descartes was quite an arrogant guy, it did make the story a bit more "spicy" and entertaining; it's always fun to watch two guys battle each other out where the protagonist (Fermat) eventually defeats the antagonist (Descartes).

[@AnotherRoof]

Glad you enjoyed! I honestly felt the exact same way -- I started writing this video intending to make it solely about the Folium, but when I discovered Descartes' trash talk I knew I had to tell that story!

[@drippyeuler]

Damn, nice story. Descartes should have offered a curve and asked the tangent at a point with slope 0. Can't use the point v if the tangent doesn't cross the x axis.

[@stephaneduhamel7706]

At a point with slope 0, it's very easy to find the tangent. It's gradient is obviously 0, because it must have the same slope as the curve.

[@RexxSchneider]

But Descartes' method wouldn't work since the normal would have an infinite slope.

[@flambambam3578]

One could follow the argument that the slope must be zero if v does not exist, for the slope must be parallel to the x-axis.

[@pyropulseIXXI]

@@RexxSchneider It works since the slope is 0 you id**t

[@pyropulseIXXI]

Is this comment sarcastic? The slope is obviously 0, and the method still works; you just use basic geometry And you've f*cked up; v still crossed the x-axis if the tangent doesn't cross the x-axis, since v comes from the normal line, you absolute dunce

[@liammargetts]

I think Fermat created a kind of proto-calculus, since his method does indeed have all the hallmarks of differentiation.

[@robharwood3538]

Well worth the length! Much appreciated for the depth of research and clarity of presentation.

[@SiqueScarface]

René Descartes was a master of derision. Just the first sentence of his "Method" is genius: Of all things, good sense is the most fairly distributed: everyone thinks he is so well supplied with it that even those who are the hardest to satisfy in every other respect never desire more of it than they already have. And Pierre de Fermat was a master of derivation, as his Method proves.

[@JavierRuizGonzalez]

Fermat's method immediately resonates on the memory I had of the derivative of a function, some 40 years ago. I had an enormous sense of satisfaction following this video. Great effort! Thanks much!

[@aguyontheinternet8436]

It's amazing that as you read more and more of the letters between Descartes and Fermat, you lose the ability to understand whether Fermat is being sarcastic. He could very well just be a math fan.

[@golddddus]

Descartes is the inventor of the Cartesian coordinate system. And that is one of the greatest mathematical discoveries of all time. If we add to that his "Discourse on Method" where the scientific method is established, Fermi's words seem sincere and not sarcastic.😎

[@maalikserebryakov]

Mohammed hijab debunked descartes

[@bjornfeuerbacher5514]

@@maalikserebryakov This is about Descartes' mathematical work, not about his philosophy. Mohammed Hijab debunked nothing of Descartes' mathematical work!

[@CliffSedge-nu5fv]

​@@maalikserebryakov Princess Elizabeth of Bohemia also debunked Descartes' philosophy. So what? His philosophy of dualism was stupid.

[@_.LZ._]

​@@maalikserebryakovmohammed hijab is a clown who doesn't know what he's talking about.

[@idontwantahandlethough]

I had never heard of your channel before but I'm glad I found it, this was super fascinating! Thanks dude :)

[@AnotherRoof]

Thanks for watching, tell your friends!

[@Deejaynerate]

I think the reason why Fermat's methods yield the same result is because he accidentally used the same definition that we use in modern day for derivatives, or at least a strikingly similar variant. Think about it, he's taking the original function, subtracting by an infinitesimally different (i.e. virtually the same) output of the same function, and then dividing it by that difference. How do we define derivatives? [F(x+a)-F(x)]/a, as a->0 If not the same, then it's a very similar idea.

[@marcozagaria6696]

Recently discovered your channel. This is by far your best video!

[@johnchessant3012]

I actually remember the folium of Descartes, as a standard exercise in implicit differentiation. I had no idea it had such an interesting history!

[@kika433]

At 27:26, y/(x-v) is the slope of the tangent, so you can skip solving for v and immediately see that the slope is 3x^2 = 3(1)^2 = 3

[@AJ-ss3jy]

You are correct and that’s how it is taught in schools now. But during their Leibniz-Descartes era, they strictly apply Euclid axioms in that 2 points is required to form a straight line. So they chose another point on the x-axis coordinates. So I think that’s what the video creator is going for: think like them.

[@jamesturner2914]

I love this channel. I was always pretty average at maths at school, then worked really hard for my GCSE and got an A. I enjoyed the problem solving, but found learning about history, geography and other subjects more intriguing ( I am about to graduate with honours in geography). What this channel does, and what school doesn’t is actually teach maths. Not just it’s application. The histories and etymologies are truly something i wish i could have learnt earlier. I am now hoping to read more “mathsy” books, one: as a challenge but also as something i’ve really enjoyed having explained to me !

[@Syntax753]

My two favourite mathematicians - and had no idea of this backstory! Thanks for presenting this to the world :D

[@CliffSedge-nu5fv]

The KZbin clickbait title at 20:08 was so accurate it hurts!

[@hughcaldwell1034]

"I have a truly marvelous demonstration of this, which Descartes' mind is too narrow to contain." - Fermat, I think...

[@raulyazbeck7425]

crazy good video. I already knew about their feud and have always wondered why Fermat was never (in a popular way) credited for calculus. I now know better thanks to your video!

Ah, Fermat was so close.

[@xicufwm]

"Another Roofian Theorem" actually sounds quite nice! Great video as always! I'll select a few partsof it to show to my AP Calculus students (our perdiods aren't long enough to show a 47-minute long video, unfortunately).

[@horacioguillermobrizuela4295]

Content: Maths + gossip = irresistible!. Presentation: Instructing, amusing, simply developed (¡nice handwriting!). A must-see video

[@johnwilson3918]

I started to a gag a little when you divided by 'a' on one line and setting 'a' to zero on the next. Great video. Thank you for sharing.

[@thedude882]

How on earth do you only have 12.600? subscribers? the quality you put in your videos is impressive!

[@AnotherRoof]

Thanks. Subscribe and tell your friends!

[@thedude882]

@@AnotherRoof Already sucsribed. And as a fellow math enthusiast, the least I can do is inform all three of my friends ;)

[@DOTvCROSS]

@18:24 "I speak, therefore I am Karen" Descartes as a moderator would make people quit math!

[@Mentox2]

24:47 - Wait a god damn second, did Fermat invent the concept of limit? holy shit...

[@CliffSedge-nu5fv]

He was this (limit as closeness approaches zero) close.

[@henrymarkson3758]

Thanks for the effort you put into producing this quality video. Much appreciated

[@socksygen]

Great video! Now what I wonder is, did Descartes even attempt to apply Fermat's method to the Folium himself and failed, or was he simply so arrogant and dismissive that he assumed it wouldn't work?

[@scialomy]

Happy to see you being better. I really enjoyed the historical approach, and I'd be happy to learn more. Can modern calculus prove Ferma's method and what are the conditions on the studied curve for the method to be valid?

[@avyakthaachar2.718]

This video combines math and history so well! Thank you so much for this video 🙏

[@Claudiostuff]

I'm subscribed to plenty of math channels and this is the first time seeing yours, and what a discovery! Keep it up, great quality here

[@AnotherRoof]

Welcome!

[@GertDijkhuizen]

Excellent video! It's so nice to learn about the history of mathematics this way. Never knew Fermat was that close to discovering calculus.

[@sigurd106]

You explain things very intuitively. You are a very talented teacher, thank you.

[@onkelpawel]

With every video my love for this channel grows.

[@jasoncampbell1464]

It's amazing how brilliant people can have the right intuition and solve complex problems and yet struggle to formalize that intuition into the next revolutionary idea (e.g. limits in the case of Fermat). It's unsettling that there are probably at least a couple revolutionary intuitions sitting in people's heads that we just haven't formalized yet

[@-minushyphen1two379]

“Folium of Descartes” sounds like the name of an epic weapon or legendary ancient artifact, so it’s what I named my best bow in Minecraft(because it looks a tiny bit like a bow). I put it next to the Trisectrix of Maclaurin and the Cissoid of Diocles.

[@AnotherRoof]

I love this. "Index of Nilpotence" always stuck out to me as one of the most metal / badass terms in mathematics!

[@dlevi67]

The Witch of Agnesi! (Presumably living on top of a hill...)

[@Luke-mr4ew]

Amazing how intellectual giants worked hard to make solutions that by the time you're through high school look completely inadequate. Makes me wonder what discoveries / inventions might be found that make today's complex problems easy to handle.

[@scialomy]

A general and analythic solution to Navier-Stokes equation. Worth 1 000 000 USD, IIRC.

[@theflaggeddragon9472]

@@scialomy Existence of smooth solutions for all time given smooth initial conditions. Not a formula.

[@jacksonmagas9698]

@@scialomy hardest way there is to earn a million dollars

[@bradwillard8009]

awesome content! can feel your enthusiasm

[@maxdemuynck9850]

Really good video, entertaining and informative, keep it up!

[@adithya1760]

Amazing video absolutely love it please do more stuff on the history of mathematics it's just something amazing

[@Heater-v1.0.0]

Boy, flame wars must have taken months in those days.

[@d.e.p.-j.7106]

That's a wonderful video. You explained so much history in a short time. I even approve of using functional notation to explain what was going on even though neither Fermat nor Descartes used it.

[@xbaaldyy]

This video was great. Would love more!

[@juanpina6638]

I just found one of the best math KZbin chanels. Great video, i love math history

[@AnotherRoof]

Happy to have you! My video on 1's status as a prime has lots of historical stuff

[@Jack-in-the-country]

I have no clue how anyone could read Fermat's letters as insincere. He clearly wanted only to glimpse the truth. Sure, he may have been annoyed by Descartes' lack of humility (who couldn't be?), but he clearly states that the truth is more important than a petty disagreement. Beautiful video man! Thanks for this. When i was a kid I always puzzled over the transition calculus makes from secant line to tangent line. It didn't seem physical, but ideal: the notion of "instantaneous change" as opposed to the physical notion that needs to reference two distinct points. It seems that Fermat used a primitive version of a similar ideal here when he considered that distance essentially negligible.

[@CliffSedge-nu5fv]

We often think of the discrete as an approximation for the continuous, but for the relation between calculus and physics, the continuous ideal approximates the discrete reality.

[@sergiolucas38]

Excellent video, I've never seen fermat's method before, it is great, thanks for the video :)

[@reluginbuhl]

This was interesting and well done! Thank you :)

[@soyokou.2810]

You should make a video on the Fundamental Theorem of Calculus. I believe it was Gregory and then Barrow, not Newton, who first observed the connection between area and tangents. Also, the Japanese circle principle (enri), Indian infinite series, and Archimedes' Method of Mechanical Theorems would also be good topics to show how ideas from calculus appeared and have been developed independently in the past.

[@DrDeuteron]

why not all the fundamental theorem? A series, perhaps.

[@honourabledoctoredwinmoria3126]

Barrow connected the Descartes derivative of the area under a curve to the curve, or specifically if there is a continuous function A(x) that has a tangent line a(x) defined using Descartes's Method of Normals, then A(x) is the area under the function a(x). But I don't think it was a proof of the Fundamental Theorem, because there's no indication he realized the significance of his proof or how generally the function A and a could be defined.

[@pyropulseIXXI]

@@honourabledoctoredwinmoria3126 This is all basic stuff that is in every calculus textbook, with them even giving more resources to look in depth at what these people do, and mor*ns still say this type of crap People are so d*mb and lazy that they need a teacher or someone on youtube to 'show' them stuff. I don't get it; how do I, when I was 8, go about finding this stuff on my own without anybody and learning in a vastly superior fashion and yet virtually no one else does this? It is so rare to find an actual person with real intelligence and a consciousness instead of being a soulless automaton maybe a few jimmies wil be rustled here, but it would further prove my point; an emotional response by id**ts

[@tolkienfan1972]

What an amazing story! I love the connection to Newton.

[@ColtonKinstley]

Really enjoyed this. Thank you.

[@ShaolinMonkster]

Super video. I enjoyed it as much as the previous. I want to add that the Ancient Greeks had laid foundation to Calculus before Fermat.

[@jonmoore8995]

Thank You for a great presentation.

[@Bemajster]

27:37 You can take the limit as a -> 0, but you can't take a = 0, because you divided by it.

[@thephilosophyofhorror]

Nice, happy 2023 :) Also made me think that if I ever do math videos, I will (also) use some (typed) Utopia-The creation of a Nation soundtrack! It just fits with the pleasant vortexes presented.

[@EnemyOfEldar]

This was fantastic! Great maths and history what a treat. I would credit Archimedes with the discovery of calculus!

[@Spacexioms]

4-year math major here and I love the history behind this stuff. Wish I learned it earlier

[@DrDeuteron]

as an old physicist, method comes before history. Both are fascinating.

[@stephenmorton9789]

Really excellent lecture.Many thanks

[@mrtthepianoman]

I was unaware of this rivalry. Fun video!

[@AnotherRoof]

Would you still back Descartes in a duel? :P

[@mrtthepianoman]

@@AnotherRoof I suppose it depends on the nature of the duel. It seems like Descartes strives for loftier mathematical ideals, while Fermat goes for the practical, down and dirty mathematics. It reminds me of the grief pure mathematicians tend to give physicists for their approximations and means of handling infinity... the physicists methods are not as "pure" or "rigorous" as mathematicians would like, but they work! Given the nature of historical mathematical duels, however, I suppose I would have to back Fermat.

[@AnotherRoof]

@@mrtthepianoman And that's kind of the reverse to their argumentation in which Descartes gets down and dirty while Fermat opts for loftier rebuttals!

[@haniamritdas4725]

Thanks for the great history lesson! I was irritated as a young man reading the correspondence between Fermat and Pascal, because Pascal seemed arrogant and condescending and damned him with the faint praise of the "great amusement" he and his friends had with Fermat's propositions. I thought at the time that Fermat was polite and friendly. So now it makes sense to me that Fermat was quite likely digging in subtle ways I didn't recognize in those letters. I loved your presentation of the maths, but really was pleasantly surprised to learn more about these "great" personalities!

[@HyperFocusMarshmallow]

Great story telling and great math!

[@sjswitzer1]

Excellent video! I was expecting that when you described the similarity of Fermat’s method to the derivative you’d have called back to 27:26 where the derivative of x^3 appears conspicuously in the derivation. But you covered the the topic more generally. Very nice.

[@143TYAGI]

Wonderful research into the minds of intellectual giants. And its reassuring that they suffered with the same flaws as us. Love the methods used in the precalculus era.

[@Bayerwaldler]

I thank the Google-algorithm to at last show me your content. What a thriller - and extremely educational.

[@AnotherRoof]

Praise be to the algorithm.

[@Simio_Da_Tundra]

"who used similar triangles and a generous amount of handwaving" handwaving is like fermat's middle name. he is pierre handwaving de fermat

[@pamdemonia]

The idea of the coordinate plane is so basic my own understanding of math that I can't even imagine what math is without it! Amazing.

[@tigioctet]

I very much enjoyed this video ! Thank you. Applying Fermat's method to the folium reminds me of the "dual numbers" and how they can be used to make precise the idea of something simultaneously small but non-zero. The idea is simply to introduce a symbol, say u, such that u^2 = 0. If you consider the quantity f(x+u)-f(x) for some polynomial f, all terms vanish either because of cancellations or because u^2 = 0. What you are left with is exactly the gradient of f (times u). Of course, I imagine it would have been hard, even at the time, to accept that such a mysterious number u "exists" and more importantly, that you can give it the geometric meaning of "arbitrarily small yet non-zero". And as you mention during the video, a notion of differentiablity is necessary to tackle similar examples where f is not merely an algebraic expression. Anyway, thanks again for the nice content ! Have a good day :-)

[@nikitakrim02]

Great video! I really loved how the soundtrack kind of put you in that time period. Can you please name a piece you used in the title card and through out??

[@AnotherRoof]

All music is by Danijel Zambo. That piece is called, fittingly, "Centuries Ago."

[@bjornfeuerbacher5514]

33:18 "There is no way to rearrange this" Why? After all, it's simply a cubic equation for y, and at Descartes' time, it had been known for about 100 years how to solve those (dal Ferro, Tartaglia, Cardano, ...)

[@giladstern91]

It seems like Fermat was using a version of dual numbers with "a" being the equivalent of a nonzero epsilon such that a^2=0. This would mean you can divide by a, but anything with a remaining a term would be 0 (and would map nicely to certain ways of doing calculus today). Is this actually generally true of his use of a in his method?

[@angeldude101]

I've actually see trouble properly dividing dual numbers. Double checking with Wikipedia, it mentioned that "Division of dual numbers is defined when the real part of the denominator is non-zero." Dividing by epsilon alone, like what roughly be done when using them for calculus, doesn't actually seem allowed. Differentiating with dual numbers seems to instead just magically extract the coefficient in front of ε. Really, calculus in general is hard to do without eventually getting 0/0. The trouble is how to make sense of that in a consistent way.

[@moritzmoeller7816]

Amazing history and math video

[@BartoszMilewski]

And a propos Dioptrics, Fermat's principle of least time could be considered the foundation not only of optics, but of Lagrangian mechanics, and modern quantum field theory.

[@honourabledoctoredwinmoria3126]

And Descartes's dioptics is a mess.

[@pokemonjourneysfan5925]

at 33:15, yeah we can totally rearrange this into y=f(x) bc it’s a cubic equation. But I think you’ll get 3 solutions.

[@yegorwienski1236]

My mind was blown when I realised Fermat's reasoning uses dual numbers for differential calculus, 'a' basically squares to 0.

[@longline]

Very happy about the judicious use of bricks there

[@AnkhArcRod]

I must ask, was the set square based triangle just coincidentally similar to the triangles you drew or did you plan it all out? Anyway, cool video. Really didn't know about either of those methods even though I have been doing recreational math for over 25 years!

[@800-high9]

After some thought, I realized that the general equation for the equation of the gradient line has slope = 3x^2 and intercept = -2x^3. Easy to see from writing the equation as x^3 = (3x^2) x - 2x^3