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How Euler Did It, by Ed Sandifer
- nickcw 3y agoI have read a few of these and enjoyed them greatly. Reading them you realise that Euler really did invent a huge swathe of mathematics in use today. In particular I read this one: http://eulerarchive.maa.org/hedi/HEDI-2009-02.pdf http://eulerarchive.maa.org/hedi/HEDI-2009-02.pdf And I realised that Euler had found two formulae for Pi which can be used to calculate any hex digit of Pi. I wrote this up in a paper: "In 1779 Euler discovered two formulas for π which can be used to calculate any binary digit of π without calculating the previous digits. Up until now it was believed that the first formula with the correct properties (known as a BBP-type formula) for this calculation was published by Bailey, Borwein and Plouffe in 1997." https://scholarlycommons.pacific.edu/euleriana/vol3/iss1/3/ https://scholarlycommons.pacific.edu/euleriana/vol3/iss1/3/
- seanhunter 3y agoPart of the problem is he wrote so much it has taken a while to go through it all. I believe the "Opera Omnia" project to publish all his works has been going for over a hundred years and is just about getting to the end now. So I would expect there's a huge amount that just hasn't been fully appreciated/digested.
- jacquesm 3y agoA while indeed. Euler and Bach are similar in that sense: to properly ingest their life's output you need more than one life.
- m_mueller 3y agoHow is this humanly possible? Was Euler even an order of magnitude faster at producing new math than, say, Gauss or von Neumann?
- karmakurtisaani 3y agoIf I recall correctly, Euler has the most pages of published math. Erdős has the most papers (some of them not more than a handful of sentences).
- TuringTourist 3y agoErdős has a huge amount of credits in the papers of others (hence Erdős number) because he would just travel all over the country helping people get unstuck on their work.
- chongli 3y agoEuler had scribes do a lot of the grunt work for him. His vision was quite bad and worsened throughout his life, going blind in his right eye rather early on and later developing cataracts in his left. He once joked "Now I will have fewer distractions" on his condition.
- iaseiadit 3y agoThere’s more about the effort to publish his complete works here: https://en.m.wikipedia.org/wiki/Opera_Omnia_Leonhard_Euler https://en.m.wikipedia.org/wiki/Opera_Omnia_Leonhard_Euler
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- 3abiton 3y agoYet I struggle to adjust to one note taking app to optimize my workflow. The paradoxal curse of choice.
- g9yuayon 3y agoI always feel amazed that Euler wrote faster than people could publish or understand his work, even after he became completely blind.
- jacobolus 3y ago> In particular I read this one: http://eulerarchive.maa.org/hedi/HEDI-2009-02.pdf http://eulerarchive.maa.org/hedi/HEDI-2009-02.pdf [...] wrote this up in a paper https://scholarlycommons.pacific.edu/euleriana/vol3/iss1/3/ https://scholarlycommons.pacific.edu/euleriana/vol3/iss1/3/ Neat! It's not clear that Euler ever realized anything about calculating an arbitrary binary digit, but it wouldn't have been too far a leap to get there. For what it's worth, the formula (13) your paper credits to Hutton was also known to Machin in 1706. As was the formula about which Sandifer says "Without citing any particular formula, Euler proclaims that ...". The famous "Machin formula" just happened to be the one that Jones published along with an accurate π approximation in Synopsis Palmariorum Matheseos, but Machin had worked out several others. See Tweddle, Ian (1991). "John Machin and Robert Simson on Inverse-tangent Series for π". Archive for History of Exact Sciences. 42 (1): 1–14. doi:10.1007/BF00384331. JSTOR 41133896. The transformation of the series for arctan to a faster-converging version which Sandifer discusses in the middle of that paper was first described by Newton in an unpublished monograph from 1684. See: Roy, Ranjan (2021) [1st ed. 2011]. Series and Products in the Development of Mathematics. Vol. 1 (2 ed.). Cambridge University Press. pp. 215–216, 219–220. Newton, Isaac (1971). Whiteside, Derek Thomas (ed.). The Mathematical Papers of Isaac Newton. Vol. 4, 1674–1684. Cambridge University Press. pp. 526–653.
- Paul-Craft 3y agoI just skimmed through the MAA article and I'm reading your paper right now. I think it's supremely cool that people are still getting mileage out of papers published almost 250 years ago. One other cool thing about Euler and BBP-type pi series: Euler seems to have derived his results in a manner similar to how the famous BBP formula {\displaystyle \pi =\sum _{k=0}^{\infty }\left[{\frac {1}{16^{k}}}\left({\frac {4}{8k+1}}-{\frac {2}{8k+4}}-{\frac {1}{8k+5}}-{\frac {1}{8k+6}}\right)\right]} is actually proven. A friend of mine gave the proof of the famous series result as an exercise in his honors calc 2 class one year. They had some fun with it. https://en.wikipedia.org/wiki/Bailey%E2%80%93Borwein%E2%80%93Plouffe_formula https://en.wikipedia.org/wiki/Bailey%E2%80%93Borwein%E2%80%9...
- pallas_athena 3y agohttps://en.wikipedia.org/wiki/List_of_things_named_after_Leonhard_Euler https://en.wikipedia.org/wiki/List_of_things_named_after_Leo...
- seanhunter 3y agoEuler's wikipedia page has one of the most casually jawdropping sentences I've ever read about a human being: "Euler's work averages 800 pages a year from 1725 to 1783. He also wrote over 4500 letters and hundreds of manuscripts. It has been estimated that Leonard Euler was the author of a quarter of the combined output in mathematics, physics, mechanics, astronomy, and navigation in the 18th century." A quarter of all the output in so many fields. For a whole century. When you think about everyone else who was contributing to science at that time. It's just completely staggering.
- rigid 3y agoI always stress how our public education is broken since it can't handle extreme talent very good. Important breakthroughs that advance one or more fields extremely or shifting paradigms completely, were done or prepared by whizzkids. In these times, we need every whizzkid we can get.
- truckerbill 3y agoEducation is a cog/mandarin factory in most countries. Whizzkids will educate themselves, what's needed is giving people idle time in order to pursue things. Most influential thinkers found themselves with this in some fashion. How much talent is wasted making people jump through hoops in academia/finance/ad-tech? A lot of pre-industrial thinkers were associated with the clergy because they received tax money from peasants.
- johngossman 3y agoReading Leo Szilard biography right now. School was easy and he just read all the time on his own. Which is pretty much (I’m no Szilard) how it was for me too. School boredom as a recipe for success?
- euroderf 3y agoAlternatively, school boredom as a recipe for being the class clown.
- norir 3y agoIt's interesting to me that I can't think of anyone remotely comparable to Euler in the public consciousness today. Even the work of someone like Erdos seems very esoteric by comparison and also was largely done in collaboration. Was Euler just born at the right time and picking all the low hanging fruit? Or maybe his immense creative production was a unique consequence of wealth plus limited distractions? I'm inclined to believe there are similarly talented individuals today but wonder if it is even possible to fully recognize them in the moment. Perhaps we will only discover the Eulers (and Shakespeares and Bachs) of today in a few hundred years time.
- arjun_krishna1 3y agoKarpathy! Carmack!
- bowsamic 3y agoCarmack? I love him but that can’t be a serious suggestion
- croutons 3y agoYeah Carmack is no doubt a 20x engineer, but I wouldn’t put him in the same category of prolific genius as Euler. As far as prolific engineers go, I’d say Fabrice Bellard would be closer (but still not even close) to an Euler.
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- FredPret 3y agoIt takes time for true quality to become apparent. Unfortunately the human who produced the work is long dead by then. If humans ever started living much longer, we'd probably see a different attitude towards work.
- ufocia 3y agoMaybe today's emphasis on "well rounded" education is a distraction from specialized talents.
- smokel 3y agoIs it possible that these prolific geniuses had smart people working for (or with) them, while taking most credit? I could imagine that in 300 years time people think that Elon Musk single-handedly invented the Turing machine, at the age of 16, during a weekend, while reading Hacker News. How would one go about disproving such an hypothesis? I've had similar doubts about Leonardo da Vinci, but I'm afraid Euler was actually just brilliant.
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- oglop 3y agoThere's ample evidence from the time that rather than take credit, as _many_ unethical scientists have tried (*cough Newton), Euler was quite the opposite. I think one of his strengths was in his prolific collaborations actually, or taking up questions others sent to him which he did not need to work on, but would out of curiosity or decency. I get why someone not familiar with him may think this, but when I think of the word "genius" only this man, von Neumann, Ramunajan and Grothendieck come to my mind. They simply saw the world differently.
- vacuity 3y agoEuler went blind in one eye, and remarked "now I will have fewer distractions". He eventually became almost blind entirely, but with the aid of scribes, his productivity actually increased. It's remarkable how Euler didn't let being blind hinder him. A truly astounding mathematical career.
- johngossman 3y agoIs there a book other than Bell’s “Men of Mathematics” that broadly covers the lives of a lot of mathematicians? I loved that book when I read it as an undergraduate, but I wouldn’t mind having a second opinion.
- vehicles2b 3y agoPrinceton Companion to Mathematics has a section (part) on mathematicians
- agnosticmantis 3y agoWhat I really like and admire about Euler is how masterfully he handled infinities and infinitesimals to arrive at correct conclusions (the vast majority of times anyway) even though analysis hadn’t been made rigorous yet (by Cauchy and Weierstrass and friends), so some of what he did was pure magic and took a lot of good intuition. An example of this is when, in solving the then notorious Basel problem, he factors trig functions into infinite products of (x +- k*pi) terms just by analogy with root factorization in finite polynomials.
- guimplen 3y agoYeah, the greatest Russian mathematician.
- testless 3y agoI think Swiss. He was born in Basel. [1] https://en.wikipedia.org/wiki/Leonhard_Euler https://en.wikipedia.org/wiki/Leonhard_Euler
- guimplen 3y agoHe worked in Russia more than in any other country and died in St Petersburg being a Russian citizen.
- eichin 3y agoOh, I thought this was riffing on the "shakespeare, in the original klingon" line.
- madcaptenor 3y agoI wonder what he would have considered himself - he spent more time in St Petersburg than anywhere else but was born in Basel and spent a good bit of time in Berlin. For what it's worth, the Opera Omnia are published by the Swiss Academy of Sciences.
- vkazanov 3y agoI think what parent meant was that Euler spent a few decades in Russia because of the funding provided by the empire. He spoke fluent Russian, even though there was a large German-speaking community there. But it was typical for scientists to travel far for money. Some of the Bernoullis, a family famous for mathematicians, also worked in Russian for quite a while. Does it really matter who payed 'em and what languages they spoke?
- testless 3y agoI was just surprised;-)
- oglop 3y agoEuler's textbooks were such a gift to Europe. I'm always a bit perplexed he doesn't get more credit for this given he wrote some of the first calculus books people could read, because Newton had not cared or tried to. So in many ways Euler was also just an incredible educator and largely lead the charge spreading many of these new ideas. Usually credit is given to Chatelet, but she didn't really write any textbooks, just a commentary to help spread Newton's mechanics. Euler actually wrote the first bona-fide textbook on the matter. Laplace was being literal when he said "Read Euler, read Euler, read Euler". I own a copy of his "Elements of Algebra" and it's interesting to read because he actually talks and uses the notion of infinitesimals in this basic algebra book. And it makes sense! He essentially just says "think of the biggest number, make it even bigger!!! Now, put it under 1, and just like that we 'get almost zero'" You would never see something like that now, or even then really, and yet the idea is so simple a kid understands. His writing just has such an optimistic and playful sense to it.
- mncharity 3y ago> You would never see something like that now I've been wondering if an llm might be tuned to recognize insightful explanations that are accessible and powerful, to then help train one for creating such. Trying for an AI tutor that's less like drilling textbook bogosity and superficiality, and more like tutoring by that rare someone known for their outlier-deep understanding of a field. Even if an llm only manages to serve as a delivery mechanism for human curated insights, having a deployment story of very widespread impact might help motivate a novel-y broad contribution and curation effort.
- pohl 3y agoMy favorite thing from Euler's body of work is the circle-of-fifths on steroids, the tonnetz. https://en.wikipedia.org/wiki/Tonnetz https://en.wikipedia.org/wiki/Tonnetz https://www.youtube.com/watch?v=nidHgLA2UB0 https://www.youtube.com/watch?v=nidHgLA2UB0
- chris_wot 3y agoHmmm... our corporate firewall is blocking the connection as it says the SSL cert is "insecure". Something tells me this is BS, but anyone got any ideas why it might say this?
- ketzu 3y agoFirefox says the certificate uses a disabled signature algorithm `SEC_ERROR_CERT_SIGNATURE_ALGORITHM_DISABLED` The certificate is signed using a SHA-1 signature, which is considered insecure by all major browsers afaik. Although, inspecting the certificate also gives me validity: Not After Fri, 05 Dec 2014 12:00:00 GMT Which is slightly out of date. Even the CA cert it shows me has Not After Sun, 03 Apr 2022 00:00:00 GMT