[HN Gopher] How Euler Did It, by Ed Sandifer
___________________________________________________________________
How Euler Did It, by Ed Sandifer
Author : nyc111
Score : 136 points
Date : 2024-01-24 15:17 UTC (7 hours ago)
(HTM) web link (eulerarchive.maa.org)
(TXT) w3m dump (eulerarchive.maa.org)
| nickcw wrote:
| I 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
|
| 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 p which can be used to
| calculate any binary digit of p 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/
| seanhunter wrote:
| Part 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 wrote:
| A 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 wrote:
| How is this humanly possible? Was Euler even an order of
| magnitude faster at producing new math than, say, Gauss or
| von Neumann?
| karmakurtisaani wrote:
| If I recall correctly, Euler has the most pages of
| published math. Erdos has the most papers (some of them not
| more than a handful of sentences).
| TuringTourist wrote:
| Erdos has a huge amount of credits in the papers of
| others (hence Erdos number) because he would just travel
| all over the country helping people get unstuck on their
| work.
| chongli wrote:
| Euler 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.
| 3abiton wrote:
| Yet I struggle to adjust to one note taking app to optimize my
| workflow. The paradoxal curse of choice.
| g9yuayon wrote:
| I always feel amazed that Euler wrote faster than people could
| publish or understand his work, even after he became completely
| blind.
| jacobolus wrote:
| > _In particular I read this one:_
| http://eulerarchive.maa.org/hedi/HEDI-2009-02.pdf [...] _wrote
| this up in a paper_
| 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 p 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 _p_ ". _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 wrote:
| I 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%9...
| seanhunter wrote:
| Euler'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 wrote:
| I 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 wrote:
| Education 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 wrote:
| Reading 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 wrote:
| Alternatively, school boredom as a recipe for being the
| class clown.
| Scene_Cast2 wrote:
| The issue I see these days is that every industry is
| getting more and more competitive, and leaves less and less
| time to think more broadly or creatively. Can't go off
| reading about differential geometry when you need a
| guaranteed perfect SAT, a great entrance essay (i.e. a
| strong personal story), and easy-to-gauge extracurriculars
| ("placed X in Y", not "read some smart books and had some
| interesting thoughts that don't impress the admissions
| officer"). Same goes for the industry and academia as well.
| programjames wrote:
| I think top schools are much more likely to accept that
| kid with a 1550 on the SAT who spent his downtime
| studying differential geometry instead of the SAT.
| cyberax wrote:
| One thing (in retrospect) that I love about the xUSSR
| school system, is its focus on competitive math and
| physics.
|
| There's a robust multi-level system of "Olympiads",
| starting from the neighborhood level, and going all the way
| up to the national level. Every student knows about them,
| and more importantly, "magnet schools" scoop up students
| who do well in competitions.
|
| This works really well for math, and so pretty much every
| Fields medal award ceremony has awardees from the xUSSR
| countries.
|
| I'm really surprised that this kind of system is not more
| widespread, especially in the US. After all, sports and
| competition is kinda a thing here?
| rigid wrote:
| > Whizzkids will educate themselves
|
| Only those you see becoming one.
|
| You never hear of all the "Einsteins" who never leave the
| patents office because they never got inspired for some
| passion or various other stupid reasons.
| kenjackson wrote:
| No system will handle an Euler well. It's best to recognize
| them and move them out of the system.
| rigid wrote:
| You're basically saying the system is obsolete when society
| reaches a point, where it needs no more mediocre
| generalists and only excellent specialists.
|
| I wouldn't be so pessimistic.
| pohl wrote:
| That's the least charitable interpretation, I think. A
| more charitable read is that no system should be one-
| size-fits-all, so move outliers to specialized
| subsystems.
| testless wrote:
| Let's hope education indivualizes to the need and talents of
| the individual partially via software. ChatGPT is my hope
| here.
|
| 15% failure rate is optimal for learning
|
| https://www.nature.com/articles/s41467-019-12552-4
| liendolucas wrote:
| Seriously? Oh god, please no. We need less of all that BS.
| People are thinking that are being "tutored" by AI, when in
| fact is just the output of a number crunching program.
| Reading books will get you way way closer to solid
| knowledge than the output crap of this so called "AI".
| vacuity wrote:
| I don't think such an attitude is warranted. While I'm
| skeptical about the potential of LLMs as AGI (whatever
| that means), being able to summarize subjects and even
| come up with test questions can be very valuable for
| learning. I am concerned about the confabulation aspect,
| though. I wonder if there could be a uncertainty metric
| for that.
| testless wrote:
| Happy to disagree here. I am hopeful that students get
| apps which provide them with - fast feedback loops -
| better adjustment to their learning speed - more patience
| - gamification - oppurtunity to ask endless questions
|
| Of course, only as an addition to the current mix. For
| math problems, this will be easier than for other
| contexts.
|
| I like text books for pop science or really hard things -
| like university level education. But I am surprised to
| see them as an option for basic education.
| Paul-Craft wrote:
| There's a reason so many things are named after the second
| person to discover something after Euler: mf died in 1783 and
| is still publishing papers today.
| oglop wrote:
| Wait until you find out he was totally blind the last 15ish
| years of his life _and wrote more during that period than any
| other_. Scribes would just sit and he would talk while they
| wrote. He literally died while doing a calculation in this
| manner dealing with the hot new invention of Ballooning.
| norir wrote:
| It'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 wrote:
| Karpathy! Carmack!
| bowsamic wrote:
| Carmack? I love him but that can't be a serious suggestion
| croutons wrote:
| Yeah 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.
| FredPret wrote:
| It 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 wrote:
| Maybe today's emphasis on "well rounded" education is a
| distraction from specialized talents.
| The_Colonel wrote:
| Specializing is actually the "easier" path, and thus there
| isn't a lot of low hanging fruit anymore. There are more
| opportunities in the intersections of different fields (areas
| of mathematics).
| chongli wrote:
| Euler did his master's dissertation on the philosophies of
| Descartes and Newton. He then joined the faculty at
| University of Basel in theology. I think this is ample
| evidence that he received a well-rounded education.
| jacobolus wrote:
| Not to mention his day job for a while as the chief
| cartographer in the Russian Empire.
| jterrys wrote:
| Absolutely HARD disagree. Pythagoras, Aristotle, Pascal,
| Descartes, Newton, Franklin, DaVinci, (just to name a few)
| were all polymaths. They all left lasting impacts outside of
| a singular domain. It's because they were such well rounded
| individuals that they were so successful and capable of
| revolutionary discoveries and inventions. Our world is made
| up of models that are related with other models that are
| related based off of that relatedness. It's maddening. Being
| "well rounded" is vital.
|
| Today that concept is watered down. A "well rounded"
| education is just taking a few classes that people hate and
| will blow off because to graduate they need to check some
| boxes so they can focus on doing one thing moderately well
| and finding their place as a cog in a machine that will abuse
| their ignorance. It's all mass produced conveyor belt
| education that manufactures young adults with little
| conventional wisdom. The more you lean into behaving like a
| part, the more you will be treated that way.
| Duanemclemore wrote:
| Euler was nothing if not diverse. Having a "liberal arts" /
| "great books" education from high school I have (anecdotal)
| first hand evidence in the impressive successes of my
| classmates in a WIDE variety of fields.
| feoren wrote:
| Part of it is low-hanging fruit, certainly. Euler lived at a
| time when it was still possible to "know all math". That
| breadth of knowledge is simply not possible for a single human
| anymore; the discipline of mathematics is orders of magnitude
| larger. Comparable mathematicians today like Erdos and Terence
| Tao collaborate because they really can't learn the intricate
| details of every corner of math, so they collaborate with
| people who work in those corners instead. I think these are all
| people that have a deep understanding of the interconnected
| structures within math, and that makes them incredibly
| productive. I'm not going to try to compare the "level of
| genius" between Euler, Erdos, and Tao (although I think the
| latter two would readily claim Euler wins), but once you have a
| gift like that, there's a big difference between having all of
| mathematics in your head and not.
| lp4vn wrote:
| The low hanging fruit argument only takes you so far. How
| many other mathematicians in his epoch or before were able to
| pick as many low hanging fruits as him?
| feoren wrote:
| By all means he was a crazy outlier generational genius. A
| few others in history, like Archimedes and Newton, have
| been accused of "not leaving anything for anyone else to
| discover" as well. The question was: why do we not seem to
| see these crazy outliers anymore? The answer is certainly
| not that truly exceptional people simply stopped being born
| after the year 1800. The nature of what it could mean to
| "know everything" about a field has completely changed.
| lupire wrote:
| Also they lived in a time with far less education and
| communication.
|
| Newton and Leibniz discovered calculus simultaneously. If
| calculus were a hot new idea now, dozens or hundreds of
| people would be discovering it simultaneously.
|
| Look at NN/LLM AI for an example.
| norir wrote:
| To be clear, I am not saying that Erdos and Tao are less
| talented than Euler. That seems impossible to say. But the
| magic of Euler is that his fingerprints are all over so many
| of the fundamental things that can be understood by a bright
| high school student but were mostly unknown before him. The
| work of figures like Erdos and Tao seems far, far less
| accessible in its present form at least and thus more limited
| in its overall impact.
| feoren wrote:
| Part of that, too, is the "founding father" effect. Most
| humans alive today are related to Genghis Khan and/or
| Charlemagne. It just takes time for new ideas to dissipate
| and cross-breed.
| zyklu5 wrote:
| I find the notion of 'low-hanging fruit' in such contexts
| profoundly ahistorical. If graph theory was low-hanging why
| did it take thousands of years since Sumer or ancient Egypt?
| Or consider something from number theory: every other batch
| of students in a math camp I'm familiar with has someone who
| has 'proved' quadratic reciprocity for themselves -- and how
| could they not ? -- since childhood they have been immersed
| in a culture which points at it; while it took Euler roughly
| 40 years to even formulate the idea and then Gauss to prove
| it. It's not that - to use an anachronistic term - class
| field theoretic phenomena was not known to other cultures
| thousands of years ago.
|
| (Btw there are many contemporary mathematicians at least at
| the level of Terence Tao but for some reason haven't been
| blessed by lay popularity -- mostly because Tao's math looks
| more school math like / familiar to non-math people than say
| Peter Scholze's)
| chongli wrote:
| It's low-hanging fruit because it depended on a bunch of
| mathematics and technology that Euler benefitted from:
| algebra, the printing press, and mass-produced paper. Sure,
| the Ancient Egyptians had papyrus but that is nothing
| compared to the volume of paper Euler had available to him.
|
| Euler lived nearly three centuries after the invention of
| the printing press. He had vast numbers of books available
| to him and essentially unlimited paper to write on. He also
| had been tutored in algebra which remains the most
| important development in the history of mathematics. The
| abstract manipulation of symbols made possible by algebra
| is such an enormous leap over the geometric methods of the
| ancient mathematicians. It allows one to solve countless
| problems trivially in seconds which would take days to
| solve geometrically.
| feoren wrote:
| > If graph theory was low-hanging why did it take thousands
| of years since Sumer or ancient Egypt?
|
| Because it _wasn 't_ low hanging thousands of years ago. It
| was only low hanging after an enormous body of foundational
| work was laid down over those thousands of years. And Euler
| _knew all of it_. It 's no longer possible to know all of
| mathematics.
|
| > every other batch of students in a math camp I'm familiar
| with has someone who has 'proved' quadratic reciprocity for
| themselves
|
| This is exactly my point though: things get easier to
| understand over time as the more foundational mathematics
| gets laid out to prepare for them. Nowadays some of this
| stuff is considered basic. It's very "low hanging fruit"
| now, it's just that those summer-camp kids aren't making
| the discovery for the very first time. What point exactly
| are you defending here?
|
| > Btw there are many contemporary mathematicians at least
| at the level of Terence Tao but for some reason haven't
| been blessed by lay popularity
|
| I'm not sure why this needs to devolve into a contest.
| Terence Tao, Peter Scholze, whoever: they can't _know all
| math_ anymore, like Euler did. That is ultimately why there
| are no more Eulers.
| Duanemclemore wrote:
| If not "low-hanging fruit" at least "first mover advantage."
| testless wrote:
| Ramanujan died early. He is another one which comes to mind.
| vacuity wrote:
| Riemann died young, too. I only really know him for the
| Riemann sum formulation of integrals (I have yet to learn
| complex analysis), but I wouldn't be surprised if he has some
| popular reach through the Riemann hypothesis.
| Cyph0n wrote:
| Galois is yet another.
| testless wrote:
| Is Euler really well known in public? Einstein is way more well
| known than Euler, I'd guess. Even if you will ask most famous
| mathematician, it is unlikely that he is named. Probably
| Pythagoras.
|
| I have no data to back it up.
| gregw134 wrote:
| Most people can't name any mathematicians, but amongst those
| that can I think he's top 5, behind more famous names like
| Newton, Archimedes and Euclid
| Tainnor wrote:
| And Gauss.
| norir wrote:
| Sure, I didn't really mean the general public but meant
| scientific/mathematically oriented public. When I studied
| math in college, I found Euler everywhere to the point that
| it was hard not to stop and wonder how the hell this guy had
| so many fundamental discoveries. By contrast, I have
| awareness of present day figures like Tao and Erdos, but I
| don't really understand their work and perceive it to be
| specialized enough that it is unlikely to be widely
| understood in the way that say e^(pi * i) + 1 = 0 is.
| lordnacho wrote:
| This is purely an indictment of modern mathematics education.
| Anyone doing ordinary high school math should have had their
| ears full of Euler's contributions for at least a couple of
| years towards the end. Math teachers who don't mention Euler
| need to be put onto a mock Konigsberg and forced to search
| for a solution.
| OldGuyInTheClub wrote:
| Euler is in a class of one. Judging by the volume of work and
| the pleasure he takes in collaborations, Terence Tao seems to
| be having a positive impact. I am not a mathematician but I
| hear good things about the quality of his work.
| karmakurtisaani wrote:
| > I am not a mathematician but I hear good things about the
| quality of his work.
|
| Not to be rude, but saying this about a Fields medallist is
| somehow very funny.
| OldGuyInTheClub wrote:
| I can see that. But, there are often controversies about
| who gets what award, even up to the Nobels. Since there are
| very highly qualified people posting here, I just wanted to
| be clear that I have no ability to understand Tao's work
| and am going by acclaim.
| CamperBob2 wrote:
| Stephen Wolfram's agent on line 2...
| chongli wrote:
| Von Neumann is the closest comparison I can think of. A child
| prodigy who made a very large number of contributions to
| mathematics, computer science, physics, and engineering.
| testless wrote:
| For modern mathematics, I think Alexander Grothendieck was one
| of the most influencial talents in the last century. He unified
| large branches of mathematics in a very short period. He was
| more a bird than a frog [1], however.
|
| [1] Bird vs. Frogs.
| https://www.ams.org/notices/200902/rtx090200212p.pdf
| smokel wrote:
| Is 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.
| oglop wrote:
| There'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 wrote:
| Euler 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 wrote:
| Is 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 wrote:
| Princeton Companion to Mathematics has a section (part) on
| mathematicians
| agnosticmantis wrote:
| What 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 wrote:
| Yeah, the greatest Russian mathematician.
| testless wrote:
| I think Swiss. He was born in Basel.
|
| [1] https://en.wikipedia.org/wiki/Leonhard_Euler
| guimplen wrote:
| He worked in Russia more than in any other country and died
| in St Petersburg being a Russian citizen.
| vkazanov wrote:
| I 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?
| oglop wrote:
| Euler'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.
| pohl wrote:
| My favorite thing from Euler's body of work is the circle-of-
| fifths on steroids, the tonnetz.
|
| https://en.wikipedia.org/wiki/Tonnetz
|
| https://www.youtube.com/watch?v=nidHgLA2UB0
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(page generated 2024-01-24 23:00 UTC)