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."
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to a faster-converging version which Sandifer discusses in the middle of that paper was first described by Newton in an unpublished monograph from 1684.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.
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.
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
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.
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.
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 Erdős 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, Erdős, 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.
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)
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.
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?
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.
Reminds me of the early days of computer science, the 60s and 70s.
I once casually flipped through the CS journal papers archived in my university library, and those two decades were really fascinating.
1) All of the papers were simple, clear, and easy to follow. No greek symbols, no fancy maths, just straighforward pseudocode.
2) The algorithms being presented were new at the time, but were rather trivial, and could have been invented by any of us. They just hadn't been invented, formalised, and written down yet. It was basically a race starting from zero, and nobody had taken very many steps yet.
3) Every new algorithm was such a huge leap that it unlocked many new avenues of advancement for other algorithms. There was a period where it was a race of publishing these follow-up developments nearly as fast as people could type.
For a modern example of this, look at the pace of progress of generative AI such as LLMs and Diffusion Models. The first papers were almost trivial, but that "one clever trick" unlocked many more ideas and the rate of publication over the last twelve months has been insane. A lot of low-hanging fruit, a lot of road-blocks suddenly removed, etc...
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.
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.
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 Königsberg and forced to search for a solution.
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.
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.
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.
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.
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).
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.
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.
Edward Witten bestrides mathematical physics in an Euler-likes way. In math/physics circles, ask the smartest person you know who is the smartest person they know and sooner or later all roads lead to him.
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.
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.
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.
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.
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.
Yep. But to be really scared of the man you can consider that during the later years of his life he lost eyesight, first in one eye, then went completely blind. All without slowing down.
The anecdote goes that with his eyesight already severely deteriorated, he would dictate papers with a grandkid or two in his lap and a cat on his shoulder.
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.
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.
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.
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.
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.
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.
Readit News
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 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/
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.
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...
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(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)
I once casually flipped through the CS journal papers archived in my university library, and those two decades were really fascinating.
1) All of the papers were simple, clear, and easy to follow. No greek symbols, no fancy maths, just straighforward pseudocode.
2) The algorithms being presented were new at the time, but were rather trivial, and could have been invented by any of us. They just hadn't been invented, formalised, and written down yet. It was basically a race starting from zero, and nobody had taken very many steps yet.
3) Every new algorithm was such a huge leap that it unlocked many new avenues of advancement for other algorithms. There was a period where it was a race of publishing these follow-up developments nearly as fast as people could type.
For a modern example of this, look at the pace of progress of generative AI such as LLMs and Diffusion Models. The first papers were almost trivial, but that "one clever trick" unlocked many more ideas and the rate of publication over the last twelve months has been insane. A lot of low-hanging fruit, a lot of road-blocks suddenly removed, etc...
If humans ever started living much longer, we'd probably see a different attitude towards work.
I have no data to back it up.
[1] Bird vs. Frogs. https://www.ams.org/notices/200902/rtx090200212p.pdf
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.
Not to be rude, but saying this about a Fields medallist is somehow very funny.
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.
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.
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.
In these times, we need every whizzkid we can get.
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.
15% failure rate is optimal for learning
https://www.nature.com/articles/s41467-019-12552-4
The anecdote goes that with his eyesight already severely deteriorated, he would dictate papers with a grandkid or two in his lap and a cat on his shoulder.
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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.
https://en.wikipedia.org/wiki/Tonnetz
https://www.youtube.com/watch?v=nidHgLA2UB0
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.
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.
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