Hacker News Reader

Comment by nickcw | Hacker News Reader

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 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/

View on HN

> 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 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to a faster-converging version which Sandifer discusses in the middle of that paper was first described by Newton in an unpublished monograph from 1684.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.

A while indeed. Euler and Bach are similar in that sense: to properly ingest their life's output you need more than one life.

How is this humanly possible? Was Euler even an order of magnitude faster at producing new math than, say, Gauss or von Neumann?

If 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).

Erdős 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.

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.

There’s more about the effort to publish his complete works here: https://en.m.wikipedia.org/wiki/Opera_Omnia_Leonhard_Euler

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...

Yet I struggle to adjust to one note taking app to optimize my workflow. The paradoxal curse of choice.

https://en.wikipedia.org/wiki/List_of_things_named_after_Leo...

I always feel amazed that Euler wrote faster than people could publish or understand his work, even after he became completely blind.

Reply on news.ycombinator.com