What Hacker News says

yesenadamMar 2021

He was writing one paper a week even after he became blind!

"So prolific was he that the journal of the St. Petersburg Academy was still publishing the backlog of his papers a full 48 years after his death." – Euler: The Master Of Us All

p.s. Last year I read his Introductio in analysin infinitorum (1748), "one of the most influential mathematics books of all time", which made the function concept basic in mathematics. Polynomials, trig functions, exponential functions, the logarithm function, etc etc - doing amazing things with infinite series. It was hard to believe it wasn't a thoroughly modernised text, everything looks so modern, but it's just that he introduced a lot of our notation. And he explains things so helpfully and modestly. A joy to read, highly recommended.

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gcanyonJan 2025

1. Socrates criticized writing itself: in Plato's Phaedrus he said it would "create forgetfulness in the learners' souls, because they will not use their memories" (274e-275b)

2. Leonard Euler criticized the use of logarithm tables in calculating: in his 1748 "Introductio in analysin infinitorum" he insisted on deriving logarithms from first principles

3. William Thomson (Lord Kelvin) initially dismissed mechanical calculators, stating in an 1878 lecture at Glasgow University that they would make students "neglect the cultivation of their reasoning powers"

4. Henry Ford in his autobiography "My Life and Work" (1922) quoted a farmer who told him in 1907 that gasoline tractors would "make boys lazy and good for nothing" and they'd "never learn to farm"

5. In 1877, the New York Times published concerns from teachers about students using pencils with attached erasers, claiming it would make them "careless" because they wouldn't have to think before writing. The editorial warned it would "destroy the discipline of learning"

6. In "Elements of Arithmetic," (1846) Augustus De Morgan criticized the use of pre-printed multiplication tables, saying students who relied on them would become "mere calculative mechanism" instead of understanding numbers

7. In his 1906 paper "The Menace of Mechanical Music," John Philip Sousa attacked the phonograph writing that it would make people stop learning instruments because "the infant will be taught by machinery" and musical education would become "unnecessary"

8. In his 1985 autobiography "Surely You're Joking, Mr. Feynman!" Richard Feynman expressed concern about pocket calculators and students losing the ability to estimate and understand mathematical relationships

I could go on (Claude wrote 15 of them!). Twenty years from now (assuming AI hasn't killed us all) we'll look back and think that working with an LLM isn't the crutch people think it is now.

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robert-wallisSep 2019

I recently learned:

> It has been claimed that Euler's identity appears in his monumental work of mathematical analysis published in 1748, Introductio in analysin infinitorum. However, it is questionable whether this particular concept can be attributed to Euler himself, as he may never have expressed it. https://en.m.wikipedia.org/wiki/Euler's_identity

But it is still cool.

https://youtu.be/yPl64xi_ZZA

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ChinjutAug 2016

One can actually derive the Euler product for sine perfectly rigorously quite quickly, in just the manner Euler did!

(Euler is often said to have glibly moved from "sin(x)/x has roots at nonzero integer multiples of π" to "sin(x)/x = the product of (1 - x/(kπ)) for each nonzero integer k" in solving the Basel problem, simply ignoring the problem of the fact that many other functions have precisely the same zeros.

But Euler was not, in fact, always so glib in presenting this argument! For example, in Volume 1, Chapter 9 of his Introductio in Analysin Infinitorum (translation by Ian Bruce available at http://www.17centurymaths.com/contents/introductiontoanalysi...), we see that Euler argues for the product formula by noting (what would in modern notation be) that sin(x) = the imaginary part of e^(ix) = the imaginary part of (1 + ix/N)^N for infinitely large N, so that a factorization of sin(x) can be extracted from a factorization of the polynomial Im[(1 + ix/N)^N] for large N.

I'll write out in modern style the extraction of that factorization, but all the ideas are already present in the Euler:)

Let us denote Im[(1 + ix/N)^N] by f_N(x). Note that f_N is a polynomial of degree either N - 1 (if N is even) or N (if N is odd), whose degree 1 term is 1. Furthermore, its roots occur where x/N is the tangent of a multiple of π/N. Putting these together, we get that f_N(x)/x = the product of 1 - x/(N tan(kπ/N)) over the nonzero integers k in (-N/2, N/2).

As N approaches infinity, N tan(kπ/N) approaches kπ. Thus, we have that sin(x)/x = the product of 1 - x/(kπ) over the nonzero integers k.

[This last step is slightly glib, in that we've commuted limits without justification. We can rectify that by bundling together the factors where k differs only in sign, saying f_N(x)/x = the product of 1 - (x/(N tan(kπ/N)))^2 over k in (0, N/2). Now we note that the movement of the factors toward their limit is monotonic in N (considering the k-th factor to be 1 when N/2 <= k), which is sufficient justification for commuting the limits.]

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eric-huDec 2011

Leonhard Euler:

Born: 1707

completes masters in philosophy: 1723

lost vision in right eye: 1735 Solved 7 bridges problem: 1736

publish Introductio in analysin infinitorum, volume 1: 1748 [2]

Euler-Bernoulli Beam Theory: 1750 Euler's Formula (geometry): 1750 [1]

publish Institutiones calculi differentialis: 1755 [2]

near total blindness: 1766

publish Theoria motuum lunae: 1768

[1] http://www.ics.uci.edu/~eppstein/junkyard/euler/ [2] http://eulerarchive.maa.org/historica/euler-timeline.html

There's plenty more that's left out. After his near-total blindness, he averaged publishing one mathematical paper per week in 1775. To assume you or I are like Euler is hubris. To assume otherwise is doing a disservice to humanity.

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