| ▲ | How Isaac Newton discovered the binomial power series (2022)(quantamagazine.org) |
| 90 points by FromTheArchives 5 days ago | 18 comments |
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| ▲ | svat 2 days ago | parent | next [-] |
| The binomial theorem (though here Newton is still talking “powers of 11”) is apparently very deep. The algebraic geometer S. S. Abhyankar, in his article “Historical Ramblings in Algebraic Geometry and Related Algebra” that won a couple of writing awards, speaking of “high-school algebra”, “college algebra” and “university algebra”, gives as his thesis: > The method of high-school algebra is powerful, beautiful and accessible. So let us not be overwhelmed by the groups-ring-fields or the functorial arrows of the other two algebras and thereby lose sight of the power of the explicit algorithmic processes given to us by Newton, Tschirnhausen, Kronecker, and Sylvester. and goes on to write: > Personal Experience 1. In my Harvard dissertation (1956, [2]) I proved resolution of singularities of algebraic surfaces in nonzero characteristic. There I used a mixture of high-school and college algebra. After ten years, I understood the Binomial Theorem a little better and thereby learned how to replace some of the college algebra by high-school algebra; that enabled me to prove resolution for arithmetical surfaces (1965, [4]). Then replacing some more college algebra by high-school algebra enabled me to prove resolution for three-dimensional algebraic varieties in nonzero characteristic (1966, [5]). But still some college algebra has remained. > I am convinced that if one can decipher the mysteries of the Binomial Theorem and learn how to replace the remaining college algebra by high-school algebra, then one should be able to do the general resolution problem. Indeed, I could almost see a ray of light at the end of the tunnel. But this process of unlearning college algebra left me a bit exhausted; so I quit! |
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| ▲ | stevenjgarner 2 days ago | parent | prev | next [-] |
| I think the significance of the Epistola Posterior only really emerged decades after it was written, when the bitter Newton–Leibniz priority controversy erupted. It served as Newton’s key evidence, by revealing that he had the main ideas well before Leibniz’s first publications (1684). The letter bolstered the Royal Society’s eventual judgment favoring Newton’s independent discovery. |
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| ▲ | dotancohen 2 days ago | parent | prev | next [-] |
| I honestly feel that with entertainment as accessible as it is today, almost any mind that could come up with this today would be immediately distracted away. |
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| ▲ | dhosek 2 days ago | parent [-] | | I don’t. My 11-year-old son managed, without knowing algebra, to find the square root of i without guidance. His technique was not dissimilar to what Newton described. And while he’s bright, I don’t know that I would put him in a class with Newton. Even with distractions available, people are still able to focus on questions like these and come up with solutions. | | |
| ▲ | 11101010001100 a day ago | parent | next [-] | | This stuff always makes me chuckle. Magnus Carlsen learned chess without guidance, modulo his father being a rated player. | |
| ▲ | thaumasiotes 2 days ago | parent | prev [-] | | > My 11-year-old son managed, without knowing algebra, to find the square root of i without guidance. I'm curious what this means. To my mind, "without guidance" and also without knowing algebra, he wouldn't know what i was, and would therefore be unable to do any computation with it. | | |
| ▲ | dhosek 19 hours ago | parent [-] | | He knew that i was the square root of 1 and he had learned some really basic skills like the distributive law, so he managed to work out that (a+bi)(a+bi) = a²-b²+2abi, but it was entirely self-directed on his part. I had taught him some how to solve simple linear equations but he didn’t have the skills to be able to work out that he needed a=b and 2ab=1 so there were some trial and error attempts at finding values that worked. (I did the same thing as he did, but in high school sophomore year while bored in health class and with a bit more sophistication mathematically than what he had.) |
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| ▲ | DoctorOetker 4 days ago | parent | prev [-] |
| lots of "math processing error". perhaps just render the formulas? |
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| ▲ | freehorse 2 days ago | parent | next [-] | | I dont have such errors in neither firefox or safari, all latex is rendered fine; but some of the equation blocks are images vs processed latex, and this is annoying with trying to render a dark theme with dark reader. | |
| ▲ | Icy0 2 days ago | parent | prev | next [-] | | I'm not getting any processing errors. Seems to render fine on both Chromium and Firefox. | |
| ▲ | cuttothechase 2 days ago | parent | prev | next [-] | | yep, on Google chrome it all appears as unrendered math latex variables. This article is unreadable because of this. | | | |
| ▲ | garyrob 2 days ago | parent | prev | next [-] | | Works for me in Firefox; fails in Safari. | | |
| ▲ | defanor 2 days ago | parent | next [-] | | I get those "math processing errors" in Firefox, after some time. Some of the error messages, which appear right before those: GET https://www.quantamagazine.org/wp-content/themes/quanta2024/frontend/js/mathjax/fonts/HTML-CSS/TeX/png/imagedata.js?V=2.7.0 NS_ERROR_CORRUPTED_CONTENT
The resource from “https://www.quantamagazine.org/wp-content/themes/quanta2024/frontend/js/mathjax/fonts/HTML-CSS/TeX/png/imagedata.js?V=2.7.0” was blocked due to MIME type (“text/html”) mismatch (X-Content-Type-Options: nosniff).
Loading failed for the <script> with source “https://www.quantamagazine.org/wp-content/themes/quanta2024/frontend/js/mathjax/fonts/HTML-CSS/TeX/png/imagedata.js?V=2.7.0”. how-isaac-newton-discovered-the-binomial-power-series-20220831:1:1
Uncaught TypeError: c.FONTDATA.FONTS.MathJax_Main[8212][5] is undefined
The URL leads to error 404. | |
| ▲ | saagarjha 2 days ago | parent | prev [-] | | Works fine for me in Safari. |
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| ▲ | bell-cot 2 days ago | parent | prev [-] | | If you have a js blocker, try blocking the js from disqus.com. |
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