How Newton discovered Taylor series (but didn't tell anyone)

In this video we explore the independent discovery of Taylor series by Isaac Newton and the manuscript which "never left Newtons hands".
Chapters
0:00 - Introduction
01:55 - James Gregory's discovery
04:22 - Isaac Newton's discovery
06:23 - Outro
In-video References:
1. Meijering, E., 2002. A chronology of interpolation: from ancient astronomy to modern signal and image processing. Proceedings of the IEEE, 90(3), pp.319-342.
2. Feigenbaum, L., 1985. Brook Taylor and the method of increments. Archive for history of exact sciences, 34(1), pp.1-140.
FAQ : How do you make these animations?
Animations are mostly made in Apple Keynote which has lots of functionality for animating shapes, lines, curves and text (as well as really good LaTeX). For some of the more complex animations, I use the Manim library. Editing and voiceover work in DaVinci Resolve.
Supporting the Channel.
If you would like to support me in making free mathematics tutorials then you can make a small donation over at
www.buymeacoffee.com/DrWillWood
Thank you so much, I hope you find the content useful.

[@topdog5252]

It’s so tragic what happened to James Gregory. I heard from a video of historian of mathematics, Jacqueline Stedall that Gregory never published his discoveries because he kept on waiting for Newton to publish first, and of course Newton never published and so Gregory just died in the end having never published and now people are less aware of his work. This also means it is harder for historians to work out exactly what he did.

[@umami0247]

So at least two others used this series yet didn't get credit for doing so. Technically this should be the Gregory Newton Taylor series? Or the GNT series for short.

[@DrWillWood]

It goes even further than this! Feigenbaum (1985) lists Leibniz, Johann Bernoulli, and de Moivre as 3 more who "anticipated" Taylor series. Bernoulli outright accused Taylor of plagiarism!!

[@RudraC]

@@DrWillWood ruclips.net/video/22uZ3D5AgaE/видео.html

[@BarkanUgurlu]

Very intriguing. Thanks a lot for this video, I truly enjoyed it.

[@jamesbentonticer4706]

Makes me wonder what great maths discoveries are out there but the authors just don't publish or tell anyone.

[@Infinium]

Such a good video, thanks soo much for sharing!

[@DrWillWood]

thanks a lot!

[@dvir-ross]

fascinating!

[@kensandale243]

The method is quite clever, but not actually a true proof.
The method assumes that you *can* represent a function by a polynomial series. If you can, then that indeed is the series. But how do you know that you can?
There are indeed functions that cannot be represented that way. Often though, in practice, they are obvious. What often happens is that some nth derivative at x =0 is undefined. For example, you cannot represent y = 1/x by a Taylor series about x =0. The general rule, though is not so trivial.

[@DrWillWood]

Absolutely! thanks for making that point. I believe Taylor also made the same assumption, certainly he made the assumption that the limit exists. From Feigenbaum (1985) (link in description) "Neither the convergence nor the validity of [the] series is mentioned". Feigenbaum also gives a great quote from Felix Klein "We have here, in fact, a passage to the limit of unexampled audacity". I suppose it was a time when methods were evolving too fast for analysis to keep up!

[@BrollyyLSSJ]

It's good to note that even if the function is smooth (all the derivatives are defined), it's still not enough to equate it with its Taylor series (en.m.wikipedia.org/wiki/Non-analytic_smooth_function)

[@kensandale243]

@@BrollyyLSSJ "t's good to note that even if the function is smooth (all the derivatives are defined), it's still not enough to equate it with its Taylor series "
Yes, I was aware of regions of convergence, and mentioned somewhat subtly that the situation was more complicated than just the derivatives existing.

[@nafisiqbal4316]

Thank you very much for this content. I enjoyed it very much.
But I have a question about the proof of Taylor Series. To prove the series, we need to use the power rule of derivatives, thus we have to prove the power rule before proving the Taylor series. To prove the power rule, we need to use generalized binomial theorem. Now in most textbooks that I have read, "generalized" binomial theorem is proved, primarily, in two ways. First one is the method of induction, which is actually a proof for natural number, not for the general case. And the second method uses power rule, or in many books, the Taylor series itself. So proving Taylor series becomes a case of round-about logic. Atleast one of the theorems (general binomial/power rule/Taylor series) has to be a postulate. Right? Or am I wrong somewhere? If one of them is a postulate, which one would that be?

[@DrWillWood]

Thank you! interesting point, I cant see a way to derive Taylor series without the power rule for differentiation. In my opinion I think the integer version of the proof of the power derivative is fine since for Taylor series the exponents are natural numbers. then it would be fair to use Taylor series in any proof, including of the generalised binomial theorem! though admittedly I haven't seen the proof.

[@faithlesshound5621]

I'm glad not to be the only one to think that this whole field, at least as demonstrated to beginners, seems to involve circular reasoning.

[@nafisiqbal4316]

@@DrWillWood Thank you very much!

[@goblin5003]

I don’t get where is the contradiction exactly
Can any of you help me?
Or maybe i’m misunderstanding what the generalized binomial theorem is???

[@hhhhhh0175]

Generalized binomial theorem is completely fine to use for Taylor series since all the powers are natural numbers. To prove the power rule for non-integers, you can use the generalized product rule for rational powers > 0 (and then continuity of exponentiation for reals? I'm not sure)
(Also, if you have exp(x) and ln(x), proving the power rule for real N is trivial, but I don't know nearly enough analysis to determine whether or not that's circular reasoning)

[@tjalferes]

Thanks Doc

[@robert-skibelo]

Most interesting. Thanks for posting. BTW, you mispronounce "corollary".

[@bryanfuentes1452]

imagine people having difficulty understanding these stuffs. Inventing it is another level.

[@haronka]

Newton literally wouldn't have told anything for the World if it wasn't for others discovering the same stuff, it looked like this for literally everything:
Other person: Hey, I discovered this
Newton: I discovered it years ago, but haven't told anyone, I have proof
The World: WOW, NEWTON DISCOVERED IT!!!
Other person: Cries

[@bwoy12345]

New proof method just dropped by Turnbull: Proof by Error

[@mobilephil244]

The entire history of maths is the story of duplication, and wasted careers, so none of this surprises me. What DOES intrigue me is that 1) back in/before the ice age, the human brain easily had all the same functionality of modern humans. 2) Maths and physics prodigies have always popped up. 3) How many times in ancient history was some of modern maths and physics discovered by geniuses that probably didn't even have a name let alone any written language to record their findings. I mean, Ramanujan could have lived at ay time in the last 2 million years.

[@doraemon402]

Newton did not discover Taylor series, there's ample historical evidence of that.

[@mastershooter64]

nobita disagrees

[@rjms06]

care providing any

[@gw7624]

There's ample evidence you're a complete and utter blockhead.

[@RudraC]

ruclips.net/video/22uZ3D5AgaE/видео.html

[@kylenetherwood8734]

Honestly, if you keep your discovery a secret, you don't deserve the credit.

[@RudraC]

Taken from India , madhavacharya's work.

[@gw7624]

Bullplop.