# definition of derivative

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26 minutes ago, wtf said:

Studiot, I was very confused by your post.

First, yes dx and dy are not infinitesimals. I misread that part of your post.

But you said that "The quantities   δx   and  δyare infinitesimals. (Newton called them fluxions, not infinitesimals)"

I have two problems here. One, what are δx and δy? I looked back through the thread and could not find that notation defined. Clarify please?

Second, Newton called the derivative a fluxion. dx and dx aren't fluxions. The limit of delta-y over delta-x is the fluxion. Of course Newton didn't have the formal concept of limit but his intuition was pretty close.

Then you tried to argue that Newton wrote a book on fluxions. Um, yeah, he did. What does that have to do with what we're talking about? What we call the derivative, Newton called a fluxion. Neither derivatives nor fluxions are infinitesimal.

Finally, Newton tried several different approaches to clarifying what he meant by (what we now call) the limit of the difference quotient. He did NOT really espouse infinitesimals in the same way Leibniz did. That's the part that is historically arguable -- what Newton thought about infinitesimals.

To be clear:

* Fluxions are derivatives, not infinitesimals. (And fluents are integrals).

* Newton didn't really use infinitesimals as such in the strong way Leibniz did.

* Newton wrote books. But fluxions aren't infinitesimals. Nor did Newton think about dy and dx as infinitesimals. Not (as I understand it) in as explicit a way as Leibniz did.

I have never tried to read Leibnitz.

What do you understand by evanescent increments?

Quote

# Evanescent Increments

What are evanescent increments?

Part of the beauty of Calculus is– I don’t really know.  Not tangible, supremely debatable– Newton named them as a measure of the ratio of fluxions.

Maybe I'm wrong about some detail but

I don't know where the attribution of the ratio being the fluxion has come from, my sources seem to clearly indicate that Newton considered  ${\delta x}$  and

${\delta y}$ as fluxions.

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28 minutes ago, studiot said:

I have never tried to read Leibnitz.

What do you understand by evanescent increments?

Maybe I'm wrong about some detail but

I don't know where the attribution of the ratio being the fluxion has come from, my sources seem to clearly indicate that Newton considered  δx   and

δy as fluxions.

Studiot you still haven't told me what the notation δy means. I don't know what you mean by that notation.

Secondly I don't think we could have a sensible conversation about what Newton meant when he wrote something down in Latin that some historian translated as evanescent increments. We don't know what Newton was thinking. He was most likely thinking like a physicist. "It doesn't make mathematical sense but it lets me explain the apple falling on earth and the planets moving in the heavens by the same simple principles. So I'll just use it, and let the mathematicians try to sort it out for the next couple of centuries."

I do know that over his career, he explained his fluxions in several different ways. That shows he was well aware of the logical problem of a lack of rigorous foundation. However, my understanding is simply that whereas Leibniz was "Infinitesimals, dude!", Newton was more like "top and bottom close to zero, ultimate ratio is what I call the fluxion."

But we're not historians of science. A lot of people have written a lot of books about every detail of Newton's thought. In the end all I want to know is what you mean by curly delta so I can have some idea what you're talking about.

Um ... just realized this. Do you mean delta-x and delta-y? What's usually marked up as $\Delta$x and $\Delta$y?

Edited by wtf

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2 minutes ago, wtf said:

Studiot you still haven't told me what the notation δy means. I don't know what you mean by that notation.

I am waiting for your definition of an infinitesimal, that I asked for a while back.

I offered my best one.

2 minutes ago, wtf said:

Secondly I don't think we could have a sensible conversation about what Newton meant when he wrote something down in Latin that some historian translated as evanescent increments.

Going to a grammar school I did Latin. (That was the English definition of a grammar school)

evanescent increments was not translated by 'some historian'.

It was part of a very famous attack on Newton by the Church of his day.

7 minutes ago, wtf said:

He was most likely thinking like a physicist.

Yes I believe I said something similar in my first post.

7 minutes ago, wtf said:

Um ... just realized this. Do you mean delta-x and delta-y? What's usually written $\Delta$?

I don't know if you mean  $\Delta$ ?

But this is connected to Newton thinking like a physicist (Which he was in all but name)

William Playfair, the accredited inventor of line graphs, pie charts etc was just being born when Newton was in his grave twenty odd years.
Newton and his contemporaries worked from tabulations.
Newton developed an advanced calculus of finite differences, characterised by the use of upper case delta to denote a finite difference.
These were fixed values and most decidedly not infinitesimal; they were (and still are) sometimes quite large in value.

Newton used these to fill in or interpolate gaps in his tables, but I thought you knew all this.

So it is not a great step from big(ish) differences to small differences characterised by lower case delta, and thence to differences as small as desired.

Later mathematicians extended this idea to the 'epsilon - delta' construction you will find in many modern higher level texts on analysis; again I'm sure you already know this.