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Most popular sources credit Newton and Leibniz with the creation and the discovery of calculus. However there are many things that are normally regarded as a part of calculus (such as the notion of a limit with its $\epsilon$-$\delta$ definition) that seem to have been developed only much later (in this case in the late $18$th and early $19$th century).

Hence the question - what is it that Newton and Leibniz discovered?

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Stellar question! –  nbubis Oct 15 '13 at 15:43
A mistake, done by Newton and Leibniz both, has resulted in changing the course of history of science in wrong direction. To solve a differential equation, which is so important in applications, is really a challenge because of that initial mistake. According to Newton and Leibniz, second order derivative of a function is the first derivative of the first derivative of the function. This means that a second derivative is an iterated limit. Similarly higher order derivatives are simply iterated limits of higher order. This is the main reason that to solve an arbitrary differential equation is a –  R K Sinha Oct 16 '13 at 6:09
@RKSinha, My experience has been that those that claim there are mistakes in Newton and Leibniz are mistaken themselves in most cases. –  user72694 Oct 16 '13 at 13:42
@Timotej, There was no epsilon-delta definition of limit in the 18th century. It first appeared in Bolzano in 1817 but was ignored. Heine noticed it about 50 years later and Weierstrass made systematic use of it around 1870. –  user72694 Oct 16 '13 at 13:45
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5 Answers

To understand the relation between Leibniz's theory, on the one hand, and modern ϵ-δ definitions and constructions of the continuum, on the other, it is helpful to make the following distinction. The distinction is between the following two items.

(1) On the one hand, we have the procedures employed by Leibniz and other pioneers of the calculus, and their procedural moves in deriving various new results such as Leibniz's derivation of the product formula for differentiation, and Euler's various solutions of the Basel problem, to give some elementary examples. These procedures can be characterized as syntactic material because at this stage in the development of mathematics, its practitioners were not overly concerned with semantic issues (see below).

(2) On the other hand, we have the issue of the ontology of mathematical objects, i.e., what are the basic objects mathematics uses, how one axiomatizes their behavior and justifies their "existence" relative to appropriate foundational theories. These semantic issues only began to emerge starting around 1870, and resulted in the modern foundations for analysis in particular and much of modern mathematics in general.

To answer your question, I would say that the contributions of Leibniz and others would today be viewed as centering on the syntactic side of the mathematical endeavor. They developed the calculus in the sense that they understood its procedures and syntax, which remain today largely as they have been developed by Leibniz, Euler, Lagrange, and others.

The semantic side of the story did not emerge until much later. This observation is sometimes expressed by claiming that the pioneers of the calculus were not "rigorous" but this I believe is a misleading statement: Gauss and Dirichlet similarly did not have access to modern sematic theories and in this sense would also not be "rigorous", but in fact there are almost no errors in their work. For this reason it is more meaningful to acknowledge the syntactic contribution of the pioneers of the calculus than to apply the essentially vague straitjacket of "mathematical rigor" in analyzing their work.

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This is actually quite a complicated question, since it spans two whole careers.

Some say calculus was not discovered by Newton and Leibniz because Archimedes and others did it first. That's a somewhat simple-minded view. Archimedes solve a whole slew of problems that would now be done by integral calculus, and his methods had things in common with what's now taught in calculus ("now" = since about 300 years ago), but his concepts were in a number of ways different, and I don't think he had anything like the "fundamental theorem".

I'm fairly sure Leibniz introduced the "Leibniz" notation, in which $dy$ and $dx$ are corresponding infinitely small increments of $y$ and $x$, and the integral notation $\int f(x)\,dx$. I suspect Newton and Leibniz were the first to systematically exploit the fundamental theorem. And the word "systematic" is also important here: Newton and Leibniz made the computation of derivatives and integrals systematic.

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About the notation, en.wikipedia.org/wiki/Integral#Historical_notation –  Asaf Karagila Oct 10 '13 at 22:59
What do you mean by "systematic"? –  Timotej Oct 10 '13 at 23:05
Part of what I mean by systematic is that they showed that many problems that might have seemed disparate were instances of a common idea. –  Michael Hardy Oct 11 '13 at 15:57
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Of course Archimedes had "exhaustion" principles to compute areas and surfaces of curved bodies using limiting arguments, like the so-called Cavalieri principle.

The essential point of Newton's and Leibniz' calculus is that they had the notion of "function". This notion is immediately tied to Descartes' idea of "coordinate system", which freed people of considering things like $x^2$ as pieces of area, and the like. Furthermore it was now allowed to consider time $t$ as a variable entering quantitative discussions.

Calculus was put on the map when Newton and Leibniz discovered that any function $f$ has certain natural concomitant functions $f'$, $f''$, as well as antiderivatives, and that the connection between these can be interpreted in a geometrical or physical way.

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Also that you can compute these derivatives and antiderivatives explicitly in many of the cases people care about. –  Zarrax Oct 15 '13 at 16:09
To an extent, people care about the functions you can find derivatives and antiderivatives of explicitly. –  Michael Oct 16 '13 at 6:16
@Christian, Actually exhaustion methods are the "opposite" of Cavalieri's principle. The former is Archimedean methodology; the latter is the methodology of "indivisibles". –  user72694 Oct 16 '13 at 15:14
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I would like to suggest a book written by an italian mathematician (Lucio Russo), namely "The forgotten revolution: how science was born in 300 BC and why it had to be reborn". It contains a lot of interesting things some of which are connected to the story of infinitesimal calculus spreading a (different) light on the works by Newton, Leibniz and Co.

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Can you give some examples of such light-speading? Is this "Lucia Russo" by any chance? Do you have a link for the book? –  user72694 Oct 18 '13 at 9:08
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from my understanding, Newton's binomial said that the delta x should always have lim -> 0 because it's in the denominator, but once you got rid of it down there, delta x=0 is possible (and then you got rid of all the delta x in the numerator), thus, making the computation of derivatives systematic and easy.

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So, you're saying: Newton said that $\Delta x$ should always have $\lim \to 0$, because it's in the denominator of $$\frac{f(x+\Delta x) - f(x)}{\Delta x}.$$ But in the limit as $\Delta x \to 0$, this quotient approaches the derivative, so we can compute derivatives. I think this is what you mean, but I still don't see how this answers the question. –  Jesse Madnick Oct 11 '13 at 7:10
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