What is the difference between Calculus of Newton and that of Leibniz?

Question

Are there any differences between the study of Calculus done by Newton as compared to that done by Leibniz? If yes, please mention point by point.

Answer 1

Newton's notation, Leibniz's notation and Lagrange's notation are all in use today to some extent. They are, respectively:

$$\dot{f} = \frac{df}{dt}=f'(t)$$ $$\ddot{f} = \frac{d^2f}{dt^2}=f''(t)$$

You can find more notation examples on Wikipedia.

The standard integral($\displaystyle\int_0^\infty f dt$) notation was developed by Leibniz as well. Newton did not have a standard notation for integration.

I have read from "The Information" by James Gleick the following: According to Babbage, who eventually took the Lucasian Professorship at Cambridge which Newton held, Newton's notation crippled mathematical development. He worked as an undergraduate to institute Leibniz's notation as it is used today at Cambridge, despite the distaste the university still had because of the Newton/Leibniz conflict. This notation is a lot more useful that Newton's for most cases. It does, however, imply that it can be treated as a simple fraction which is incorrect.

Answer 2

You should definitely take a look at the second chapter of Arnold's Huygens & Barrow, Newton & Hooke. The late Prof. Arnold summarized therein the difference between Newton's approach to mathematical analysis and Leibniz's as follows:

Newton's analysis was the application of power series to the study of motion... For Leibniz, ... analysis was a more formal algebraic study of differential rings.

Arnold's overview of Leibniz's contributions to the theme is spiced up with a non-negligible number of thought-provoking remarks:

In the work of other geometers--e.g., Huygens and Barrow--many objects connected with a given curve also appeared [for example: abscissa, ordinate, tangent, the slope of the tangent, the area of a curvilinear figure, the subtangent, the normal, the subnormal, and so on]... Leibniz, with his individual tendency to universality [he considered necessary to discover the so-called characteristic, something universal, that unites everything in science and contains all answers to all questions], decided that all these quantities should be considered in the same way. For this he introduced a single term for any of the quantities connected with a given curve and fulfilling some function in relation to the given curve--the term function...

Thus, according to Leibniz many functions were associated with a curve. Newton had another term--fluent--which denoted a flowing quantity, a variable quantity, and hence associated with motion. On the basis of Pascal's studies and his own arguments Leibniz quite rapidly developed formal analysis in the form in which we now know it. That is, in a form specially suitable to teach analysis by people who do not understand it to people who will never understand it... Leibniz quite rapidly established the formal rules for operating with infinitesimals, whose meaning is obscure.

Leibniz's method was as follows. He assumed that the whole of mathematics, like the whole of science, is found inside us, and by means of philosophy alone we can hit upon everything if we attentively take heed of processes that occur inside our mind. By this method he discovered various laws and sometimes very successfully. For example, he discovered that $d(x+y) = dx+dy$, and this remarkable discovery immediately forced him to think about what the differential of a product is. In accordance with the universality of his thoughts he rapidly came to the conclusion that differentiation [had to be] a ring homomorphism, that is, that the formula $d(xy) = dx dy$ must hold. But after some time he verified that this leads to some unpleasant consequences, and found the correct formula $d(xy) = xdy + y dx$, which is now called Leibniz's rule. None of the inductively thinking mathematicians--neither Barrow nor Newton, who as a consequence was called an empirical ass in the Marxist literature--could [have ever gotten] Leibniz's original hypothesis into his head, since to such a person it was quite obvious what the differential of a product is, from a simple drawing...

Answer 3

Beyond the issue of notation, Newton experimented with a number of foundational approaches. One of the earliest ones involved infinitesimals, whereas later he shied away from them because of philosophical resistance of his contemporaries, often stemming from sensitive religious considerations closely related to inter-denominational quarrels. Leibniz also was aware of the quarrels, but he used infinitesimals and differentials systematically in developing the calculus, and for this reason was more successful in attracting followers and stimulating research--or what he called the Ars Inveniendi.

Answer 4

From Loemker's translation,

"Leibniz's reasoning, though it strives for a broader application of the law of inverse squares than to gravity alone, is less general than Newton's (Principia, Book I, Propositions I, 2, 14), since it presupposes harmonic motion."

Leibniz, Gottfried Wilhelm Philosophical Papers and Letters : A Selection / Translated and Edited, with an Introduction by Leroy E. Loemker. 2d ed. Dordrecht : D. Reidel, 1970. p.362

Answer 5

I'm not a historian, but I have to answer this because the previous answers have gotten it completely wrong. Leibniz's $\displaystyle \frac{df}{dx}$ is not equal to $f'(x)$.

I think there are two main differences between Newton's calculus and Leibniz's:

• Newton's calculus is about functions. Leibniz's calculus is about relations defined by constraints.
• In Newton's calculus, there is (what would now be called) a limit built into every operation. In Leibniz's calculus, the limit is a separate operation.

Both points, and the second one especially, seem to be poorly understood today. When people write what looks like Leibniz notation today, they're almost always doing Newtonian functional calculus in a weird and inappropriate notation. Then they complain that it's inconsistent and blame Leibniz. It isn't Leibniz's fault.

In actual Leibniz calculus, when you write $y=x^3$, that isn't a definition of $y$ as a function of $x$, it's a constraint on valid $(x,y)$ tuples. The locus of valid points may happen to be the graph of a function, but nonfunctional constraints like $x^2+y^2=1$ are equally acceptable.

$d$ applied to an expression gives you the difference between its value at the current point and at some other point that satisfies the constraints, where the other point depends only on the current point and not on the expression. In other words, if $(x,y)$ satisfies the constraints then so does $(x+dx,y+dy)$. Supposing we use the constraint $y=x^3$, we therefore have

$$\begin{eqnarray} y+dy &=& (x+dx)^3 \\ dy &=& 3x^2dx+3x\,dx^2+dx^3 \\ dy/dx &=& 3x^2+3x\,dx+dx^2 \end{eqnarray}$$

This holds for any two points satisfying the constraint, if $dx\ne 0$.

$dy/dx$ is not equal to $3x^2$ (unless $dx=-3x$). But they are equal in an appropriate limit. Leibniz wrote $dy/dx \mathrel{{}_{\ulcorner\!\urcorner}} 3x^2$ for this. (I've seen this relation called adequality. LaTeX (and MathJax) apparently doesn't have a symbol for it. See the source for the terrible hack I used to write it, which isn't my invention.)

Because $dx$ and $dy$ are defined everywhere the constraint is satisfied, we also have

$$\begin{eqnarray} \frac{dy}{dx} + d\Big(\frac{dy}{dx}\Big) &=& 3(x{+}dx)^2+3(x{+}dx)(dx{+}ddx)+(dx{+}ddx)^2 \\ \frac{d(dy/dx)}{dx} &=& 6x + 6dx + 5ddx + 3x\,ddx/dx + (ddx)^2/dx \end{eqnarray}$$

This holds for any three points satisfying the constraint. If you question the legitimacy of this, one way of looking at it is to imagine that $x$ and $y$ are functions of an arbitrary parameter and $(df)(t)=f(t{+}δt)-f(t)$. The expression is then a correct statement about $x$ and $y$ evaluated at $t$, $t{+}δt$, and $t{+}2δt$, and these are arbitrary points since the parameter is arbitrary.

Clearly it's a PITA to carry all these terms around, especially when you know they're going to disappear in a limit. But if you know that they're going to disappear then you can drop them early. That doesn't make the calculus nonrigorous, any more than omitting tedious intermediate algebraic steps (as I just did) makes it nonrigorous. What does make it nonrigorous is failing to observe a distinction between equality and equality-in-the-limit, because then you can prove nonsense like $dx=-3x$.

Leibniz wrote $d^2x$ as a shorthand for $ddx=d(dx)$, but he didn't write $\displaystyle\frac{d^2y}{dx^2}$ (at least according to Wikipedia). I suspect he would've considered it an abomination, because it doesn't mean $(d^2y)/(dx^2)$, even though that is a meaningful quantity in his notation. It means $d(dy/dx)/dx$. In effect, the scope of the top left $d$ operator extends out of the "numerator", into the "denominator", and stops halfway through the product represented by $dx^2$. It's as though $fx^2$ meant not $f(x^2)$, not $f(x)^2$, but $f(x)\cdot x$. I assume that this notation was invented by people who thought that $\displaystyle\frac{d}{dx}$ was just a weird way of writing a functional derivative.

I blame the modern misunderstanding of Leibniz's notation on the lack of a standard way to write anonymous functions. If you define $f(x)=x^3$ then $\dot f$ or $f'$ is concise and convenient, but people don't like having to name everything. Instead of inventing a sensible, composable notation like $D(λx.x^3)$ or $(x\mapsto x^3)'$, they decided to write $\displaystyle\frac{d}{dx} x^3 = 3x^2$, which is inconsistent with Leibniz's notation and which still left them with no way of writing an anonymous function unless they happened to be taking the derivative of it.

Answer 6

From a practical point of view, the notation was vastly different.

A particular sore point for me is that the Leibniz notation lets you incorrectly work with derivatives as though they were a mathematical fraction. Unfortunately this 'works out' a lot of the time so its still used, even in college courses, today.

I don't think there is anything wrong with shortcuts, up to the point that they don't interfere with understanding. In this case, I do believe it creates a misunderstanding of the subject matter. This alone I think puts Newtons notation above Leibniz's.