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

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.

• – Danu Oct 29 '14 at 8:29

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 alot more useful that Newton's for most cases. It does, however, imply that it can be treated as a simple fraction which is incorrect.

• It does, however, imply that it can be treated as a simple fraction which is incorrect. Not true. For a good discussion of this, see Blaszczyk, Katz, and Sherry, Ten Misconceptions from the History of Analysis and Their Debunking, arxiv.org/abs/1202.4153 . See also en.wikipedia.org/wiki/Non-standard_analysis . As explained in the Blaszczyk paper, Leibniz basically got this completely right, including what in NSA is now referred to as the distinction between the quotient dy/dx and the derivative, which is the standard part of that quotient. – Ben Crowell Dec 22 '14 at 3:45

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.

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...

• Arnold's claim that Leibniz "came to the conclusion" that $d(xy)=dxdy$ is an error that has been extensively discussed elsewhere. Leibniz did not make such a claim but on the contrary asked whether this was true. And sure enough he came to the conclusion that it was not so, soon enough. Arnold's sarcastic tone probably stems from his distrust (following Berkeley and Cantor?) of infinitesimals, which is also obvious in some absurd claims he makes here as to the alleged "obscurity" of their meaning. – Mikhail Katz Apr 7 '16 at 9:48

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.

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