# Why do we use Leibniz's “version” of calculus instead of Newton's?

I understand that they invented calculus independently at roughly the same time, but why do we use Leibniz's terminology/notation rather than Newton's?

For example, why don't we use "fluxion" and "fluent"? Instead, we use Leibniz's derivatives.

What are the historical reasons for this?

There are many notations for derivatives as the concept has been expanded in many different ways. For example, there is also Heaviside's operational D. This is also used for the Frechet and Gateaux derivative (which is implicitly used in the notation for tangent bundles in differential geometry).

Newton chose a notation for ease of use. As a physicist he was mostly interested in the first and second derivatives of time. Since the dependent variable implicitly understood, there is no need for the notation to reflect this. Hence he needed only to indicate the degree of the derivative. This is an integer. Buy since we are only interested in the first two, 1 & 2, we don't have to indicate the degree by a numerical prefix (as they do in some notations), we can simply indicate it by a single or double dot. It's quicker a d more convenient.

When one is interested in the calculus for its own sake, then a more comprehensive notation is necessary. This should indicate the degree, the dependent and independent variables. Thus Liebniz's notation is more natural here.

Had Newton been more interested in geometry than physics, and Liebniz more interested in physics than geometry it would have been likely we would have seen their notations being swapped. In other words, the notation associated with their names reflected their interests.

(It's worth adding that doing differential geometry with intuitionistic logic allows the introduction of infinitesimals that is much closer to how Newton envisaged them, his fluxions, rather than the traditional epsilon-delta techniques of traditional analysis. Moreover, they generalise to the infinite-dimensional context seamlessly, unlike the usual calculus for which there are many differing techniques).

Many textbooks in English did use Newton's notation and terminology for a long time. For example, see Hutton, 1807, https://archive.org/details/acoursemathemat02huttgoog , which uses the dot notation and terms like "fluent." We still do use elements of Newton's notation in many fields. For example, in physics, if you have a function of position and time, it's common to use dots for time derivatives and primes for spatial derivatives.

Leibniz's notation has some objective advantages. Unlike Newton's notation, it makes it easy to do dimensional analysis, and it works well when you have lots of different variables that you might be differentiating or integrating with respect to. It works whether you want to think in terms of variables or functions, limits or infinitesimals.

Newton's notation was unclearly presented, and many people didn't understand how he intended it to be used. He had a notation $$o$$ for an infinitesimal change in the independent variable, so that if $$x$$ depends on $$t$$, then what in Leibniz notation would be written as $$dx$$ would be notated in Newton's notation as $$\dot{x}o$$. But he had a shorthand convention of writing this as $$\dot{x}$$, omitting the $$o$$ when context made it clear that what was intended was an infinitesimal change in $$x$$. This confused his readers. There is a discussion of this in Boyer, https://archive.org/details/TheHistoryOfTheCalculusAndItsConceptualDevelopment at p. 201.

• The o and O notation is still actually used in mathematical analysis to describe the limiting behaviour of function when one isn't interested in the precise behaviour. I recall finding it actually a useful notation. – Mozibur Ullah Jul 7 at 19:08
• @MoziburUllah: You're talking about a completely different "o" notation. There is no connection between the one you're talking about and the one I'm talking about. If you're still confused on this point, you may want to read my answer more carefully, as well as looking at the reference I provided to Boyer. – Ben Crowell Jul 7 at 22:09