# Tag Info

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It is a play of words by Charles Babbage. Deism was a religious belief or rather a movement promoting the idea that God exists but it does not interfere with whatever happens in this world. This old philosophy according to the Wikipedia "...asserts God's existence as the cause of all things, and admits its perfection (and usually the existence of ...

25

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 ...

17

You say: "A well-known and specific example is that Leibniz is less well regarded than Newton for his calculus". Well known?? I think this is just incorrect. Leibniz version of calculus lead to an explosive development of calculus in continental Europe. Think of l'Hopitale, Bernoulli's, Euler and many others. While calculus in England experienced ...

17

Mauro Allegranza's comment pretty much says it all but to elaborate a bit one could mention that Leibniz came to mathematics rather late in his intellectual career and was essentially a self-educated scholar. His older colleague Huygens encouraged him to pursue mathematics, and his encouragement (on many occasions) was instrumental in Leibniz's development. ...

14

It is discussed in multiple manuscripts, letters and publications from 1675 to 1701. According to Fracois Ziegler's post on MO Did Leibniz really get the Leibniz rule wrong?, Leibniz originally thought $d(uv)=du\,dv$ in a special case, but corrected his mistake the same month in the manuscript Methodi tangentium inversae exempla (November 11, 1675). Later ...

10

Several factors come together to suggest that the idea that "English mathematics [was] ever significantly behind -- by say 50 years, 100 years, or even centuries" (i.e. in the post-Newtonian 18th or early 19th centuries) is at best a sweeping over-generalization, although something very like it has clearly become a received view. Two recent valuable ...

8

I think I can answer the last bit. From Wikipedia: On the death of Queen Anne in 1714, Elector George Louis became King George I of Great Britain, under the terms of the 1701 Act of Settlement. Even though Leibniz had done much to bring about this happy event, it was not to be his hour of glory. Despite the intercession of the Princess of Wales, Caroline ...

8

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 ...

8

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 ...

8

The first answer is excellent but just for context on the actual math: Newton notation for derivative of f(x): $\dot f(x)$ Leibniz notation for derivative of f(x): $\frac {df}{dx}$ Newton's notation is fine for very basic single variable derivation and generally the derivative for the described cases but the Leibniz notation is general purpose and ...

7

The reference is probably to a treatise sent to Huygens on 5 October 1691, where Leibniz says (and illustrates with several examples) that "Whenever the subtangent [$=y/y'$, but it would also work for just the tangent $y'$] is a product of two quantities or formulas, of which one is given purely in terms of the abscissa $x$, and the other in terms of the ...

6

This rule is, indeed, due to Leibniz, although it was Johann Bernoulli who realized its broader implications, and there is an interesting story to its discovery. It is told in Chapter 3 of Families of Curves and the Origins of Partial Differentiation by Engelsman. The rule appears in a Leibniz's 1697 letter to Bernoulli, as a side result in their long ...

6

Leibniz states the product rule in his first paper on the calculus (1684). It's in the middle of the fist page (page 467) as can be seen here: https://www.maa.org/press/periodicals/convergence/mathematical-treasure-leibnizs-papers-on-calculus-differential-calculus and also in English translation (top of page 2) here: http://www.17centurymaths.com/contents/...

5

The motivation for applying derivatives to polynomials over general fields is their use in detecting multiple roots: if $K$ is a general field, a polynomial $f(x)$ in $K[x]$ has no repeated roots if and only if $f(x)$ is relatively prime to $f'(x)$ in $K[x]$. Formal derivatives on polynomials in $K[x]$ for a general field $K$ were introduced by Steinitz in ...

4

I think the two other answers here when combined offer a complete explanation of Knuth's word game. He calls Leibniz notation "d-ism" to combine the English word "deism" with Leibniz's use of the prefix $d$ in $dx$ to suggest an infinitesimal change in $x$. So Leibniz would write $dx/dt$ for the rate of change of $x$ with respect to time $t$. Believing in ...

3

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 ...

3

It appears that they are right in the "metaphysical" sense, here is a passage from Leibniz's letter to Bernoulli (1699):"I concede an infinite multitude, but this multitude forms neither a number nor one whole. It only means that there are more terms than can be designated by a number; just as there is for instance a multitude or complex of all numbers; but ...

3

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 ...

3

Leibniz did indeed describe a mechanism for tracing the solution to any differential equation of the form dy/dx=f(x) using tractional motion. For a detailed explanation, see Viktor Blåsjö, The myth of Leibniz’s proof of the fundamental theorem of calculus, Nieuw Archief voor Wiskunde, ser. 5, 16(1), 2015, 46–50, especially Figure 6, available for free here: ...

2

Leibniz bragged as early as 1671 about solving the problem of longitude, among a long list of other major problems, with the help of his "combinatory art" At the time he was euphoric about the sweeping capabilities of his "characteristica universalis" to solve any intellectual problems. Unfortunately, his letter does not give many details: "I will ...

2

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 ...

1

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 ...

1

The question reads like an answer (Conifold’s remark). So, answer in the form of a question: Open a modern textbook on calculus or differential equations. To whom are the theorems and methods due? (The result is not wholly uniform, but there is a trend. Need it be attributed to notation, “crisis,” or anything? Does people not discovering things require ...

1

Yes, Leibniz did design a famous computing machine called the stepped-reckoner. It was a (digital) computer, but not an analog computer. Don't confuse digital with electronic. The difference is only that digital computers use discrete values for computation, whereas analog computers use continuous values, for example real numbers in case of Tractrix. I ...

1

It is an interesting question what Leibniz proposed, but I suppose his proposal was not practical: it is not mentioned in the histories of navigation, which probably means that it was never implemented. Your proposal is also not practical for many reasons. Determination of ship's position by tracking the speed and direction was a principal method of ...

1

Richard Arthur's article can be complemented with a Leibnizian fragment edited into: Gottfried Wilhelm Leibniz, The Labyrinth of the Continuum: Writings on the Continuum Problem, 1672-1686, edited by Richard Arthur (2001), page 82: Infinite numbers (ca.April 1676). Consider the conclusion, page 99: Whenever it is said that a certain infinite series of ...

1

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 ...

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