# Tag Info

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### How did Isaac Newton write the integral symbol?

Newton used both vertical bars ($\overset{|}{x}$) and rectangles ($\boxed{x}$) to denote integrals in his Quadratura curvarum published in 1704. Here, the bar notation is used on the bottom of page 9 ...
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### Who invented the Leibnitz notation $\frac{d^2y}{dx^2}$ for the *second* derivative?

Leibniz did use this notation for instance in his paper Supplementum geometriae practicae, Acta Eruditorum, April 1693, p. 179 (Google Books link):

### Did Newton know about non-inertial frames?

TL;DR Yes, but... 1) Inertial frames To say that Newton had the modern conception of even inertial frames (based on the laws of motion), is an overstatement. Theoretically, he did not need them ...
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### How was curvature originally defined and calculated?

Apollonius (c. 262–190 BC) "calculated" curvature of conic sections implicitly when solving the problem of drawing normals to them in book V of Conica, but he did not think of it as a ...

### Who first used the word "calculus", and what did it describe?

According to Carl B. Boyer, "The history of the calculus and its conceptual development", Dover Publications 1959, page 98, The improved notation led also to methods which were so much more facile ...
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### Where did Leibniz explore the product rule of differential calculus?

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 ...

### Is Spivak right in what he says about Galileo?

Yes, indeed when trying to obtain the law of falling bodies, Galileo's first conjecture was that the speed is proportional to the distance traveled. After some contemplation, Galileo understood that ...
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### Riemann's Contribution to Integration

The following is a slightly edited version of my 31 January 2003 sci.math post archived at google groups. Riemann  introduced his integral in his December 1853 Habilitationsschrift thesis. In his ...
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### Did Benjamin Franklin know calculus?

Apparently, no. Boman's biography Benjamin Franklin's Numbers specifically focuses on Franklin's mathematical activities, and calculus is not among them. He was deep into magic squares, and ...
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### How were derivatives of trigonometric functions first discovered?

Did you try looking in any books on the history of calculus? The following is taken from "The Historical Development of the Calculus" by C. H. Edwards (p. 205 ff). The inverse sine function (for ...
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### Did Michel Rolle say that the calculus is "a collection of ingenious fallacies"?

"Did Rolle ever say/write any such thing (as that the calculus was 'a collection of ingenious fallacies')?" Michel Rolle (France, 1652-1719) certainly did attack the mathematical basis of the ...
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### Who discovered the power rule for derivatives?

Cavalieri was presumably the first to state the "power rule" for areas under parabolas with positive integer exponents, but he only derived it up to $n=4$, beyond that his methods became intractable. "...
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### How did Newton prove the generalised binomial theorem?

According to William Dunham in his Journey Through Genius [Newton's Binomial Theorem is] not a "theorem" in the sense of Euclid or Archimedes in that Newton did not furnish a complete proof. Yet ...
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### Dirichlet integral's history

It arose in Dirichlet’s famous proof of the convergence of Fourier series (1829, p. 161),1 then again in his “discontinuous factor” method to compute integrals (1839; 1904, pp. 193-195, 353-385)2 and ...
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### What book did Maria Gaetana Agnesi write which contained both differential and integral calculus?

The book is a work in two volumes: Maria Gaetana Agnesi, "Instituzioni Analitiche ad uso della gioventù italiana", Milan 1748. The title roughly translates to "Foundations of Analysis ...

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

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 ...

### Historical development of power series

Some power series, like the geometric progression were indeed encountered since the ancient times, but the first person who used them systematically was I. Newton. Actually Newton considered this his ...

### Who introduced the notation $y|_{x=a}$?

You can see : Giuseppe Peano , Lezioni di Analisi Infinitesimale, 2 vols., 1893, page 17 : $$[f(x)]_{x=a}=f(a).$$ Not sure it is the earliest... but Peano was a prolific "inventor of notations". ...
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### Who first considered the $f$ in $f(x)$ as an object in itself, and who decided to call it a function?

Cantor 1895 is predated at least by Dedekind in §2 of Was sind und was sollen die Zahlen? (1888) (translation): 21. Erklärung *). Unter einer Abbildung $\varphi$ eines Systems $S$ wird ein Gesetz ...
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### Is the prime notation for derivatives $f'$ due to Euler?

I agree, as there is further evidence that Lagrange got his primes from Euler: 1. Everyone since Cajori (1923, p. 6; 1929, p. 207) quotes Théorie des fonctions analytiques (1797) for the (sic) “new” ...

### $\frac{dy}{dx}$ versus $\frac{{\mathrm d}y}{{\mathrm d}x}$

One already finds upright d's in Lacroix's Traité élémentaire de calcul différentiel et de calcul intégral (1802). I don't now if this is the earliest, but it is interesting to note that the ...

### How influential was the Kerala school to European development in Calculus?

In my work with primary sources in such authors as Fermat and Leibniz, I have occasionally come across references to earlier works by Arab mathematicians, but have never seen references to work by the ...

### Was English mathematics behind Europe by many years because of Newton's notation?

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 ...
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### Why is differentiation under the integral sign named the Leibniz rule?

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 ...
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### What algebra problem did Serge Lang give to calculus students?

In 1969 Lang wrote an article for the Columbia Daily Spectator, Don't Blame Us if You Flunk Math (Volume III, Number 4, December 8, 1969). The phrasing of the subline illustrates how much the times ...
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### What was the real need of divergence and curl operators?

These operations arose from the study of quaternions see e.g. Thomson and Tait's Treatise on Natural Philosophy, that should probably have information on the sort of math. Stokes's theorem originated ...

### Who discovered the indeterminate forms like 0/0?

Special cases were handled algebraically even before the "l'Hopital's" rule, which appears in l'Hopital's 1696 transcription of tips on calculus he purchased (literally) from Johann Bernoulli in 1694, ...
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### Who invented differential calculus for rational functions?

The differential calculus for rational functions can be reduced to finding roots of polynomials. Such methods were used by Fermat and Descartes. See problems 12.3-12.5 of my history of mathematics ...