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

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 ...
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  • 736
26 votes
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Who calculated for the first time the volume (and surface area) of the sphere exactly?

This is one of those questions that is much trickier than it appears, many different people contributed to the formulas as we write them today. The short answer, that doesn't really do justice to ...
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23 votes
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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):
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18 votes
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Why is Leibniz less well regarded?

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 ...
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17 votes
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Is there a 'lost calculus'?

As a matter of fact, there was something now called "lost calculus" or "algebraic calculus" in the 17th century, that avoided concepts like limits or infinitesimals, which where problematic at the ...
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17 votes
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Are there any theorems that become "lost" and discarded over time?

What most mathematicians are doing at a given time is determined by current fashion to a very large extent. I do not know a (fashion-independent) criterion by which a theorem can be "useful", but ...
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14 votes

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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14 votes
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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 ...
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13 votes

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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13 votes
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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 [6] introduced his integral in his December 1853 Habilitationsschrift thesis. In his ...
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13 votes
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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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12 votes

Who calculated for the first time the volume (and surface area) of the sphere exactly?

Archimedes calculated the exact formulas (in the way that the ancient Greeks gave formulas) in his book On the Sphere and Cylinder. This was not "experimental": He gave a full geometric proof, ...
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12 votes

Was the Riemann Integral the first integration theory?

The first rigorous integration theory in due to Eudoxus and Archimedes. It is called the method of exhaustion, and it allowed Archimedes to find the volumes of the balls, pyramids, cones, areas of ...
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12 votes
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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 ...
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12 votes
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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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12 votes
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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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  • 2,809
11 votes
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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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11 votes
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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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10 votes

Why did Newton want lines to be generated by continued motion of points rather than by apposition of parts?

Problem: classical geometry is not happy with infinitesimals Newton is systematically trying to avoid basing calculus on infinitesimal geometric quantities. We can see this from how he emphasizes ...
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10 votes

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 ...
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10 votes
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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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10 votes

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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10 votes
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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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10 votes
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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” ...
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10 votes

$\frac{dy}{dx}$ 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 ...
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10 votes

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 ...
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  • 3,639
10 votes

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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  • 2,809
10 votes

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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10 votes
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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 ...
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  • 2,521
9 votes
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Why is calculus missing from Newton's Principia?

There are too separate issues here. The method of fluxions and fluents, Newton's version of calculus, is amply represented in Newton's extant papers, starting with 1669 On Analysis by Equations with ...
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