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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 history, is that only Euler presented volume formulas in this form in his textbooks after 1737. The principal step was no doubt made by Archimedes in On Sphere and ...

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 ...

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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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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

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 time. It was developed by Descartes, Hudde and others, and is described in Suzuki's award winning paper The Lost Calculus (1637-1670): Tangency and Optimization ...

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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 theorems can be beautiful. Some deep and beautiful theories can be out of fashion for many years. Sometimes they came into fashion again. Some examples. ...

14

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 in application than the cumbrous geometrical procedures of Archimedes, of which they were modifications, that these methods were eventually recognized as ...

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 ...

13

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 anticipated some basic ideas of statistics and population dynamics. In a 1772 letter to Joseph Priestley he described what he called Moral or Prudential Algebra for ...

12

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 segments of parabolas etc. (By the way, it was proved in 20th century that one cannot find the volume of a pyramid by pure geometric methods, some form of integral ...

12

"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 infinitesimal calculus. I haven't found the exact phrase attributed to him by the authors quoted in the question, but there are plenty of broadly similar attacking ...

12

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 thesis he also gave an example, correctly verified, of a Riemann integrable function whose discontinuities form a dense set. Riemann's thesis wasn't widely known ...

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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, rigorous for its time period. He considered this his greatest work. He asked that a diagram representing his proof be inscribed on his tomb. This was apparently done ...

11

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 property of a curve, and his "calculations" are constructions of segments. The first person to "see" curvature was Oresme (c. 1320-1382), ...

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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. "Presumably", because the timeline is spotty due to absence of publications, and Fermat might have done it earlier. In any case Fermat was likely the first to ...

11

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 radian angles in the first quadrant) can be related to an area under an arc of the unit circle, which is $y = \sqrt{1-x^2}$ in the first quadrant. Newton knew from ...

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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 this cannot be the case and eventually came with the correct law. Good source on Galileo: S. Drake, Galileo at work. (There are many editions). From the modern ...

11

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 in his “jump function” proof of the central limit theorem (1846; 2011, pp. 69-74).3 Apparently first evaluated by Euler in De valoribus integralium a termino ...

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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 that his method is consistent with the "ancient" standard of rigor: To institute an Analysis after this manner in finite Quantities... is consonant to ...

10

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 main mathematical discovery: that any equation (algebraic, differential, etc.) can be solved by substituting a power series with undetermined coefficients, and ...

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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". Regarding : how they express "$y$ under the condition that $x=2$" see e.g. page 34 [shortened] : let $y$ the natural logarithm of $x$ [...] and $f(x) = \... 10 Cantor 1895 is predated at least by Dedekind in §2 of Was sind und was sollen die Zahlen? (1888) (translation): 21. Erklärung *). Unter einer Abbildung$\varphi$eines Systems$S$wird ein Gesetz verstanden, nach welchem zu jedem bestimmten Element$s$von$S$ein bestimmtes Ding gehört, welches das Bild von$s$heißt und mit$\varphi(s)$bezeichnet wird; ... 10 I agree, as there is further evidence that Lagrange got his primes from Euler: 1. Everyone since Cajori (1923, p. 6; 1929, p. 207) quotes Théorie des fonctions analytiques (1797) for the (sic) “new” notation$\ f'x,\ f''x,\ f'''x,\ f^{\scriptsize{\mathrm{IV}}}x$. But Euler’s much earlier Institutionum calculi integralis vol. 3 (1770) had already ... 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 ... 10 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, see Indeterminate Forms Revisited, by Boas. For example, Descartes's method of finding tangents involved resolving "indeterminate forms" like$0/0$, see Is ... 9 The sign of the integral was proposed by Leibniz (for indefinite integral). It is a stylized letter$S$, and$S$stands for the "sum". The things seen in your picture are probably metal parts of the house construction which can be seen on many old houses built of stones. They have nothing to do with the integration symbol:-) Remark. The modern notation for ... 9 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 an Infinite Number of Terms sent as a letter to John Collins, and disseminated by him to multiple correspondents, including Leibniz. The dotted shorthand was ... 9 [EDIT: check the (*) for a potential earlier reference in Lagrange] The first mathematician (who used the methods of characteristics for differential equations) seems to be Paul Charpit de Villecourt (?-1784). According to I. Grattan-Guinness et S. Engelsman, The manuscripts of Paul Charpit, Historia Mathematica, vol. 9, 1982, p. 65-75: He presented his ... 9 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 his insight and intuition served him well enough to devise a germane formula... Further, his inherent belief in the persistence of patterns suggested to him that ... 9 Gibbs (1889, p. 140):$ \qquad \dfrac{d\,\log\mathrm V}{d\,\log p} = - \dfrac{d\,\log n}{d\,\log\lambda} $Riemann (1868, p. 89):$ \qquad \dfrac{d^2y}{dx^2}-\dfrac1{\alpha\alpha}\dfrac{d^2y}{dt^2}=4\dfrac{d\smash[t]{\dfrac{dy}{d(x+\alpha t)}}}{d(x-\alpha t)} $Hamilton (1843, p. 362):$ \quad\ \dfrac d{d\log t}\,(1+t)^{-m}\,\mathrm T_m \$ Jacobi (1841, p....

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