3
votes
1answer
70 views

Ancient calculus or thorough observation

Integration. It's the simplest way on earth with which we can derive any formula like surface area or volume of symmetrical shapes and solids (square, circle, cube etc.). But what I've been hearing is ...
0
votes
2answers
91 views

Why do some sources call calculus, “the calculus”?

No need to cite specific sources since I think it's a fairly common thing to see. What's up with that? Thank you Edit: I've seen it in several places. Here's where I'm currently looking at it at: ...
9
votes
2answers
180 views

Newton's “Famous Blunder”?

On page $225$ of Isaac Newton on Mathematical Certainty and Method by Niccolo Guicciardini (see here for a link), I read In the following demonstration... Newton made a famous blunder... He wrote, ...
2
votes
1answer
105 views

Reference Request - Early Calculus Papers

Question: I am looking for good references on the early calculus papers. Optimally, I want emphasis on the mathematics itself and I want that mathematics to be translated into modern terminology and ...
4
votes
0answers
75 views

Origin of the family name de l'Hôpital? [closed]

The "de" is because he was a nobleman. The circumflex accent is to show that in Old French, there was an s, so that the word is clearly "hospital." But is it known why his family was called that? ...
13
votes
1answer
284 views

l'Hopital's questionable premise?

Historians widely report that l'Hopital's 1696 book Analyse des Infiniment Petits pour l'Intelligence des Lignes Courbes contains a questionable premise expressed by an equation of type $x+dx=x$ ...
5
votes
1answer
82 views

Who introduced the finite difference notation using $\Delta$?

We all know that Leibniz introduced the differential notation $dx, dy$, and that in developing his calculus for infinitesimal differences he was inspired by his previous work on finite diffences. ...
5
votes
1answer
71 views

eulers original derivation for the Euler–Maclaurin formula?

Please does someone know a good description of how Euler did derive his summation formula? Thank you!
15
votes
10answers
1k views

Challenge: Demonstrate a Contradiction in Leibniz' differential notation

I want to know if the Leibniz differential notation actually leads to contradictions - I am starting to think it does not. And just to eliminate the most commonly showcased 'difficulty': For the ...
3
votes
1answer
163 views

Errors of Euler interpretation?

To complement the recent post on Euler's errors, I would pose the following question: what common errors of Euler interpetation appear in the literature? What errors are attributed to Euler's work in ...
4
votes
5answers
237 views

How did Newton and Leibniz actually do calculus?

How did Leibniz know to write derivatives as $$\frac{dy}{dx}$$ so that everything would work out? For example, the chain rule: $$\frac{dy}{dz}=\frac{dy}{dx}\frac{dx}{dz}$$ Integration by Parts: ...
4
votes
2answers
144 views

Works on Calculus by Newton and Leibniz (primary sources)

I'm trying to find PDFs or hard copies of the following works from the dawn of calculus. Does anyone know where I could find English translations of them? Newton - De analysi per aequationes numero ...
4
votes
0answers
73 views

Was the Weierstrass function constructed or discovered?

Reading Halmos' I want to be a mathematician, he mentions a continuous function without a tangent. Naturally, I was curious to see how such a function could possibly exist, and I imagined it to be ...
4
votes
1answer
66 views

What is the history of this theorem about the finite sum of a polynomial?

I discovered and proved the following theorem back in high school, and have waited patiently to hear something about throughout my college career (which is nearing it's end, hope to have finished my ...
7
votes
6answers
344 views

Evaluating the reception of (epsilon, delta) definitions

There is much discussion both in the education community and the mathematics community concerning the challenge of (epsilon, delta) type definitions in real analysis and the student reception of it. ...
3
votes
1answer
309 views

sin(x) infinite product formula: how did Euler prove it?

I know that $\sin(x)$ can be expressed as an infinite product, and I've seen proofs of it (e.g. Infinite product of sine function). I found How was Euler able to create an infinite product for sinc by ...
14
votes
2answers
753 views

Are infinitesimals dangerous?

Amir Alexander is a historian of mathematics. His new book is entitled "Infinitesimal: How a Dangerous Mathematical Theory Shaped the Modern World". See here. Two questions: (1) In what sense are ...
3
votes
1answer
105 views

Important results of calculus before Newton and Leibniz?

We have all come to know that calculus was invented by Newton and Leibniz, right? But many calculus results were already proven by the time. I have read that Fermat already found how to calculate ...
5
votes
1answer
97 views

the word “derivative”

When did the word "derivative" come into use in calculus, and why? As in Can the word "derive" be used to mean "take the derivative of"? the word "derivative" in normal English ...
3
votes
2answers
225 views

History of Calculus

Newton/ Leibniz invented calculus on approximately 1680's. Cauchy/Weierstrauss defined the $\epsilon - \delta$ definition of a limit in approximately 1820's. So how did they define derivatives? I ...
21
votes
3answers
771 views

Who came up with the $\varepsilon$-$\delta$ definitions and the axioms in Real Analysis?

I've seen a lot of definitions of things like boundary points, accumulation points, continuity, etc, and axioms of the real numbers. But I have a hard time accepting these as "true" definitions or ...
0
votes
2answers
130 views

Roberval's Method and Tangent Construction involving parabola $y^2=4ax$

Problem: Let $u$ denote the distance of a moving a point $P$ on the parabola $y^{2}=4px$ from the directrix $x=-p$ and from the focus $\left(p,0\right)$. If the point moves in such a way that ...
14
votes
3answers
458 views

Who gave you the epsilon?

Who gave you the epsilon? is the title of an article by J. Grabiner on Cauchy from the 1980s, and the implied answer is "Cauchy". On the other hand, historian I. Grattan-Guinness points out in his ...
23
votes
6answers
810 views

What did Newton and Leibniz actually discover?

Most popular sources credit Newton and Leibniz with the creation and the discovery of calculus. However there are many things that are normally regarded as a part of calculus (such as the notion of a ...
2
votes
2answers
234 views

Origin of the Power Rule Proof: Who first proved the power rule?

Who was the first person to prove the power rule for derivatives? The person could have proved the power rule using limits and the binomial theorem or difference of two nth powers, or the implicit ...
1
vote
0answers
62 views

what is a connection between two simple yet important economics and math formula: elasticity

what makes it interesing to define them in mathematics? what is a connection between two simple yet important economics and math formula: elasticity? Something interesting to read: ...
0
votes
0answers
83 views

Which link is there between Calculus and Quadrature?

I'm trying to find the origin of the integration process. To do that I'm studying "De Analysi" by Newton. I would like to know the process that lead Newton to the Rule I $\ref{Rule I}$ below, and if ...
12
votes
3answers
230 views

Why “integralis” over “summatorius”?

It is written that Johann Bernoulli suggested to Leibniz that he (Leibniz) change the name of his calculus from "calculus summatorius" to "calculus integralis", but I cannot find their correspondence ...
25
votes
4answers
2k views

Is mathematical history written by the victors?

The question is the title of a recent piece in the Notices of the American Mathematical Society, by twelve authors (of which I am one). The contention is that traditional history of mathematics is ...
0
votes
1answer
96 views

Johann Bernoulli did not fully understand logarithms?

This wikipedia article makes the claim: "Bernoulli's correspondence with Euler (who also knew the above equation) shows that Bernoulli did not fully understand logarithms." This is found under ...
16
votes
1answer
250 views

Who is buried in Weierstrass' tomb?

The tangent half-angle substitution often used to anti-differentiate rational functions of sine and cosine, and also sometimes used to find closed-form solutions of some differential equations, is ...
11
votes
4answers
222 views

Who was the first to prove $\lim_{x \to 0} \frac{\sin{x}}{x}=1 $?

Who was the first to prove $\lim_{x \to 0}\frac{\sin{x}}{x}=1$?
4
votes
2answers
99 views

Is the validity of measuring area by approximation an assumption of calculus?

The assumption that if you subdivide an area into more and more sub intervals, the approximation gets better and better. Has this been formally proved, or is it just intuition? Thanks!
16
votes
1answer
466 views

A curious theorem by Peano

Let $f$ be defined on $[a,b]$ and there differentiable. Show that for every $ \epsilon>0 $ there exists a partition $\, a=a_0<a_1<...<a_n=b \,$ of $ \,[a,b] \,$ so that $$\left|\frac ...
2
votes
0answers
136 views

Is Risch's algorithm powerful enough to determine any integral of a function have a closed form or not?

Is Risch's algorithm powerful enough to determine any integral of a function have a closed form or not? Is there a historic piece of reference that support your answer? ...
1
vote
3answers
468 views

How did Euler and Bernoulli prove this limit?

Prove that the lim as x approaches infinity of $(1+1/x)^x$ exists, and prove this without assuming any prior knowledge of $e$.
10
votes
4answers
570 views

Is the theory of dual numbers strong enough to develop real analysis, and does it resemble Newton's historical method for doing calculus?

I've been interested in non-standard analysis recently. I was reading up on it and noticed the following interesting comment on the Wikipedia page about hyperreal numbers, right after giving an ...
2
votes
1answer
179 views

What is the purpose of defining the notion of inflection point?

What is the purpose of defining inflection point? I know that it is defined to be the point where the second derivative is zero and the second derivative sign changes. It has to have some purpose ...
16
votes
4answers
758 views

How did the ancients view *infinitesimals*?

With some category/topos theory we can now put infinitesimals on a rigorous ground, as in Bell's A Primer of Infinitesimal Analysis, where the author introduces $\epsilon$ satisfying \begin{equation} ...
2
votes
0answers
284 views

On the geometric arguments used in Newton's *Principia Mathematica Naturalis Philosophae*

When one reads Newton's Principia Mathematica, one is immediately aware of the complexity of the synthetic geometry that he uses to prove his propositions. This I understand because all of the ...
10
votes
1answer
335 views

Old versus New enunciation of Taylor's Theorem.

I am studying from Spivak' Calculus, and he states Taylor's Theorem as follows: THEOREM Let $f',\cdots,f^{(n+1)}$ be defined on $[a,x]$ and let $R_{n,a}(x)$ be defined by ...
5
votes
4answers
478 views

Euler and infinity

What do people mean when they say that Euler treated infinity differently? I read in various books that, today, mathematicians would not approve of Euler's methods and his proofs lacked rigor. Can ...
1
vote
0answers
112 views

History of calculus-based optimization

I would like to know: - who started with calculus-based optimization problems and when it was, - if there is a book focusing on history of ellipses/ conic sections - if someone ever tried to ...
3
votes
1answer
160 views

Who came up with the Euler-Lagrange equation first?

Could someone explain who came up with the specific equation first? http://en.wikipedia.org/wiki/Euler-Lagrange makes it sound like Lagrange got it first, in 1755, then sent it to Euler. but: ...
0
votes
2answers
189 views

Why was Newton concerned with finding tangents?

While trying to teach myself calculus, I stumbled upon a BBC documentary called The Birth of Calculus. In the documentary, the narrator explains that Newton and other contemporary mathematicians were ...
11
votes
3answers
2k views

how exactly did calculus change our understanding of the world?

I am taking calculus course and I keep wondering if this is really necessary. I know it is the cornerstone of modern science but what I don't understand is why and how. Was it impossible to pursue ...
6
votes
2answers
295 views

Why does Maclaurin get his own polynomial?

Why is a Taylor polynomial centered around $0$ called a Maclaurin polynomial? It's only a special case of the Taylor polynomial, and it is calculated the exact same way as a Taylor polynomial centered ...
8
votes
3answers
753 views

Volumes of cones, spheres, and cylinders

Given a sphere with radius r, a cone with radius r and height 2r, and a cylinder with radius r and height 2r, the sum of the volume of the cone and sphere is equal to the volume of the cylinder. If we ...
3
votes
3answers
536 views

Oldest books on Calculus

What are some of the oldest books available on Calculus? I'm curious to see how the old teaching and explanatory styles compare to modern ones.
8
votes
2answers
178 views

integrating the secant function, who figured this out?

I was looking at how the secant function is integrated. The process is not obvious, and I don't expect it to be but I wanted to know if anyone knows who figured this out. Here's what I'm talking ...