Use this tag for questions concerning history of mathematics, historical primacies of results, and evolution of terminology, symbols, and notations.

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7
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1answer
244 views

Chinese estimate for $\pi$. Were they lucky?

The famous chinese estimate $\pi\approx\frac{355}{113}$ is good. I think that is too good. As a continued fraction: $$\pi=[3:7,15,1,292,\ldots]$$ That $292$ is a bit too big. Is there a reason for a ...
8
votes
1answer
203 views

How did Kurt Gödel's Incompleteness Theorem affect the mathematical world?

Hi I am looking not to understand the Incompleteness Theorem, but to find out more about how and what this has effected the mathematics world. I am in high school, in Honors Algebra II, and I am ...
3
votes
2answers
70 views

How did Fourier arrive at the following regarding his series and coefficients?

I am reading Karen Saxe's "Beginning Functional Analysis." Perhaps it is poor exposition on her part, but she states: ...Fourier begins with an arbitrary function $f$ on the interval from $-\pi$ ...
3
votes
1answer
102 views

History of terminology: sheaves, presheaves, etc.

I've been looking at some old notes (1970s) on Riemann surfaces, trying to match up terminology with modern definitions (at least going by Wikipedia). The notes use the same terms as Gunning's ...
10
votes
0answers
331 views

How does a Lehmer Sieve work?

http://en.wikipedia.org/wiki/Lehmer_sieve Apparently a Lehmer Sieve was a mechanical device that used chains and pulleys to factor numbers and solve diophantine equations. It once was able to factor ...
4
votes
2answers
103 views

How did self-similarity come into mathematics?

As far as I understand the interest in self-similarity was born outside of mathematics. The textbooks I came across give a few objects as examples (tree, broccoli, river, etc) yet it's clear that the ...
2
votes
0answers
86 views

What did “logarithm” initially mean? [duplicate]

I just read that logarithms were not initially defined in terms of their inverse relationship to exponential functions (and that Euler was the first to develop them in this way). So how were they ...
5
votes
2answers
821 views

Why the name “square root”?

Why do we say that $\sqrt{a}$ is a square root of $a$? Is this because $\sqrt{a}$ is a root of the function $f(x)=x^2-a$? Cubic root similarly? Thanks in advance
30
votes
6answers
1k views

Why are integrals called integrals?

What is the historical background for this term? I cannot quite see what is integral about an integral, even if we go back to the viewing it as the area under a curve. It seems a strange choice of ...
1
vote
1answer
180 views

Why do different countries/regions have different methods of counting large numbers?

When we start counting large quantities of $10's$, the number system varies by country/region: Europe/US: $10^3$ (thousand, million, billion are all multiples of $10^3$) Japan/China/Korea: $10^4$ ...
6
votes
1answer
207 views

History of the terms “prime” and “irreducible” in Ring Theory.

In ring theory, a nonzero, nonunit element $p$ of a integral domain is called irreducible if $p=ab$ implies that exactly one of $a$ and $b$ is a unit, and it's called prime if $p\mid ab$ implies that ...
4
votes
2answers
209 views

Why did the ancients hate the Parallel Postulate?

I am reading this book, Gödel's Proof, by James R. Newman, at location 117 (Kindle), it says, For various reasons, this axiom, (through a point outside a given line only one parallel to the line ...
11
votes
3answers
771 views

History of notation: “!”

Does anyone know where the factorial "!" symbol came from? I can't decide if it is my favorite or least favorite notation in mathematics...
3
votes
1answer
80 views

Translation of Paolo Ruffini's work on Galois theory

Paolo Ruffini famously wrote a work providing the first proof of the unsolvability of the quintic with the extraordinary title "Teoria Generale delle Equazioni, in cui si dimostra impossibile la ...
2
votes
0answers
57 views

Which came first, energy minimization or pde?

I'm interested in a historical perspective on pde. I would like to know more about the original derivation of pde. It seems like d'Alembert was working on the one dimensional wave equation $$ ...
2
votes
1answer
64 views

How did Napier rounded his logarithms?

How did Napier round his logarithms? Wikipedia says: By repeated subtractions Napier calculated $(1 − 10^{−7})^L$ for $L$ ranging from 1 to 100. The result for $L=100$ is approximately $0.99999 = ...
3
votes
3answers
114 views

Colloquialisms in Math Terminology

What are some of your favorite colloquial sounding names for mathematical objects, proofs, and so on? For example, manifolds are often described using an atlas and a neighborhood describes a small ...
3
votes
1answer
53 views

Why half coversed or coversed trigonometric functions are being deprecated?

As you can see here there are some names for some trigonometric functions that I can't find in any text or math related papers today. In my opinion this kind of approach will also make it easier to ...
3
votes
1answer
121 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 ...
0
votes
0answers
26 views

How well-known are these contra-Bernoulli inequalities?

The standard and extremely useful Bernoulli inequality states that $(1+x)^n \ge 1+nx$ for positive integer $n$ and $x \ge 0$. I have needed an inequality of the form $(1+x)^n \le 1+c(n)x$ where $x$ ...
1
vote
2answers
140 views

What's the intuition behind definition of chaotic function?

I read books A First Course in Discrete Dynamical Systems by Richard A. Holmgren and An Introduction to Chaotic Dynamical Systems by Robert L. Devaney. I want to understand which concepts of "chaos" ...
1
vote
3answers
111 views

Mathematical logic and foundations of mathematics in the 20th century

I would like some references regarding the foundations of mathematics in the 20th century, and mathematical logic, e.g. (1) I want to understand what happened to the foundation, what originated the ...
5
votes
1answer
328 views

What is “Bourbaki's style in mathematics”?

I know Nicolas Bourbaki "is the pseudonym of a group of (mainly) French mathematicians who publish an authoritative account of contemporary mathematics." But what characterizes "Bourbaki's style in ...
1
vote
1answer
67 views

Infinite Product Identity for Hyperbolic Sine

Prove $\prod_{n\in\mathbb{N}\backslash\left\{ 0\right\} }\left(1+\left(\frac{\alpha}{\pi n}\right)^{2}\right)=\frac{\sinh\left(\alpha\right)}{\alpha}$. I saw this formula in a book and have no idea ...
5
votes
2answers
208 views

In Whitehead and Russell's PM, does not identity imply existance?

At the end of ✳96.48, $ \sim(w=\overset{\smile}{R}‘max_R‘J_R‘x)$ is chosen over $ w\neq\overset{\smile}{R}‘max_R‘J_R‘x$, on account of the latter's implication of existence. But ✳13.02 states that ...
2
votes
1answer
98 views

Differences in how mathematical results are proved in the time of Euclid and in the twentieth century [closed]

What is the difference in the manner demonstrated in Euclidean time and as demonstrated in the twentieth century?
14
votes
2answers
776 views

Why is analysis called “analysis”?

Just as the topic says, how did the name "analysis" come to denote the specific mathematical branch dealing with limits and stuff? The term "analysis" seems very generic compared to the words for the ...
9
votes
1answer
221 views

Ramanujan's personification of small positive integers

I dimly recall reading somewhere (perhaps in "The Man Who Knew Infinity"?) that Ramanujan associated personalities (perhaps it was mystical personalities, e.g. specific gods and goddesses?) with small ...
2
votes
1answer
109 views

The meaning of Differentials in Integration

This is further to the questions discussed in a previous post Here is an example of what I mean: Suppose that $C$ is a closed path in the plane and consider the line integral of $xy\,dx+x^2\,dy$ over ...
11
votes
5answers
382 views

How did Euler realize $x^4-4x^3+2x^2+4x+4=(x^2-(2+\alpha)x+1+\sqrt{7}+\alpha)(x^2-(2-\alpha)x+1+\sqrt{7}-\alpha)$?

How did Euler find this factorization? $$\small x^4 − 4x^3 + 2x^2 + 4x + 4=(x^2-(2+\alpha)x+1+\sqrt{7}+\alpha)(x^2-(2-\alpha)x+1+\sqrt{7}-\alpha)$$ where $\alpha = \sqrt{4+2\sqrt{7}}$ I know that ...
3
votes
1answer
111 views

Fibonacci's proof that $x^3+2x^2+10x=20$ has no solution in radicals?

I read on a poster today that Fibonacci showed that $x^3+2x^2+10x=20$ has no solution expressible in radicals, way back when. I couldn't find the proof anywhere. Does anyone know where I can find it? ...
13
votes
1answer
308 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
109 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. ...
70
votes
24answers
11k views

Mathematicians ahead of their time?

In every field there's always that person who's just years ahead of their time. For instance, Paul Morphy (born 1837) is said to have retired from chess because he found no one to match his technique ...
2
votes
1answer
107 views

History of the Basel problem

Why were the people of the $18$th century interested in the Basel problem? (The Basel problem asks for the value of $\sum\limits_{n=1}^{\infty}\frac{1}{n^2}$).
5
votes
1answer
105 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!
14
votes
1answer
569 views

History of Lagrange Multipliers

How did Lagrange discover Lagrange multipliers? Also, was it related to his work on the calculus of variations? And how did he originally understand/implement the technique?
8
votes
2answers
172 views

Theory vs problems in modern math

Quick background: I'm a fourth year undergraduate entering graduate school next year. I am trying to identify areas of mathematical research in which there tends to be more emphasis on developing new ...
1
vote
0answers
62 views

History Question re Euler's Constant $\gamma$

What used to be be called "Euler's Constant" (http://en.wikipedia.org/wiki/Eulers_constant) is now frequently called the "Euler-Mascheroni Constant". I have tried to find out what contribution ...
8
votes
0answers
136 views

Did Field's Medalist Klaus Roth suffer from test anxiety?

I remember hearing the story that Fields Medalist Klaus Roth was convinced that he could not pass a qualifying exam when he was a graduate student. He was then given a so called practice exam for him ...
1
vote
2answers
81 views

Uses of Jacobian of a map on $\mathbb{R}^n$.

For a map $f:\mathbb{R}^n\to\mathbb{R}^n$, Jacobian matrix of $f$ is defined as $$\begin{bmatrix} \frac{\partial f_1}{x_1}& \frac{\partial f_1}{\partial x_2}& \ldots \frac{\partial ...
4
votes
1answer
118 views

In Whitehead and Russell's PM, is there a typo in ✳72.23?

It seems that where $\gamma$ appears at the end of line 1, 4, 5, there should be a $z$ instead, i.e. $✳72.23\hspace{10pt} \vdash : R,S \in 1 \rightarrow Cls .\supset. ...
17
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
167 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 ...
2
votes
3answers
123 views

History of infinite series representations of $\sin(x)$ and $\cos(x)$

When did the famous infinite series representations for $\sin(x)$ and $\cos(x)$ came about? To be specific when did people realise that the ratio of the two sides of a right triangle with one angle ...
24
votes
5answers
2k views

Euler's errors?

What mathematical errors is Leonhard Euler known to have made? PS: As I wrote in a comment below: "However, I would not consider proof to be an error merely because it's not a proof by present-day ...
8
votes
1answer
118 views

First usage of the symbol ∈

Concerning a book [1] I am reading the symbol $\in$ was first used by Giuseppe Peano and is the first letter $\epsilon$ (epsilon) of the word ἐστί (means "is"). Does anyone know in which work of Peano ...
9
votes
1answer
333 views

Why is the permanent of interest for complexity theorists?

Studying a bit about the determinant and the permanent, I'm told that although both concepts have very similar formulas, the permanent was of not much interest historically - it was until later that ...
7
votes
1answer
211 views

What cubic problems did Tartaglia and Fior pose to each other?

I have been researching the history of finding roots to general polynomials and the story of solving for the roots of cubic polynomials ($ax^3+bx^2+cx+d=0$) lead me to find several sources describing ...
3
votes
1answer
77 views

Enlightening books giving a guided tour of mathematics, in a style that Gian-Carlo Rota would not mind?

I am currently reading Gian-Carlo Rota's Indiscrete Thoughts. What more can I say apart from "the man can write"? (In other words, you should really read it if you are interested in mathematics.) I ...