Questions tagged [math-history]

Use this tag for questions concerning history of mathematics, historical primacies of results, and evolution of terminology, symbols, and notations. Consider if History of Science and Mathematics Stack Exchange is a better place to ask your question.

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Example of objects which were thought to exist infinitely but later proved to be finite

Is there any example of objects which were thought to exist infinitely but later proved to be finite? I feel like this question was already asked, but I couldn't find the exact duplicate. My research: ...
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Historical reference to the group cohomology of the integers mod 4

I am looking for a historical reference for the first calculation of the group cohomology of the integers mod 4, or perhaps finite cyclic groups. I know that already in his 1953 paper in the Comment. ...
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What did Thue prove when, concerning rational approximations?

A follow-up to When did Liouville come up with the first transcendental numbers? In 1909, Thue showed that if $\alpha\in\mathbb R$ is algebraic of degree $n$ and $s>\frac12n+1$, and if $c$ is any ...
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Abstract symmetric definition of duality in linear algebra?

In his Linear Algebra, 4th ed. from 1975, Greub presents (p. 65) an abstract, symmetric definition of duality, in which two vector spaces $E^*,E$ over a field $\Gamma$ are said to be dual if there is ...
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Did Euler produce any Russian text?

Crossposted on History of Science and Mathematics SE Wikipedia says that Euler (1707 - 1783) "mastered Russian and settled into life in Saint Petersburg" in 1727. Did he produce any Russian ...
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Where can I find this “generalization” of Ramanujan?

I read at Wikipedia that In his notebooks, Ramanujan generalized the Euler product for the zeta function as $\prod_p(x-p^{-s})\approx\frac{1}{Li_s(x)}$ I can also see that Wikipedia mentions at the ...
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Is there any English translation of this Gergonne paper?

This is the paper: “Variétés. Essai de dialectique rationnelle”. Annales de Mathématiques pures et appliquées, tome 7 (1816-1817), p. 189-228 (“Varieties. Essay about rational dialectic”, By J.D. ...
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Uses of Vieta Jumping in research mathematics?

Vieta jumping has been a prominent method for solving Diophantine equations since 1988. It was popularized when it was used to solve an IMO problem, but has it been applied to research mathematics, ...
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How Eratosthenes measured the Earth's circumference

When Eratosthenes measured the Earth's circumference, he provided $250,000$ stadia as its measure equal to about $25,000 $miles. Since the radian is a dimensionless number, the radian couldn't have ...
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Beltrami's Essay on the Interpretation of Non-Euclidean Geometry

I am reading the Essay of the title written by Beltrami in Italian and I found a specific point of the essay which in my opinion could be fully clarified only if compared with its translations. At the ...
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Who first explicitly wrote the determinant identity $\det(1+AB) = \det(1+BA)$?

Though this identity can be easily proved, I am wondering who first explicitly write it in such a simple and elegant form? I check several textbooks on linear algebra but find no evidence (see below ...
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Historical perspective on non-constructive reasoning in decidability proofs

In the 1920s, the Entscheidungsproblem or decision problem was a key area of investigation. This is the question of whether the satisfiable formulas of first order logic (or some fragment thereof) ...
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Sign of permutation who discovered that

I am reading a book about mathematical games. One game is the 15 puzzle https://en.wikipedia.org/wiki/15_puzzle The answer whether it is solvable or not, was solved by using sign of permutations. My ...
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Who was the first to restrict Taylor series limits to dual sided and why?

I've been reading Taylor's original paper, https://books.google.com/books?id=r-Gq9YyZYXYC&pg=PP3#v=onepage&q&f=false (There are also translations, http://17centurymaths.com/contents/...
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need English translation of Cantor's Über die Ausdehnung eines Satzes aus der Theorie der Trigonometrischen Reihen [closed]

I need English translation of Cantor's Über die Ausdehnung eines Satzes aus der Theorie der Trigonometrischen Reihen, I was unable to find it online. I want to know what Cantor said about the ...
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Translation of Jordan Curve Theorem Original Proof?

I am searching for an English translation of Jordan's Original Proof of the JCT. So far I have found this paper by Hales that seeks to "bring the terminology and language up to date" and fix ...
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1answer
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How to expand this expression in series?

From Poisson's first Memoire sur la distribution d'electricite (1812), $\S$39. Suppose $b$ is very small relative to $c-a$ where $a,b,c$ are positive real scalars. Consider the equations \begin{align*}...
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History of the exponential function and its derivative

Who was first to discover that the derivative of the exponential function is the exponential function itself. Can someone recommend me any good resources where I can read and learn more about the ...
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Is there such a thing as synthetic geometry of plane curves?

Conic sections on the plane have the same properties to objects of synthetic geometry as the proofs in, say, Apollonius of Perga's synthetic geometry book on conics. This means algebraically-defined ...
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Why does Skolem's decision procedure for first order logic reject satisfiable formulas?

In his (1928), Skolem describes a decision procedure for first order formulas with the prefix $\forall \exists$. We can think of Skolem’s procedure as a sort of infinite truth table, where at each ...
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1answer
43 views

what is the real Intuition behind the Standard deviation [duplicate]

A typical statistics course will define the standard deviation as "the average of the difference between the data set and the mean ". So if we tried to describe the definition mathematically ...
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question on Riemann's 1859 paper, page 2

This is the paper I am referring to. (After fighting through page one with success) I don't get the start of page 2 at all. I don't understand the motivation and connection to the previous expression ...
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Why are the symbols $\cdot$ and $+$ used for the operations of Boolean algebra?

The symbols $\cdot$ and $+$ are often used to denote Boolean product and sum, but they make some of the system's properties, like distributivity over $\cdot$, counter-intuitive: $$a+(b\cdot{c})=(a+b)\...
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Bourbaki: Use of tau choice operator?

I have a question about evaluating truth of the sign assemblies in Bourbaki Theory of Sets. It's my understanding that when there are duplicate assemblies each starting with the choice operator $\tau$,...
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Is there a native notation for base 60/sexagesimal numbers?

I have been reading about the Babylonian-derived sexagesimal or base-60 number system. Is there a standard notation for base-60 numbers that is analogous to that used by binary (0-1), octal (0-7) and ...
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1answer
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Consistency vs. non-contradiction in the algeraic logic tradition

In his introduction to (Skolem 1970), Wang claims that Skolem's conflations of consistency (non-contradiction) and satisfiability are "explained by the fact that Skolem is in the [algebraic] ...
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The origin of “The Lost Boarding Pass Problem”

There are several post here concerning "The Lost Boarding Pass Problem": The Lost Boarding Pass Advanced Advanced airplane problem Taking Seats on a Plane Generalize airplane problem But ...
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Why does Hadamard claim that Pascal could have discovered non-Euclidean geometry

In The psychology of invention in the mathematical field, p. 53, Hadamard makes the following claim: Is his point that there must, in any axiomatic theory, be undefined terms, and if you write the ...
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Longest-ever time between problem posing and solving?

Guiness World Record claims that Goldbach's conjecture is the oldest unsolved problem. A natural related question is what solved problem went unsolved for the longest time. In other words, of all the ...
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How to prove the Vandermonde's determinant for a $3\times 3$ matrix when rows and columns have been swapped?

The problem is as follows: The following determinant is named after french mathematician Alexandre-Théophile Vandermonde who lived in the late 18th century. Prove this determinant $\left|\begin{...
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108 views

Facts or Theorems Similar to Pigeonhole Principle

Are there any facts or theorems in mathematics having similar energy with pigeonhole principle? What I mean by similar energy is the statement is simple, "trivial" as you don't need to prove ...
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who were the amateur mathematicians had won the field medal?

Who were the amateur mathematicians had won the field medal? I search the field medal winners. I didn't find any single amateur mathematician who won the field medal. All are from academics ...
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Focusing Property of Parabolas (Historical Question)

I'm curious about the historical development of the idea that parabolas redirect incoming light to the focus. I'm curious how this was discovered, and whether it was discovered mathematically or ...
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Why is square root by long division found so?

We were taught the long division method of finding square root in junior classes. The logic behind the method used to be unclear, it remains so even now! However, we learnt and practiced the ...
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1answer
115 views

Bourbaki: Quantification Expansion

I'm reading Theory of Sets where existential quantification is defined: a) $$ \exists x R = (\tau_x(R)|x)R $$ and I have questions regarding the expansion for an example relation: b) $$ \exists y \...
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1answer
75 views

($\varepsilon$, $\delta$)-definition of limit by Weierstrass [closed]

I am looking for the original ($\varepsilon$, $\delta$)-definition of limit by Weierstrass, but I cannot find an exact quote or a reference. I saw that somewhere it was claimed that this definition ...
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How was the 1° angle constructed for the first time before protractors became a thing? [closed]

It is a widely known fact that any arbitrary angle cannot be constructed using the ancient Greek method of only using a compass and an unmarked straightedge. However, between then and now, we have ...
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The historical development of iterative methods before the era of the computer

Newton (early 18th century) and Gauss (early 19th century) both developed, among others in that time, iterative methods to find solutions to computational problems. Today these methods are immediately ...
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Cauchy sequences: why not a shorter, simpler definition?

Any ideas on what made $$\forall \varepsilon>0,\exists n_0 \in \mathbb{N}\text{ such that }m,n \geq n_0 \implies |x_m-x_n|<\varepsilon$$ prevail over the equivalent and arguably simpler (for ...
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The Infinitely Nested Radicals Problem and Ramanujan's wondrous formula

In mathematics, a nested-radical is any expression where a radical (or root sign) is nested inside another radical, eg. $\sqrt{2 + \sqrt{3}}$. By extension, an Infinitely nested radical (aka, a ...
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What are the origin and reason of the name matrix?

I wondered why the term matrix had been chosen as the name for what otherwise looks like a table of numbers. Is there any originator and (speculative) reasons why this term gained traction? Do we know ...
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Difficult equations with fractions and variables

i tried but i'm stuck it seems too difficult for me why is this even true? please. help me. For any $\alpha \in \mathbb{R}$ and $n \in \mathbb{N}$ $$ \sum_{p=1}^n \frac{\sin(\alpha+ 2\pi p / n) }{x^2-...
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Why cant we apply the square root of a negative number to equate more imaginary numbers?

In mathematics, polynomials like $x^2-1$ would have a clear solution of $x=\pm 1$. However, without complex numbers could you solve $x^2+1$? No, there would be no possible solution without adding a ...
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Does the Chain Rule in Leibniz notation cancel terms?

I understand this notation is now a differential operator and this is the limit of a quotient, but Leibniz regarded $\frac {dy}{dx}$ as a quotient. In Leibniz's theory where $\frac {dy}{dx}$ is a ...
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Practical significance of $\frac{17}{21}$=$\frac{1}{2}$+$\frac{1}{6}$+$\frac{1}{7}$ [closed]

In an article of ancient maths I found that Babylonians also had a great idea of maths. For practicle purposes, if I want to divide 17 grain bushels among 21 workers, the equation would be $\frac{17}{...
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Examples of two inequivalent definitions which were once thought to be equivalent

I once thought two definitions that I made in proof theory were equivalent, but then I discovered a counterexample. So I am now led to wonder, have there ever been cases in the history of mathematics ...
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Story behind the proof of hard theorems [closed]

I would like to know the story of how the proofs of some "hard" theorems were reached. More specifically, how did the author get to the "right" ideas (especially when the proof is ...
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What algorithm is Newton using in the “De analysi” to extract the square root of a polynomial?

I'm reading a 1745 English translation of Newton's De analysi (apparently the most up-to-date there is, surprisingly). The Latin is here. In this tract he shows how to use the integral power rule for ...
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Equivalence of minor epicycle and eccentric

In epicycle-deferent astronomy, adding a second minor epicycle to account for observational discrepancies is mathematically equivalent to shifting the deferent into a so-called eccentric, or a circle ...
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English translation of Riemann's complete works

Browsing in the library I came across with the mathematical (and some philosophical) papers of Riemann, collected by Weber and Dedekind in the original German (although published by Dover). But ...

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