# Questions tagged [mathematics]

For questions about the quantitative study of topics such as numbers, structure, space, and change, carried out by investigating patterns

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### Original description of point sources and point spread functions

I already asked this question in the Astronomy community, but there it was recommended to me to also try my luck here. I would like to know the original description of point sources and point spread ...
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1 vote
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### Did Russell's paradox affect Peano's construction in the Formulario?

Russell's letter to Frege in 1902 was after the first edition of the Formulario Mathematico but before the fifth one (1908). So I wondered if the discovery of Russell's paradox had made Peano to edit ...
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### Cauchy integral formula [duplicate]

There is an integral Cauchy formula. It would be nice to know who exactly (for the first time) added Cauchy's name to a formula that Cauchy was not the author of. Who do you think it could be? But ...
1 vote
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### New mathematics theory vs new mathematical theorem

On the Academia site, there is a recent question that asks about obtaining reviewers for a "new theory". I'm only an amateur mathematician, not a professional, and the question got me ...
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### Origin of Riemann-Stieltjes Integral

What need (if there was any) created Riemann-Stieltjes integral? What did Riemann-Stieltjes integral want to attain?
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### Use of the verb "induct" in proofs by mathematical induction

Occasionally, in a proof by mathematical induction, the writer will say something like, "We induct on $n$" or "We induct on the number of vertices." This usage of the verb induct ...
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### In which work did Euler invent the Euler Substitutions for a quadratic composed into a radical?

A famous technique in the modification of integrands is the set of “euler substitutions” that provide substitutions for the structure $$\sqrt{ax^2 +bx+c}$$ That is a fairly common occurence in ...
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### Early results on the Fourier transform

Published tables of Fourier transform pairs have been available for many years. One such example is the paper by George Campbell in the Bell Systems Technical Journal in 1928. Most such tables simply ...
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### Origin of the usage of $\lambda$ to represent eigenvalues

I'm curious whether anyone knows how $\lambda$ came to be used to represent eigenvalues and or who (if anyone) was responsible for the convention. I've looked through a couple of books on the history ...
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### What is the title of the 1676 Memoir in which Leibniz first used the Chain Rule?

On Wikipedia it says: "The chain rule seems to have first been used by Gottfried Wilhelm Leibniz. He used it to calculate the derivative. He first mentioned it in a 1676 memoir [ Chain Rule ...
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### The Originator of Cobweb Diagrams

A cobweb diagram is a visualization tool that allows one to qualitatively study the iterates of a self-map of the real line based on the graph of the function; here is an example: (Here the map is ...
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### Where did the index of a subgroup notation $[G:H]$ begin to be used?

In texts of algebra, the cardinality of cosets is written in $[G:H]$ or $|G:H|$. Where did this notation originate? The history about $G/H$ can be found here. $[G:H]$ is called index of a subgroup. ...
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### Where can I find the early proofs for the simplicity of $\text{PSL}(n,q)$?

I am doing a research project on the history of group theory, and want to know about the early developments on the subject. There are plenty of proofs about why the projective special linear group ...
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### History of group actions as their own structures

I'm interested in when (and how) the modern idea of a group action developed and how group actions became their own algebraic structures. As far as I can tell in the 19th century group actions were ...
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### Was not chess notation an earlier sort of Cartesian Coordinates?

I realize modern algebraic notation is fairly new but for as long as there have been recorded chess games, every square I think was referred to using two coordinates, in old descriptive notation a ...
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### The history and origin of the Argument Principle ( or Cauchy's argument principle)

I am looking for a book that discusses The history and origin of the Argument Principle ( or Cauchy's argument principle) Thanks!
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### Katz's symbol 兄 for Gauss-Manin connections

In his famous 1970 paper , Nicholas Katz used the character 兄 for the Gauss-Manin connection. I have always been curious about the history behind this symbol. Question: What motivated Katz to use ...
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### When was the first time/s that sheaves entered algebra and algebraic geometry?

I'm interested in knowing about the first published texts in which sheaf-theoretic methods were used in algebra and/or in algebraic geometry. The oldest instance I am aware of is J.-P. Serre, ...
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### How and when did the dedicated study of locally compact groups begin?

How and when did the dedicated study of locally compact groups begin? Specific instances from literature, recorded stories, etc., may help supplement the answers. There seems to be no reason why I ...
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### When was the calculus first part of college curriculum in USA?

Or I guess the Colonies, if it happened before 1776? I know that mathematics tended to be both applied and emphasized things like taking fifth roots. Also, I think long ago high school and college may ...
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### Is there a theorem proof whose accuracy is doubted because it is short?

Is there a theorem proof whose accuracy is doubted because it is short? He told me while chatting with a friend of mine. It's about a mathematician who proves a difficult theorem very briefly and ...
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### What was known about Chebyshev polynomials in 1900?

Around 1900, was it widely known that the Chebyshev polynomial $T_n(X)$ satisfies the identity $$T_n(X) \circ \frac{X+X^{-1}}{2} = \frac{X^n+X^{-n}}2?$$ And also, would one expect top-notch ...
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### Did the ancient Greeks know that "most" cube roots are irrational?

It is common knowledge that the Pythagoreans discovered irrational numbers (or incommensurability), for example if the hypotenuse of an isosceless right triangle is compared with one of the legs or ...
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### Leonhard Euler's Mathematical Proof of God

There is a famous legend inspired by Euler's arguments with secular philosophers over religion, which is set during Euler's second stint at the St. Petersburg Academy. The French philosopher Denis ...
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### Did Riemann invent the Riemann curvature tensor?

I'm pretty sure they weren't using tensors in the modern sense at that point, but to what extent did Riemann lay out the structure or significance underlying his eponymous tensor? In his habilitation ...
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### Poincaré's definition of mathematics (?)

Poincaré is said to have given this definition of mathematics: It is the art of giving the same name to different objects. I can not believe this quote is from Poincaré. Did Poincaré really say this? ...
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### The origin and use of the term "equianharmonic" (elliptic function)

In Weierstrass notation, the principal elliptic function $\wp$ is a solution of the differential equation $$(\wp')^2= 4\wp^3 -g_2\wp -g_3.$$ The case when $g_3=0$ is called lemniscatic (it ...
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### Influence of Edmund Landau's list of four problems

At the 1912 International Congress of Mathematicians, Edmund Landau listed four basic problems about prime numbers: Goldbach's conjecture, the twin prime conjecture, Legendre's conjecture (that there ...
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I am interested in knowing what is the original source/author of the following identities: $\tan{\frac{\alpha}{2}}\tan{\frac{\beta}{2}}+\tan{\frac{\alpha}{2}}\tan{\frac{\gamma}{2}}+\tan{\frac{\beta}{2}... 0 votes 0 answers 28 views ### Were Priestly and Davey wrong about about "Order: A Theory with A View" being written by Ivan Rival? The textbook Introduction to Lattices and Order by Priestly and Davey mentions in its first edition preface: In a proselytizing article, Order: a. theory with a. view [in Kla.ssifikation und Ord:nung,... • 299 2 votes 0 answers 66 views ### What was probability like before Kolmogorov? How was probability studied or understood before Kolmogorov axiomatized it? I mean Bernoulli and Gauss had already started inventing and using distributions way before Kolmogorov came into the scene. ... • 121 2 votes 0 answers 145 views ### Zermelo's or Fraenkel's early consideration of something equivalent to countable Replacement I have now claimed a few times on the internet, based on something (sensible!) I read, that at some point in the 1920s, that Zermelo at one point considered as a set theoretic axiom (schema) something ... 1 vote 1 answer 195 views ### Why did Clairaut's theorem take so long to prove? I was reading the Wikipedia on Clairaut's Theorem (Symmetry of second derivatives) and the article accounts the significant amount of time and failed proofs occurred before the theorem was made fully ... • 11 0 votes 0 answers 42 views ### What did George Boole contribute to order theory? The Wikipedia page says for order theory says: As explained before, orders are ubiquitous in mathematics. However, earliest explicit mentionings of partial orders are probably to be found not before ... • 299 0 votes 1 answer 46 views ### What is Bertrand Russell citing Gilman for in "On the Notion of Order"? In the first few pages of "On the Notion of Order", Bertrand Russell has the footnote: The following account of the genesis of order is virtually identical with that of Mr. B. I. Gilman, ... • 299 1 vote 1 answer 68 views ### When did Abel publish his test for the convergence of series? Did Abel published of testing the convergence of series? If so, when did he published it. Also, did he offer a proof of the test? Or did he simply stated the test? • 125 1 vote 1 answer 171 views ### Was religion not present at all in the Astronomy of the Ancient Greeks? I attended a lecture on the history of Astronomy and Mathematics, and I was somewhat puzzled by how scientific the early Greeks were. Yes, I am aware that they have many of the greatest mathematicians ... • 133 2 votes 0 answers 118 views ### The first commutative diagram Cf. https://golem.ph.utexas.edu/category/2013/07/the_first_commutative_diagram.html Is this the first ever commutative diagram? By commutative diagram I mean anything that shows an equality (or ... 0 votes 0 answers 107 views ### What are the second and third "Hebrew Algebras" Mary Everest Boole refer to? In the book "Philosophy and Fun of Algebra", Mary Everest Boole has a chapter dedicated to what she calls "the first Hebrew algebra" (Chapter 6). She later refers to the "... • 111 3 votes 1 answer 107 views ### What original mathematics arose from Jacquard loom and similar weaving devices? Research mathematics is seldom directly applicable, but the questions we study are motivated by interplay with contemporary demands. As a combinatorialist, it's apparent that my field has benefited ... • 131 4 votes 2 answers 177 views ### Engineering problems leading to mathematical research I would like to know examples of mathematical research which had its origin in engineering problems (I am mainly interested in mechanical engineering and civil engineering) and which was actually ... • 5,506 5 votes 0 answers 238 views ### Does "Metatron's cube" have a history and a serious name in geometry? This is a figure that I saw while going down the rabbit hole of "Sacred Geometry" back when conspiracy theories and related nonsense were relatively harmless and fun to laugh at. A book ... • 183 5 votes 6 answers 524 views ### Are there any well known mathematicians who were fascists? I only recently learned that Pascual Jordan, a well known physicist, with significant contributions to the development of early quantum mechanics was a paid up member of the Nazi Party. He in fact ... • 3,510 1 vote 0 answers 36 views ### Why was Cauchy studying Schur polynomials and related topics? Let$\lambda_1 \geq \lambda_2 \geq \cdots \geq \lambda_n$be nonnegative integers. The Schur polynomial$s_{\lambda}(x_1, \ldots, x_n)$can be defined as the ratio $$s_{\lambda}(x_1, x_2, \ldots, x_n) ... 6 votes 3 answers 277 views ### When did bounties and prize money for open mathematical problems start being a thing? I'm a science/math journalist [ger] and currently I'm working on an article about the culture of prize money/bounties for solving open mathematical problems (Millennium Prize Problems and such). One ... • 61 3 votes 1 answer 126 views ### Are there any arithmetic problems studied by Euler still open? Fermat's last theorem, which Euler had studied in the case of certain exponents, was only solved in the 1990s. Also, a counterexample to Euler's sum of powers conjecture has been found quite recently (... 4 votes 0 answers 110 views ### Did Rayleigh or Ritz prove the Rayleigh–Ritz theorem? The maximum eigenvalue of a real symmetric (or complex Hermitian) matrix is given as the maximum of the associated the quadratic form:$$ \lambda_{\rm max}(A) = \max_{\|x\| = 1} x^*Ax. \tag{1}$$This ... • 141 1 vote 1 answer 141 views ### How to build a protractor without a protractor? We all know how to use a protractor, it is taught in elementary school. However, I was wondering what type of knowledge is required to build one from scratch. For instance, was the understanding of$\...
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Given a measure space $(\Omega,\Sigma,\mu)$ and sequence of pointwise non-decreasing, non-negative, measurable functions $\{f_n\}_{n=1}^{\infty}$ on it, Lebesgue's monotone convergence theorem says ...
The following well-known formula for pythagorean triples is commonly called Euclid's formula: If $a, b, c$ are three natural numbers with $a,c$ odd, $b$ even, $\gcd(a,b,c)=1$ and $a^2+b^2=c^2$, then ...