# All Questions

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### When was Euler's log-sine integral first computed by real methods?

In Sec. 2.4 of Inside Interesting Integrals (2015), Paul J Nahin says of $$I:=\int_0^{\pi/2}\ln (a\sin x)dx=\int_0^{\pi/2}\ln (a\cos x)dx$$that: For many years it was commonly claimed in textbooks ...
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### Reference - Schwarz's Proof of Clairaut's Theorem

Where can I find a copy (online) of Schwarz's paper that proved Clairaut's theorem for mixed partial derivatives? His paper is: Schwarz, H. A., "Communication", Archives des Sciences Physiques et ...
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### What are the modern connections of the Pentagramma Mirificum studied by Gauss?

In the last years, i read a lot about a mathematical object that was discovered by John Napier in 1620 and explored much more deeply by Gauss, who called this "Pentagramma Mirificum" (latin for "the ...
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### How do you rate Stephen Wolfram's physics papers?

Stephen Wolfram was a child prodigy of sorts who published his first physics papers in peer reviewed journals at 17 at around 1975. His academic work seems very impressive before he apparently quits ...
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### What inspired Nicholson to break water into hydrogen and oxygen with electricity?

It seems a bit of a stretch to go from a battery to electrolysis but this fellow appears to have thought of it only a few weeks after Volta invented the "pile" battery. I am wondering if he knew ...
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### What was the aperture of the first Galileo's telescope?

Galileo used a refractor i.e. no mirror: but what was the aperture?
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### Question about Gauss's contributions to the theory of electric circuits

This question is a continuation of my previously-posted question: Several questions about Gauss's contributions to electromagnetism. I wrote it after user vonbrand asked me to split my original ...
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### Why statistical moments are called moments?

According to the Jeff Miller's Earliest Known Uses of the Words of Mathematics "Moment was taken into Statistics from Mechanics by Karl Pearson when he treated the frequency-curve (or observation ...
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### Failures in math

I would like to have help in producing examples of mathematicians that, in some sense I'll explain below,turned their career into failure. I am mainly interested in examples from XIX and XXth century. ...
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### Where to find a list of laurates of the Volta prize?

Hearing to episode 3 of Stephen Fry's podcast Great Leap Years, he describes this very prestigious French prize for scientist of 19th century called the Volta prize (le prix Volta in French). The ...
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### English translation of Lagrange's Théorie des fonctions analytiques?

I've done some looking around and came up with nothing. There apparently was a German translation done by August Leopold Crelle, but I couldn't find anything else. If anyone else knows of an English ...
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### When, and by whom was the Schwarzschild metric first taken to be valid for all radii greater than zero?

The metric was originally defined to be valid only from the surface of a black hole outward but somewhere along the line it was extended inward to include the region under the event horizon. This ...
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### History of group theory character tables (as used in physics and chemistry)

Does anyone know who started using the symbols A, B, E, T (First column, left) for showing irreducible representations of symmetry groups? In older maths books I see capital gamma. Herein A= totally ...
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### Who discovered the expansion for factorial as successive difference of integers?

Who discovered the following theorem $$\sum_{r=0}^{n}\binom{n}{r}(-1)^r(n-r)^n=n!$$
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### From a historical perspective why is beta decay associated with an anti neutrino instead of a neutrino?

I know it sounds like a silly question but didn't both Pauli and Fermi hint at Beta decay missing a particle to carry away a small amount of it's energy or mass but why is this an anti neutrino ? The ...
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### Who did say that anyone who discover a new particle should be fined instead of receiving a prize?

I am almost sure I read once that a famous physicist said that anyone who discover a new particle should be fined instead of receiving a prize. The context was that at the time there was more and more ...
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### What are some good metrics for intellectual progress (of all sorts)?

My thinking about this topic is vague, and I'm looking to clarify it. I'm not sure what "intellectual progress" is or if that's even a useful abstraction, but it seems like it should include things ...
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### The Greeks did not discover “a single scientific law”

The title is drawn from a sentence in a Jim Holt article, "The Dangerous Idea of the Infinitesimal," now a chapter in his book collection.1 I found this a striking claim, and perhaps true, as the ...
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### Did Srinivasa Ramanujan have a surviving sibling?

Wiki (https://en.wikipedia.org/wiki/Srinivasa_Ramanujan) says 'After his death, his brother Tirunarayanan chronicled Ramanujan's remaining handwritten notes consisting of formulae on singular moduli, ...
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### Checking a Godel quote on *Principia Mathematica*

Is there serious doubt of whether the first edition of Russell and Whitehead's Principia Mathematica used the ramified theory of types? I am travelling and cannot easily check sources but I do easily ...
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### Sphericity of Earth from lunar eclipses - is Aristotle's argument valid?

Aristotle is often credited with proving the sphericity of Earth from the fact that the shadow of the Earth on the moon during lunar eclipses is always an arc of a round circle (as opposed to arcs of ...
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### Was the telegraph system of 1859 powered by AC or DC and how extensive was it?

I was actually curious how the Solar Coronal event of 1859 known as the Carrington Event affected the telegraph system. Wiki does not seem to have much detail of this but there is mention of it in a ...
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### Etymology of Some Terms Used in Ratio and Proportion in Old Algebra Textbooks

In older algebra textbooks for high school (mainly 19th century) four properties of ratio and proportions were used. The properties were Invertendo, Alternendo, Componendo, and Dividendo. This ...
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### Historical evidence for claim that we use base 10 because of the number of fingers

It is a common belief that we use a base 10 representation of integers because we have 10 fingers. Does there exist historical evidence which supports the claim that this is true and that the number ...
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### Coverage of a topic by scientific research through time

I'm interesting of how scientific research covers a specific topic in a time. Scientific field: biology, cellulosic ethanol production Like: ...
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### Query 31 in Newton's opticks and the second law of thermodynamics

Query 31 in Newton's opticks contains the following words: "But by reason of the Tenacity of Fluids, and Attrition of their Parts, and the Weakness of Elasticity in Solids, Motion is much more apt to ...
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### Origin of the smooth but nowhere real analytic function built with dyadic rationals

I found the following interesting function and its analysis at Non-analytic smooth function article in Wikipedia. I include a screen capture below for those who don't wish to navigate away: I could ...
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### Statistical Power as a Microscope Metaphor

An answer on this question on the cross validated stack exchange compared statistical power to a microscope, such that "in order to see small things you need a powerful microscope" is analogous to "in ...
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### Can we identify Paul Benacerraf in these photos

This question is about Paul Benacerraf, who worked on the philosophy of mathematics, and wrote the 1965 essay What numbers could not be (see: Benacerraf's identification problem). He was at Princeton ...
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### Do these trigonometric identities belong to Antonio Cagnoli?

I'm new to this stack community, please bear with me as I try to explain my question properly. Recently I came across with these trigonometric identities (where $\omega + \phi + \psi = 180^\circ$): ...
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### Notation $n=efr$ in algebraic number theory

When $\Bbb Q \subset K$ is a field extension of finite degree and when $p \in \Bbb Z$ is a prime number, the ideal $p O_K$ decomposes uniquely as a product $\prod_{i=1}^r P_i^{e_i}$ of prime ideals of ...
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### What were the first practical applications of Newton's mechanics on engine design?

I'm interested in when and how Isaac Newton's mechanics first came to be applied practically in engine design. Did for example Thomas Newcomen take advantage on Newtons discoveries when he invented ...
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### Law of the Unconscious Statistician - history of the term?

The "Law of the Unconscious Statistician" states that, for a random variable $X$ with density $f_X(x)$ and a function of it $h(X)$ we have that $$E[h(X)] = \int_{-\infty}^{\infty} f_X(x)h(x) dx$$ ...
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### Historically how it was discovered that we need fields to describe matter?

This question is from one historical perspective. The question is: how physicists historically found out that one needs quantum fields to describe matter? Being more detailed. Let us consider the ...
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### Foundational crises in non-Western historical mathematical communities

In Foundations of Set Theory by Fraenkel, Bar-Hillel, and Levy (1973), the authors argue that there have been three distinct periods of crisis in the foundations of mathematics. The first was ...
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### Why did Noether's theorem take so long to show up?

Obviously like they say hindsight is 20/20, but it seems to me that all the ingredients for Noether's theorem were in place more than a hundred years before its publication, and to be honest it is not ...
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### When was relativistic mass first observed?

I counldn't find precise info on who/ when successfully observed relativistic mass increase. The only thing I found by a Google search is that it was first confirmed in 1908 in experiments with ...
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### Why was Courant's “Methods of Mathematical Physics” suppressed, by the Germans, during WW2?

In the preface to Methods of Mathematical Physics Richard Courant, the author, wrote that the book was suppressed by the National Socialist rulers(Nazi) of Germany. Hence, my question. Thanks.
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### Several questions about Gauss's contributions to electromagnetism

First of all, i appologize if i'm asking too many questions about Gauss (and some will also say "not very interesting questions"); i know i might appear ridiculus - but i really think it's an ...
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### What did Albert Einstein mean by “psychical entities”?

I first saw this quote on p. 26: Philip Johnson-Laird BA PhD Psychology (UCL), Stuart Professor of Psychology Emeritus at Princeton. (Author isn't a logician.) How We Reason (1st edn 2008). ...
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### On the birthdate of Gotthold Eisenstein

The birthdate of Gotthold Eisenstein is Apr. 16, 1823 as is stated in the Wikipedia. But a letter(whose recipient is Gauss) of Encke on Oct. 11, 1852 clearly states that the birthdate of Eisenstein is ...
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### Help translate from German a quote by Hermann Weyl in Space Time Matter

I would like to find an accurate translation to the following quote from Space Time Matter: Man muß gegen diese Orgien des Formalismus, mit dem man heute sogar die Techniker zu belästigen beginnt, ...
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### Why Mendel's experiments didn't draw attention until 20th century?

In The Gene: An Intimate History, author concludes the Mendel's pea experiments publication by these words: Mendel himself requested forty reprints, which he mailed, heavily annotated, to many ...
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### How did Gauss's electrodynamics law preempt the later developement of retarded potentials theory in physics?

My question refers to a fragment on electrodynamics written by Gauss at around 1835. In this short note (see Gauss's werke, volume V, p. 616-617) Gauss wrote down a fundamental equation which ...
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### What was Gauss's theoretical work related to his invention of the “Double-Gauss lens”?

Gauss's contributions (Dioptriche Untersuchungen, 1840) to the "classical" optical theory (Gaussian optics, i.e calculations made under the paraxial approximation) are well known, but long before he ...
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### Where does the letter S in “$S$-units” and in localization $S^{-1} R$ come from?

In number theory, we may encounter the notion of $S$-unit, $S$-integer, etc. where $S$ is a finite set of prime numbers (for simplicity). For instance, if $S = \{2,3\}$ then the $S$-integers are the ...
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### Did Aristotle note that ships disappear over the horizon hull-first?

I have run across several references to Aristotle's arguments for a spherical earth which claim that he noted that ships sink over horizon hull-first. For instance, Isaac Asimov writes in his essay ...