76

normalis already meant right-angled in classical Latin; for example, angulus normalis appears in the first century text De institutione oratoria (volume XI, paragraph 3.141) by Marcus Fabius Quintilianus.
In a commentary on this text from the fifteenth century this early use of the word "normalis" is explained as "rectus", see screenshot:
"Angulus ...

answered Jun 26 '14 at 7:29

Carlo Beenakker

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62

Let me mention as a counterpoint that there is less need for
new terminology than one might expect. Mathematical exposition
is often more successful and clearer without new terminology, and
one should consider whether one needs any new terminology at all.
It seems to be a typical beginner's mistake to name everything in
sight, introducing all kinds of fancy ...

54

Formal, adj. Relating to or involving outward form or structure, often in contrast to content or meaning.
In mathematics, a formal argument is one that manipulates the form of an expression without analysing the interpretation of that expression. For instance, one may formally interchange a limit and an integral (or a summation and an integral, etc.) ...

50

Even more is true for this theorem. Check out this drawing from Arseniy Akopyan wonderful book of Geometry in Figures (Second, extended edition, 2017). On page 65 we find Figure 4.7.29)
In the foreword, Arseniy Akopyan writes
"It is commonly very hard to determine who the author of a certain
result is."
He nevertheless provides source for many of the ...

answered Oct 26 '17 at 15:56

Moritz Firsching

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49

I found this footnote on page 195 of Fricke and Klein's Vorlesungen über die Theorie der automorphen Functionen (1897):
Translation:
The term "homomorphic" seems more appropriate than the previously$^\ast$ used "isomorphic", because it refers not to "equality" but to "similarity" of two groups. The term "isomorphism" will therefore from now on be used in ...

answered Sep 3 '17 at 19:15

Carlo Beenakker

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48

Very briefly: until work of Hans Maass c. 1949, "modular" or "automorphic" both referred to holomorphic functions invariant-up-to-cocycle (that is, invariant holomorphic sections of a bundle) on a quotient $\Gamma\backslash X$. For $X$ the upper half-plane, these were ellipic modular forms, visible since the 19th century. For $X$ a product of upper half-...

46

Although just beyond your 50-year scope, this may be of interest. Among the series $\mathsf A_n, \mathsf B_n, \mathsf C_n, \mathsf D_n$ in the Cartan-Killing classification of simple Lie groups, everyone (I believe) always agreed to call $\mathsf A_n$ the special linear group, $\mathbf{SL}(n)$, and $\mathsf B_n$ and $\mathsf D_n$ the special orthogonal ...

45

Quantum algebra is an umbrella term used to describe a number of different mathematical ideas, all of which are linked back to the original realisation that in quantum physics, one finds noncommutativity. The areas now encompassed by the term "quantum algebra" are not necessarily directly or obviously related to each other (and this is even more true for ...

42

Your saying "elliptic functions are the functions on elliptic curves over $\mathbb C$" is somewhat misleading, I think. First came elliptic integrals measuring arc-length on an ellipse. These are generalizations of the inverse trig functions (take the ellipse to be a circle). The inverse functions to the elliptic integrals are elliptic functions. It was ...

39

"$65537$-gon" is the name. Likewise "$257$-gon":
writing (let alone saying) something like "diacosipentacontaheptagon"
serves less to communicate $-$ if indeed it succeeds in communicating at all $-$
than to show off.

answered Jun 1 '18 at 1:17

Noam D. Elkies

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37

Gordon Royle is right, I'm living in La Vacquerie. The reference to GWU is not correct: that is the workplace of my colleague Bill Schmitt. The US pronunciation is indeed "cray-poe", but in France it tends to become "crah-poe".
Henry

36

The subject known for decades as recursion theory, studying the class of recursive functions and the recursively enumerable (r.e.) sets and degrees, is now known almost universally, especially amongst the newer generation, as computability theory, studying the computable functions and the computably enumerable (c.e.) sets and degrees. This change, led by ...

36

The name of the curve is cochleoid (= shell-shaped rather than cardioid = heart-shaped).
I compare the two below (gold = cochleoid, blue = cardioid). The distinction shell/heart refers to the additional windings remarked upon by მამუკა ჯიბლაძე , without these windings the two shapes would be qualitatively the same.

answered Sep 15 at 20:27

Carlo Beenakker

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35

In full generality, there provably isn't any method for complete simplification (i.e., bringing an expression into a canonical simplest form). Simplifying should have two key properties: it should be algorithmic, and simplifying two different expressions for the same thing should give the same simplified form. If you have a simplification method with these ...

33

Jakob Nielsen's 1921 paper in Danish, with the title "Om Regning med ikke-kommutative Factorer og den Anvendelse i Gruppetoerien", is where he proved that subgroups of free groups are free. I don't know about the original Danish, but the English translation by Anne W. Neumann is available in Nielsen's collected works, with translated title "On calculation ...

33

Not an answer - just want to note that the curve has more hidden branches. They can be seen looking at the parametric equation
$$
x=\frac{\sin(\varphi)}\varphi,\quad y=\frac{1-\cos(\varphi)}\varphi
$$
where $\varphi=2\theta$

30

Since I am being asked the same question repeatedly and since the given answers are not quite correct, I post another answer despite the thread being so old.
According to a talk by Domingo Luna around 1985, the term spherical variety is not derived from spheres, at least not directly. Firstly, spheres are way too atypical, e.g., their compactification ...

29

An example of a failure to change notation is the movement by Eilenberg, Jacobson, Herstein and others to replace function notation $f(x)$ with $xf$ and then have the composition $fg$ mean first do $f$ and then $g$. The notation has the advantage that diagrams $X\xrightarrow{f}Y\xrightarrow{g} Z$ don't have to be flipped around. It also has the property ...

29

The origin of all these uses is very different. Joe Silverman explained the genesis of the sequence ellipse $\rightarrow$ elliptic integral $\rightarrow$ elliptic function $\rightarrow$ elliptic curve.
Another large class of occurrences of the word "elliptic" is connected with
"trichotomies", that is classifications of some objects into three classes. Such ...

answered May 8 at 14:04

Alexandre Eremenko

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28

Apparently first introduced by Weierstrass in Winter 1862/63 lectures published by H. A. Schwarz (1881, 1885, 1892, 1893), §9:
Mit der Sigma-Function $\mathfrak Su$ ist die Pe-Function $\wp u=\wp(u\mid\omega,\omega')=\wp(u;g_2,g_3)$ durch die Gleichung
$$
\wp u=-\frac{d^2}{du^2}\log\mathfrak S u=\frac{(\mathfrak S'u)^2-\mathfrak S u\mathfrak S''u}{\...

28

Another approach is to present both as theorems, but to present only A as a `marquee result' in the introduction, mentioning that it follows from the stronger but more technical B.

27

Van Vu and I coined the term in our book because there did not seem to be a widely adopted name for it previously. (Gowers, for instance, refers to "number of additive quadruples" rather than "additive energy", but this seemed to be too unwieldy to use for our purposes.) I think we settled on "energy" due to the vaguely quadratic nature of the expression, ...

27

Let me address the question "what happens if some name it has already been used but you don't agree with the choice?", by giving a recent example from (mathematical) physics. The 2012 experiment that discovered a "Majorana fermion" in a superconductor attracted much attention because it would be a realisation of a non-Abelian anyon. The name was a red ...

25

I am not a specialist in either etymology nor the english language (I am not a native speaker of english as well) but since the words scholium and porism have both greek origins, I thought it might be of some interest to add some info on how these words have been used in both ancient and modern greek:
The word "porism" comes from the greek word "Πόρισμα" and ...

answered Feb 1 '17 at 2:21

Konstantinos Kanakoglou

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25

This is not at all intended as a complete answer to the question, but one criterion that feels important is that for a bijection $f$ to count as explicit, one shouldn't need to know in advance that there exists a bijection in order to prove that $f$ is a well-defined bijection. So for example if you order the elements of two sets $A$ and $B$ in some way that ...

24

My name is Martin Crapo, and have always pronounced it Cray-poe. The name evolved from Crapaud and Crepeaux. Pierre Crapo was the French boy who stowed away on his brothers French Merchant ship in the late 1600's. His brother initially refused to bring him on his trip to America from France, so young Pierre boarded the ship unbeknownst to his brother and hid....

24

This variety of groups has indeed been considered in the literature. It is known that the following conditions hold for every group $G$ satisfying the identity $[x,y]^2=1$:
$[[x,y_1,\ldots,y_m],[x,z_1,\ldots,z_n]]=1$ for all
$x,y_1,\ldots,y_m,z_1,\ldots,z_n\in G$ (see [1]).
$[[x_1,x_2],[x_3,x_4]]=[[x_{\pi(1)},x_{\pi(2)}],[x_{\pi(3)},x_{\pi(4)}]]$ for all $...

24

The abelian categories in which all short exact sequences split I would call "split abelian categories", reserving the term "semisimple abelian category" for a more restrictive condition. Roughly, perhaps an abelian category should be called "semisimple" if all objects in it are coproducts of simple objects (a nonzero object is called simple if it has no ...

answered Apr 13 '19 at 0:28

Leonid Positselski

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23

Roger Penrose's abstract index notation for tensors is a relatively modest example, but I think it fits all the criteria of the question. Around 1952, Penrose invented a personal graphical notation for tensors and tensor operations such as contraction and covariant derivatives. It's been described as "fornicating ostriches;" variations on it are referred to ...

23

There's no more serious name for the topograph, as far as I know. And Conway puts a lot of thought into his names, so I think it's best to keep it. I think it's meant to fit into a larger metaphorical system with his rivers and lakes and climbing, which make it so pleasant to study binary quadratic forms. The "topograph" refers to the topographical map ...

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