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The word elliptic appears quite often in mathematics; I will list a few occurrences below. For some of these, it is clear to me how they are related; for instance, elliptic functions (named after ellipses, see here) are the functions on elliptic curves over $\mathbb C$. For others, I do not know if there is a relationship at all.

  1. Ellipses
  2. Elliptic integrals
  3. Elliptic functions
  4. Elliptic curves
  5. Elliptic genera (in the sense of Hirzebruch)
  6. Elliptic (as opposed to parabolic or hyperbolic) isometries of the hyperbolic plane
  7. Elliptic partial differential operators, elliptic PDEs
  8. Elliptic cohomology

I am interested in the etymology of this word, in particular, the origins of the different usages listed above. More precisely, I was wondering whether there is, in a way, a single "strain" for all uses of elliptic in mathematics, going all the way back to ellipses in Euclidean geometry.

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    $\begingroup$ Might be more suitable for History of Science and Mathematics SE. $\endgroup$ – Wojowu May 8 at 10:20
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    $\begingroup$ My rough understanding is that elliptic integrals arose when trying to find arc lengths of ellipses, and the these integrals can be viewed as integrals over curves of the form $y^2=$cubic or quartic, hence elliptic curves. From there you get elliptic cohomology and genera. On the flip side, elliptic operators have positive-definite associated quadratic forms, just like equations defining ellipses. It seems to be more of a tree of connections than a single strand. $\endgroup$ – Wojowu May 8 at 10:28
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    $\begingroup$ The level curves of the principal symbol of an elliptic PDE are ellipses $\endgroup$ – Thomas Rot May 8 at 10:29
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    $\begingroup$ Early calculations of the arc length of an ellipse by Euler led to elliptic integrals, then to elliptic functions (Legendre, Abel) and elliptic curves. That is one strain. The elliptic PDE's are a different strain, more closely related to the geometric shape of an ellipse. $\endgroup$ – Carlo Beenakker May 8 at 10:31
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    $\begingroup$ I think it's kind of neat to know that 'ellipses' itself is etymologically derived from 'ellipsis', an omission, and is part of a linguistic, not just mathematical, continuum from 'ellipse' to 'parabola' to 'hyperbola': an ellipse leaves out some eccentricity ($e < 1$); a parabola has just the right amount of eccentricity ($e = 1$); and a hyperbola has too much eccentricity ($e > 1$). $\endgroup$ – LSpice May 8 at 16:09
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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 noted that the integrand of an elliptic integral is (after a change of variables) of the form $dx/\sqrt{f(x)}$, where $f(x)$ is a cubic (or quartic, depending on your preference). This in turn leads to elliptic curves, which are curves of the form $y^2=f(x)$, since then the integrand is $dx/y$, and the integral is on the curve. At this point, one sees that the use of the word elliptic in elliptic curve is quite unfortunate, since the geometry of an elliptic curve is quite different from the geometry of the ellipse from which it derives its name. Further, there is the distinction between a (smooth algebraic) curve of genus $1$, and such a curve with a marked base point that serves as the identity element for its group law. This is especially important if one is working over a non-algebraically closed field, but even over $\mathbb C$, if elliptic curve includes the group law, then it presupposes the choice of a point.

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    $\begingroup$ It might be worth adding that the arc-length of a hyperbola is also given in terms of elliptic integrals, so that they just as well might have been named 'hyperbolic curves' (but that would probably have been an even worse name clash!) $\endgroup$ – Kevin Casto May 9 at 21:19
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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 classifications occur very frequently, and most of them can be traced to the simplest trichotomy "positive, negative, zero". Historically, the oldest non-trivial triple classification is the classification of irreducible plane conics (ellipse, parabola, hyperbola). Later, there was a tendency to use the words elliptic, parabolic, hyperbolic for any such trichotomy (metrics of positive, negative and zero curvature, Riemann surfaces, PDE's, singular points of dynamical systems, linear-fractional transformations, etc.)

Then, when various generalizations occur, mathematicians like to keep the term even in absence of a trichotomy. For example, according to Gromov, a Riemannian manifold is called elliptic if it receives a non-constant quasiregular map from $R^n$. Here the term elliptic comes from the theory of Riemann surfaces (which can be elliptic parabolic or hyperbolic), though in Gromov's situation there is no hyperbolic or parabolic case anymore. Similarly a dynamical system can be "hyperbolic", while the terms "elliptic, parabolic" have no such standard meaning in this case.

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    $\begingroup$ Indeed it's quite disturbing that an elliptic curve is a parabolic Riemann surface... $\endgroup$ – François Brunault May 8 at 15:06
  • $\begingroup$ Also one can have a 'hyperbolic' or a 'parabolic' group, whereas elliptic group would have no standard meaning. $\endgroup$ – Hollis Williams May 8 at 20:22
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    $\begingroup$ @Francois Brnault: It is much more disturbing from my point of view that algebraic CURVE is a Riemann SURFACE:-) $\endgroup$ – Alexandre Eremenko May 8 at 22:07
  • $\begingroup$ @Alexandre Eremenko: $\endgroup$ – Qfwfq May 9 at 0:16
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    $\begingroup$ @Qfwfq: this "cool" feature causes great inconveniences when you write a complicated text involving curves $F(x,y)=0$ and Riemann surfaces, and curves $\gamma(t)$ on these surfaces. $\endgroup$ – Alexandre Eremenko May 9 at 21:05
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As to why the conic section got called ellipse, the introductory chapter of Toomer, Diocles, On Burning Mirrors is interesting. He does not give a conclusive answer, but here's an excerpt, p. 7:

Apollonius found symptomata for all three curves, and defined them by the method of "application of areas", which was the standard Greek procedure for formulating geometrically problems which are, algebraically, equations of the second degree. In the parabola, if the ordinate is $y$ and the abscissa $x$, he represented the symptoma corresponding to equation $y^2 = px$, by saying that the rectangle of side $x$ and area equal to $y^2$ is applied (παραβάλλɛται) to the line-length $p$. In the case of hyperbola [...] he represents the relationship $y^2=x(p+\frac{p}{a}x)$ by saying (see Fig. I) that a rectangle of side $x$ and area equal to $y^2$ is applied to $p$ so that it exceeds it (ύπερβάλλɛι) by a rectangle similar to $\frac{p}{a}$. Similarly [for the ellipse] he represents $y^2=x(p-\frac{p}{a}x)$ by saying (see Fig. II) that a rectangle of side $x$ and area equal to $y^2$ is applied to $p$ so that it falls short of it (έλλείπɛι) by a rectangle similar to $\frac{p}{a}$. Hence he gives the curves the names "parabola", "hyperbola" and "ellipse" respectively.

Toomer then discusses at length "what features in this account appear difficult to sustain in the light of Diocles' treatise".

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This may not be "etymological", but may perhaps shed some light on the relationship between E/P/H things in mathematics:

A. Rastegar, EPH-classifications in Geometry, Algebra, Analysis and Arithmetic (2015).

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The origins of all of the companion terms parabola, hyperbola, and ellipse were coined by Apollonius of Perga, in his classic text "On Conic Sections." (He was born about 262 BC, approximately 25 years after the birth of Archimedes.)

The terms we use are direct descendants of the Greek words. Now, the reason they have the names they is that in the construction of conic sections produced by passing a plane through a cone, a parabola (something literally "thrown beside") differs from a hyperbola (something thrown over) and an ellipse (left out). These are in reference to a "parameter" (Apollonius's term) something "measuring alongside," a parameter exceeding, and a parameter being deficient by (leaving out), an amount.

We use those same Greek words in a variety of unrelated contexts, of course. Hyperbolae refers to statements that are over the top, excessive, and by extension unbelievable. Parabolic seems to be generally in reference to the conic section, but ellipse means something that has been left out, in the same way that Apollonius used it in reference to the construction of his conic section. Generally we think of an ellipse as a figure that can be described as an oval, with two foci. (A circle is a degenerate ellipse, because there is only one focus.) However the reason Apollonius called an ellipse "deficient" is because the same parameter was lacking, in the same way that it was excessive in the hyperbola.

The classic text "On Conic Sections" is part of the Great Books of the Western World Collection, which comprise the core of the Great Books programs taught at St. John's College (Santa Fe, NM, and Annapolis, MD), the University of Chicago, and a few other offshoots -- St. Mary's Moraga, CA for example.

Marklan

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    $\begingroup$ For a nonmathematical appearance of “parabola”. we have the word “parable”, and more: it would seem that there was a Latin verb “parabolare”, to parabolate, I suppose, which is the origin of the Italian verb “parlare” and the French “parler”. So, “parliament”, etc. $\endgroup$ – Lubin May 12 at 0:40
  • $\begingroup$ Excellent point, Lubin. I thought of parable this morning, and was going to edit my post accordingly but see that you mentioned it before I could do so. Thank you. Marklan $\endgroup$ – Marklan May 12 at 13:24

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