# The origin(s) of the word “elliptic” [closed]

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.

• 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. – Wojowu May 8 at 10:28
• The level curves of the principal symbol of an elliptic PDE are ellipses – Thomas Rot May 8 at 10:29
• 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 ($e > 1$). – LSpice May 8 at 16:09
• I answered a question like this in math.SE. – J. M. isn't a mathematician May 8 at 18:24
• I have always found it amusing that one can speak elliptically (omitting words), parabolically (in parables), or hyperbolically (exaggerating), all in contrast to speaking straight. – Timothy Chow May 8 at 22:45