TITLE: The origin(s) of the word "elliptic"
QUESTION [34 upvotes]: 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.

Ellipses
Elliptic integrals
Elliptic functions
Elliptic curves
Elliptic genera (in the sense of Hirzebruch)
Elliptic (as opposed to parabolic or hyperbolic) isometries of the hyperbolic plane
Elliptic partial differential operators, elliptic PDEs
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.

REPLY [5 votes]: 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