elleipsis

Ellipse comes from Greek elleipsis, meaning 'a falling short, a deficiency, an omission,' from elleipein — en- (in) and leipein (to leave, to fall short). The root leipein gave Greek eclipse (ekleipsis, a forsaking, a failing of light) and the grammatical term ellipsis (an omission of words understood from context). Apollonius of Perga applied elleipsis to the conic section where the relevant geometric quantity falls short of a standard — the complement to the parabola (which equals the standard) and the hyperbola (which exceeds it). He named three curves with three words for comparison: deficiency, equality, excess. The system was elegant enough to survive two thousand years without modification.

Geometrically, an ellipse is the set of all points for which the sum of distances from two fixed points (foci) is constant. This definition makes the circle a special case of the ellipse — one where the two foci coincide. The ellipse is the closed conic section, the one that curves back on itself rather than opening toward infinity. It has two axes of symmetry, a longer (major) and a shorter (minor), and its shape is described by its eccentricity — a number between zero (circle) and one (parabola at the limit). Every ellipse is a circle that has been stretched along one axis, and the stretch is precisely measured by how far the two foci have been separated.

Johannes Kepler's discovery in 1609 that planetary orbits are ellipses rather than circles was one of the pivotal moments in the history of science. The ancient assumption — shared by Ptolemy, Copernicus, and virtually every cosmologist before Kepler — was that celestial bodies move in circles or combinations of circles, because the circle was philosophically perfect. Kepler's ellipses were an aesthetic catastrophe and a physical revelation: the planets do not move in perfect circles, but they move in perfect ellipses, and the Sun sits at one focus, not at the center. The curve that Apollonius named for deficiency turned out to be the shape of the solar system.

The word ellipsis in grammar — the omission of words that can be inferred from context — shares its origin with the mathematical ellipse, and the connection is not purely accidental. Both involve a kind of shortfall: the grammatical ellipsis leaves something out; the geometric ellipse is defined by falling short of a standard. In typographical practice, the three dots of the ellipsis (…) represent omitted text. It is one of the stranger etymological coincidences in English that the mark for 'something missing' and the shape of planetary orbits descend from the same Greek word for deficiency.