parabolē

Parabola comes from Greek parabolē, meaning 'comparison, juxtaposition, a parable,' from paraballein — para- (beside) and ballein (to throw). To throw beside something: the root ballein is the same that gave Greek ballista (throwing engine), and eventually English ball and ballet. Apollonius of Perga, who systematically named all three conic sections in the third century BCE, chose parabolē for the curve now called a parabola because of a specific technical property he identified: the parabola is the conic section where the square of the ordinate equals the rectangle formed by the latus rectum and the abscissa — the curve is 'equal to' a comparison, in the algebraic sense of that word. The name was a mathematical pun that connected geometric proportion to the rhetorical device of the parable, both of which involve laying one thing beside another to illuminate a relationship.

Apollonius gave all three conic sections names based on the relationship between certain geometric quantities. The ellipse was named elleipsis (deficiency, from elleipein, to fall short) because the relevant quantity falls short of a standard. The hyperbola was named huperbolē (excess, from huperballein, to throw beyond) because it exceeds it. The parabola was named for the case where they are exactly equal — para (beside, equal to). This naming system was one of the great intellectual elegances of ancient mathematics: three related curves, three related names, each name capturing the fundamental algebraic relationship that defines the curve. The same root huperbolē gave English hyperbole (exaggerated speech), and parabolē gave English parable (a comparison, a story set beside a truth).

Galileo's work in the early seventeenth century established that projectiles move in parabolic paths — a conclusion that connected the ancient Greek geometry of conic sections to the actual physics of motion in a way Apollonius could not have anticipated. When a ball is thrown, it traces a parabola; when a cannon fires, the trajectory is parabolic. This discovery made the conic sections suddenly practical: the elegant curves that Greek mathematicians had studied for their formal properties turned out to describe the real motion of physical objects under gravity. The parabola leaped from the pages of Apollonius's Conics to the trajectories of cannonballs and eventually to the paths of spacecraft.

The parabola's geometric property — that every point on the curve is equidistant from a fixed point (the focus) and a fixed line (the directrix) — makes it the ideal shape for collecting or projecting energy. A parabolic mirror brings parallel rays to a single focus; a parabolic antenna collects radio waves from space; a parabolic reflector in a car headlight projects a parallel beam of light. Every satellite dish is a parabola, because the shape that Apollonius named from a word for comparison is also the shape that concentrates distributed signals into a single point. The Greek word for 'beside' is built into every dish on every roof.