The names we still use today — parabola, ellipse, hyperbola — are not ours: we owe them to Apollonius of Perga (c. 262–190 BC), the Greek mathematician the ancients nicknamed “the Great Geometer”.
A single family of curves
Before Apollonius people believed that the ellipse, parabola and hyperbola arose from three different kinds of cone, each cut perpendicular to a generating line. Apollonius had the decisive insight: a single cone is enough, and it is the different angles of the cutting plane that produce the three curves. When the plane is parallel to a generating line of the cone, one obtains precisely the parabola.
Why “parabola”
The names chosen by Apollonius describe a property of the areas built on the chords of the curve, and were already in use in Greek geometric language:
- ellipse (from élleipsis, “falling short”) — the constructed area falls short of a reference rectangle;
- hyperbola (from hyperbolé, “excess”) — the area exceeds that rectangle;
- parabola (from parabolé, “application”, “exact comparison”) — the area matches it exactly.
In modern language, for a parabola in the form the square of the vertical chord is directly proportional to the distance from the vertex, with neither deficit nor excess: it is this “exact comparison” that gives the curve its name.
A very long legacy
Apollonius’ Conics was a monumental treatise in eight books and remained the standard reference on conic sections for almost two thousand years. It was the tool with which, in the 17th century, Kepler described the elliptical orbits of the planets and Galileo recognized that a projectile follows a parabolic trajectory: the very curve we study here through the equation (Boyer, Katz).
Links
Topics: Parabola
Concepts: Parabola · Focus · Directrix
People: Apollonius of Perga · Menaechmus