SPIRAL

plane curve that, in general, unwinds around a point while moving ever farther from the point. Many kinds of spiral are known, the first dating from the days of ancient Greece. The curves are observed in nature, and human beings have used them in machines and in ornament, notably architecturalfor example, the whorl in an Ionic capital. Archimedes discovered the spiral that bears his name. The equation of the spiral of Archimedes is r = aq, in which a is a constant; r is the length of the radius, measured from a point O; and q is the angular position (amount of rotation) of the radius. The equiangular, or logarithmic, spiral was discovered by the French mathematician Ren Descartes in 1638; the Swiss mathematician Jakob Bernoulli in 1698 further studied its properties. Its equation is r = a exp q cot f, in which a is a constant. This spiral is related to the circle in that the circle intersects its own radii everywhere at 90; the equiangular spiral intersects its own radii everywhere at the same angle but other than 90, that angle being represented in the equation above by f. This approximate curve is observed in spider webs and, to a greater degree of accuracy in the chambered mollusk, Nautilus, and in certain flowers. Other plane spirals are Euler's, or Cornu's, or Clothoid; Cotes', Fermat's, or parabolic; lituus; Poinsot's; reciprocal, or hyperbolic; and sinusoidal.