Although Greek mathematician Archimedes did not discover the spiral that bears his name (see figure-->), he did employ it in his On Spirals (c. 225 BC) to square the circle (geometry) and trisect an angle (geometry). The equation of the spiral of Archimedes is r=aθ, in which a is a constant, r is the length of the radius from the centre, or beginning, of the spiral, and θ is the angular position (amount of rotation) of the radius. Like the grooves in a phonograph record, the distance between successive turns of the spiral is a constant—2πa, if θ is measured in radians.

The equiangular, or logarithmic (logarithm), spiral (see figure-->) was discovered by the French scientist René Descartes (Descartes, René) in 1638. In 1692 the Swiss mathematician Jakob Bernoulli (Bernoulli, Jakob) named it spira mirabilis (“miracle spiral”) for its mathematical properties; it is carved on his tomb. The general equation of the logarithmic spiral is r=ae^{θ cot b}, in which r is the radius of each turn of the spiral, a and b are constants that depend on the particular spiral, θ is the angle of rotation as the curve spirals, and e is the base of the natural logarithm. Whereas successive turns of the spiral of Archimedes are equally spaced, the distance between successive turns of the logarithmic spiral increases in a geometric progression (such as 1, 2, 4, 8,…). Among its other interesting properties, every ray from its centre intersects every turn of the spiral at a constant angle (equiangular), represented in the equation by b. Also, for b=π/2 the radius reduces to the constant a—in other words, to a circle of radius a. This approximate curve is observed in spider webs and, to a greater degree of accuracy, in the chambered mollusk, nautilus (see photograph-->), and in certain flowers.