Cotes's spiral

Cotes introduces his analysis of these curves as follows: “It is proposed to list the different types of trajectories which bodies can move along when acted on by centripetal forces in the inverse ratio of the cubes of their distances, proceeding from a given place, with given speed, and direction.” (N. b. he does not describe them as spirals).

The curves in polar coordinates, (r, θ), r > 0 are defined by one of the following five equations: A > 0, k > 0 and ε are arbitrary real number constants.

Cotes's spirals appear in classical mechanics, as the family of solutions for the motion of a particle moving under an inverse-cube central force.

The analysis is based on the method in the Principia Book 1, Proposition 42, where the path of a body is determined under an arbitrary central force, initial speed, and direction.

Depending on the initial speed and direction he determines that there are 5 different "cases" (excluding the trivial ones, the circle and straight line through the centre).

Following E. T. Whittaker, whose A Treatise on the Analytical Dynamics of Particles and Rigid Bodies (first published in 1904) only listed three of Cotes's spirals,[6] some subsequent authors have followed suit.