A parameterization of a plane curve is a map x: I --> R2 of an open Interval I = (a,b) of the real line R into R2. Such a map can be thought of as giving the position of a moving point at a certain time. For example, the parameterization x(t) = (t, 2t) descibes a line. In any time interval, the moving point moves exactly twice as far in the y-direction as in the x-direction. At time t = 0, it is at the origin.
Of particular import to this paper is the cycloid. Its parameterization is given as follows:
x(t) = (t - sin t, 1 - cos t)
and it looks like this:
The cycloid is the path traced by one point of a circle rolling on a line. It is also the solution to the tautochrone problem and the brachystochrone problem. Huygens used it to create a theoretically perfect pendulum (one whose period remained constant).