Length and duration are measures of the extent of motion. They are measured by comparing them with a uniform reference motion. Although uniform linear motion is simpler in theory, uniform circular motion is simpler in practice – and essential for unstopped motion. With one addition, the classic circular clock with hands serves as a reference motion. The addition is to mark the circumference in length units along with the duration units of the angles between the hands and the vertical.

Galileo uses horizontal uniform linear motion to mark length and duration below (from his *Dialogues Concerning Two New Sciences, Fourth Day*):

The horizontal uniform motion of a particle coming from the right at *a-b* is continued with *b-c-d-e* as the horizontal component of the particle descending with uniform acceleration *b-o-g-l-n*. Because the horizontal motion is uniform, it can measure either length or duration of the motion. The vertical component represents the dependent variable, which has the form of a parabola.

To interchange length and duration requires four steps starting with a parametric function in time: (1) replace length components with their radius, which is the stance; (2) switch time and stance; (3) expand time to include angular components; and (4) linearize stance so that it is an independent parameter. The steps (2) and (4) have the effect of inverting the parametric function (2), and then a kind of re-inversion that switches the dependent and independent status of the variables (4).

The result of this interchange process is that the equations of motion for length and duration are interchangeable without functional change. All of the equations of physics in terms of parametric functions of time may be adopted as parametric functions of stance. In that sense it would be best to abstract a functional representation that applies to both length and duration, time and stance.