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Interchange of length and duration

Length and duration are independent 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):

Galileo parabola

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 represent 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 in an equation with a parametric function of time requires five steps: (1) replace length components with their radius, which becomes the base; (2) switch time and base, that is, switch the independent and dependent variables; (3) linearize base, that is, break its dependent relation; (4) bring time under a functional relation with the new parameter, base; and (5) expand time to include angular components. Functions are inverted and the independent and dependent status of variables is switched. An inversion and a kind of re-inversion return to the same function.

In the example above, the horizontal uniform motion which was taken by Galileo to represent time is re-conceived to represent the independent length variable, base. The constant acceleration of the vertical component is re-conceived to represent the dependent duration variable with constant legerity. The quadratic sequence in units of length becomes a sequence in units of duration at a constant rate.

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 base. In that sense it would be best to abstract a functional representation that applies to both length and duration, time and base.

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