Horlogerie, Plate IV. Encyclopédie, ou Dictionnaire Raisonné des Sciences, des Arts et des Métiers. Edited by Denis Diderot and Jean le Rond d'Alembert, Paris 1768. Photograph by D Dunlop. |

^{1}so here is a generic description of how to determine elapsed time using a clock. Let particle P be characterized by some repetitive chain of events noted as

$\Psi ^{\sf{P}} = \left( \sf{\Omega}_{1}, \sf{\Omega}_{2} \ \ldots \ \sf{\Omega}_{\it{i}} \ \ldots \ \sf{\Omega}_{\it{f}} \ \ldots \ \right)$

Consider measuring $\Delta t$ the elapsed time between events $\sf{\Omega}_{\it{i}}$ and $\sf{\Omega}_{\it{f}}$. This quantity depends on the frame of reference F which is represented by another chain of events

$\Psi ^{\sf{F}} = \left( \sf{F}_{1}, \sf{F}_{2} \ \ldots \ \sf{F}_{\it{j}} \ \ldots \ \sf{F}_{\it{k}} \ \ldots \ \right)$

Since F is a reference frame, we assume that every report about P is accompanied by an observation of F so that events of F and P can be associated in pairs like

$\left\{ \sf{P}_{\it{i}}, \sf{F}_{\it{j}} \right\}$ and $\left\{ \sf{P}_{\it{f}}, \sf{F}_{\it{k}} \right\}$

To make a laboratory measurment of elapsed time first select some standard clock $\mathbf{\Theta}$ that is presumably part of the frame of reference F. Let this clock be calibrated so that it is characterized by its period $\; \hat{\tau}^{ \mathbf{\Theta}}$. Observe and compare events to determine the numbers $j$ and $k$ by counting clock cycles. Report the result as

$\Delta t = \left( k-j \right) \hat{\tau}^{ \mathbf{\Theta} }$

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