Horlogerie, Plate IX-7. 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. |

$\Psi ^{\sf{P}} = \left( \sf{P}_{ 1} , \sf{P}_{ 2} , \sf{P}_{ 3} \ \ldots \ \sf{P}_{ \it{k}} \ \ldots \ \right)$

Make the comparison using $\, \delta _{ t} \,$ their temporal orientation, to define $\, \epsilon_{ t} \,$ as the **direction-of-time** for P's events

$\epsilon_{t}^{\, \sf{P}} \equiv \, \delta _{ t}^{\, \sf{ P}} \! \cdot \delta _{ t}^{\sf{\, Earth}}$

When P is like the Earth both particles have the same temporal-orientation; $\delta _{ t}^{\, \sf{ P}} \! = \delta _{ t}^{\sf{\, Earth}} \! = \pm 1 \, \,$. If this condition obtains then

$\epsilon _{t}^{\, \sf{P}} =1$

and we say that P's events are in **historical** **order**. The historical-order is locked into the numerical-order of event-index $k$ using the following nomenclature. Let $\sf{P}_{\it{i}}$ and $\sf{P}_{ \it{f}}$ be an arbitrary pair of events in $\Psi^{ \sf{P}}$. If $\epsilon _{t}^{\, \sf{P}} =1$ and $\it{i} < \it{f} \;$, then we call $\sf{P}_{ \it{i}}$ the **initial** event, and $\sf{P}_{ \it{f}}$ the **final** event of the pair. Later we also use $\epsilon_{t}$ to make a formal definition of time.

Thermal Processes |

warming | $\large{ T_{ \it{f}} \, > T _{\it{i}} }$ |

cooling | $\large{ T_{ \it{f}} \, < T _{\it{i}} }$ |

**history**we imply it is historically ordered. Sooner or later, ice cubes melt away and fires eventually burn-out. We describe these processes as going

*forward*in history. To label initial and final events consistently within this collective understanding of order, consider using a thermal reference provided by the absorption of a tepid particle. Let $\epsilon_{t}^{\, \sf{P}} = 1$ so that P has the same temporal orientation as the Earth. Then $T$, the temperature, of P and the Earth are both on the same side of tepid. Both are either higher or lower. So they both describe the absorption in the same way; as either a warming process or a cooling process. But not one of each. The terms

*initial*and

*final*can be used uniformly for describing both P and the Earth. Then binary description extends agreement to thermally similar particles: Almost any particle is portrayed as going forward in history because almost all of our experience happens in Earth-like conditions.

Summary |

Adjective | Definition | |

Direction of Time | $\epsilon_{t}^{\, \sf{ P}}\equiv \, \delta _{ t}^{\, \sf{ P}} \! \cdot \delta _{ t}^{\sf{\, Earth}}$ | 6-6 |

Adjective | Definition | |

Historical Order | $\epsilon_{t} = 1$ | 6-7 |

Noun | Definition | |

Initial Event | The event with the lowest event-index in a historically ordered pair of events. |
6-8 |

Noun | Definition | |

Final Event | The event with the highest event-index in a historically ordered pair of events. |
6-9 |

Adjective | Definition | |

History | $\sf{\text{A chain of events that is in historical order.}}$ | 6-10 |

Verb | Definition | |

Warming Process | $T_{ \it{f}} > T _{\it{i}}$ | 6-19 |

Verb | Definition | |

Cooling Process | $T_{ \it{f}} < T _{\it{i}}$ | 6-21 |