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Cartesian Coordinates

Let particle P be described by a chain of events written as

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

Cartesian coordinates were invented by RenΓ© Descartes who is pictured in this engraving.
RenΓ© Descartes, 1596β€”1650.

where each event is characterized by   d \! \bar{r} its displacement vector where

d \! \bar{r} = ( d \! x, \, d \! y, \, d \! z )

Then the abscissa of event k is defined by

\displaystyle x_{k} \equiv x_{\mathsf{o}} + \sum_{i=1}^{k} d \! x_{i}

where x_{\mathsf{o}} is arbitrary and often set to zero. The ordinate is defined as

\displaystyle y_{k} \equiv y_{\mathsf{o}} + \sum_{i=1}^{k} d \! y_{i}

And the  z-cooordinate or applicate of event k is

\displaystyle z_{k} \equiv z_{\mathsf{o}} + \sum_{i=1}^{k} d \! z_{i}

Recall that the  z-component of the displacement  d \! z is just a simple linear function of the wavelength. So if P is isolated then it moves in regular steps along the polar-axis, and can be described by

z_{k} =  z_{\mathsf{o}} +  k \, d \! z

The three numbers  x,  y and  z are called the Cartesian coordinates of event k after the work of RenΓ© Descartes . More exactly, they are the rectangular Cartesian coordinates in a descriptive system that is centered on P. We use them to express the position of an event as

\bar{r} = ( x, y, z )

EthnoPhysics uses a finite categorical scheme of binary distinctions to describe sensation. So  N, the total number of quarks in a description, may be large but not infinite. In principle  N is finite and accordingly displacements may be small, negligible or nil, but not infinitesimal. Later we assume that  N is large enough to make an approximation to spatial continuity. Then the use of calculus may be appropriate.