"… all points of those curves which we may call geometric [are] those which admit of precise and exact measurement …"1
"… geometry is founded in mechanical practice, and is nothing but that part of universal mechanics which accurately proposes and demonstrates the art of measuring …"2
Thus mensuration is an essential notion for both Cartesian geometry and Newtonian mechanics. So to ensure that measurement is theoretically well founded, WikiMechanics defines a length as the distance between two atoms. Let these two atoms be noted by $\mathbf{A}$ and $\mathbf{B}$. Then the length $\ell$ of the distance between them is
$\ell \equiv \left\| \, \bar{r} ^{\, \mathbf{B}} - \bar{r} ^{\, \mathbf{A}} \vphantom{\sum^{2}} \right\|$
where $\bar{r}$ notes the position. Recall that the distance between any two events is determined by their positions. So this definition of length just adds the requirement that positions are anchored by atoms thus guaranteeing that they are well-defined three-dimensional quantities. A length is a measurable distance. And this implies that some minimum amount of sensory detail is required to logically discuss length. There have to be at least enough quarks involved to make a couple of atoms. This is relevant because as Benoit Mandelbrot has remarked; if the scale of a length measurement is not limited, then as it is made smaller and smaller every approximate length tends to increase steadily without bound.3 So the foregoing definition of length safeguards the possibility of answering questions like: How long is the coast of Britain?Measuring Length
$\ell = 299,792,458 \cdot \Delta t$
The elapsed time depends on the frame of reference F. So the length depends on the frame too. If F is chosen so that the atoms being measured are at rest, then the elapsed time is the proper elapsed time and noted by $\Delta t^{\ast}$. The two increments are related as $\Delta t^{\ast} = \gamma \Delta t$ where $\gamma$ is the Lorentz factor. Similarly, when $\gamma =1$ then the length is called a proper length, noted by $\ell ^{\ast}$ and given by $\ell ^{\ast} \equiv \, 299,792,458 \cdot \Delta t ^{\ast}$. So these lengths are related as
$\ell ^{\ast} = \gamma \ell$
The Lorentz factor for particles in motion is always greater than one, $\gamma ≥ 1$. So observations of moving atoms always measure a smaller length than between stationary atoms, $\ell ≤ \ell^{\ast} \,$. This effect is called length contraction.
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