Length
René Descartes, //La Géométrie,// book 2 page 319, Paris 1637. Roman font with archaic ligatures.
René Descartes, La Géométrie, book 2 page 319, Paris 1637. Roman font with archaic ligatures.
Historically, measurement is an important part of geometry. According to René DescartesXlink.png

"… all points of those curves which we may call geometric [are] those which admit of precise and exact measurement …"1

Nowadays mathematicians are less constrained, but Descartes is clear about his analytical geometry; it presumes measurement. This also seems to have agreed with Isaac NewtonXlink.png as he set down the laws of motion fifty years after the publication of the text shown in the accompanying photograph. He wrote that

"… 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 MandelbrotXlink.png 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?Xlink.png

Measuring Length

Length has been measured at least since ancient Egyptians stretched cords and knotted ropes to survey agricultural fields and construct pyramids. For hundreds of years, calibrated measurement techniques have usually required some kind of a measuring rod. An ideal measuring rod is rigid so its own length is presumably constant. To measure the length between $\mathbf{A}$ and $\mathbf{B}$ count the least number of rods that fit between them. Lengths are conventionally expressed in metresXlink.png and abbreviated as (m). The requirement for a least number is based on the historical practice of stretching a rope or surveyor's chain. More recently an optical method has been adopted to measure length. It requires a clock to determine an elapsed time $\Delta t$. To optically measure a length in meters, first measure the elapsed time in seconds for a photon to travel from $\mathbf{A}$ to $\mathbf{B}$. Then

$\ell = 299,792,458 \cdot \Delta t$

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 of the atoms in $\sf{F}$. Similarly, when $\gamma =1$ then the length is called a proper length, noted by $\ell ^{\ast}$ and given by

$\ell ^{\ast} \equiv \, 299,792,458 \cdot \Delta t ^{\ast}$

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.

Right.png Next step: making space for models of molecules.
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