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

"… 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 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 events associated with these two atoms be written as $\mathbf{A}_{\it{i}}$ and $\mathbf{B}_{\it{f}} \,$. Then the length $\ell$ of the distance between these events is

$\ell \equiv \left\| \, \bar{r} ^{\, \mathbf{B}}_{ \it{f}} - \bar{r} ^{\, \mathbf{A}}_{ \it{i}} \vphantom{\sum^{2}} \right\|$

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 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:

Length has been measured at least since ancient Egyptians stretched cords and knotted ropes to survey agricultural fields and construct pyramids. For the last few hundred years, measurement techniques have usually required a measuring rod generically noted by the letter $\sf{L}$. An ideal measuring rod is rigid so its own length $\ell^{\sf{L}}$ is presumably constant. And for calibrated measuring rods $\ell^{\sf{L}}$ is known, conventionally expressed in and abbreviated as (m). To measure the length $\ell^{\mathbf{AB}}$ between two space-time events $\mathbf{A}_{\it{i}}$ and $\mathbf{B}_{ \it{f}}$ count the least number $N$ of copies of $\sf{L}$ that can be fit between the two events. Then

$\left\| \, \bar{r} ^{\, \mathbf{B}}_{ \it{f}} - \bar{r} ^{\, \mathbf{A}}_{ \it{i}} \vphantom{\sum^{2}} \right\| = N \ell^{\sf{L}}$

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, first measure the elapsed time in seconds for a photon to travel from $\mathbf{A}_{\it{i}}$ to $\mathbf{B}_{\it{f}}$. Then $\ell ^{\mathbf{AB}}$ is given in metres by $\Delta t$ multiplied by the integer 299,792,458.

 Next step: making space for models of molecules.
page revision: 289, last edited: 15 Mar 2018 00:00