What does a ruler equation tell us? Let $l$ be a line in an incidence geometry $\{Y, L\}$ where $Y$ is a set of points and $L$ is a set of lines. Assume there is a distance function $d$ on $Y$. A function $f:l\to \mathbb{R}$ is a ruler (or coordinate system) for $l$ if, $f$ is a bijection and for each pair of points $P$ and $Q$ on $l$ we have that $|f(P)-f(Q)|=d(P,Q)$.

In the Euclidean plane we have the following types of lines, $$L_{a} = \{(a,y)\ |\ y \in \mathbb{R} \}$$
$$L_{m,b} = \{(x,y) \in \mathbb{R}^2 \ |\ y=mx+b\}$$
and there rulers are respectively,
$$f(a,y)=y$$
$$f(x,y)=x\sqrt{1+m^2}.$$

My question is, what exactly does this ruler tell us? What is the difference between a ruler and euclidean distance, i.e. $$d_E(P,Q)=\sqrt{(x_1-x_2)^2+(y_1-y_2)^2}$$
For example if I am given the line $y=2x+3$ the coordinate of $R=(1,5)$ with respect to its ruler, given above, is $f(1,5)=\sqrt{5}$. What exactly is that telling us?
 A: If you're working with the Euclidean plane, the extra information given by the ruler isn't really useful; it's basically just another way to define distance.
The ruler postulate's important in axiomatic geometry for other reasons. If you're working with a geometry that isn't the Euclidean plane, knowing that it satisfies the ruler postulate tells you more than knowing that there's a distance $d(P,Q)$ between any two points $P$ and $Q$. There's two things we gain from the ruler postulate:


*

*Many metrics are incompatible with the ruler postulate: for example, if you defined $$d(P,Q) = \begin{cases}0 & P=Q \\ 1 & P \ne Q\end{cases}$$ then that satisfies the axioms of a distance function, but doesn't give you a ruler. The ruler postulate means that you have a "Euclidean-like" distance function, in some sense.

*Because a ruler on a line $l$ is a bijection between the points of $l$ and $\mathbb R$, it guarantees the existence of lots of points on that line. For example, if you just take the subset of the Euclidean plane consisting of points whose coordinates are algebraic numbers (that is, solutions to polynomial equations), and the corresponding lines, we get a model of almost all the axioms of Euclidean geometry. But there is no ruler to be defined on lines in this geometry.


In other words, the ruler postulate is (among other things) a completeness axiom: it tells you that every line has all the points that a line in the Euclidean plane would have.
