I've been reviewing unpackings of Birkhoff's postulates and I may be out of my depth, at least as far as terminology is concerned. I think that I understand the general concept of being able to measure an angle in radians that is between two rays which share a point in common (isn't atan2 used for this?). When I read the third postulate, however, I cannot presently explain the bold part as shown below.
"Postulate III
The set of rays {ℓ, m, n, ...} through any point O can be put into 1:1 correspondence with the real numbers a (mod 2π) so that if A and B are points (not equal to O) of ℓ and m, respectively, the difference am − aℓ (mod 2π) of the numbers associated with the lines ℓ and m is ∠ AOB. Furthermore, if the point B on m varies continuously in a line r not containing the vertex O, the number am varies continuously also."
Are the "numbers associated with the lines" coordinates? Are the numbers associated with the lines different subsets of the reals?
My top priority is to see a worked example of getting an angle measurement from "associated numbers" as per the method indicated in the postulate (assuming I'm interpreting that correctly) in the hopes that it will help me understand to what the terms in the postulate are referring. Icing on the cake would be an explanation of what it means for point B to vary continuously.
See : George Birkhoff, A Set of Postulates for Plane Geometry Based on Scale and Protractor (1932).
The "primitive" (undefined) objects of B's geometry are :
Thus, the system pressupposes the existence (and properties) of real numbers.
Postulate I assumes that there are "enough" points on the line, i.e. that for each real number $r$ there is a corresponding point on the line whose distance is $r$.
Postulate III assumes that we may associate to every ray $m$ a real number $a_m$ ($\mod 2 \pi$), that is its "angle".
The difference $a_m-a_l(\mod 2 \pi)$ between two rays $m$ and $l$ with "common origin" $O$ is equal to the real number "measuring" the angle $∠ AOB$.