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Postulate L-1. (Linear Ruler Postulate) Given any two distinct points, there is a 1-1 correspondence, called a ruler, between all points with the real numbers that sends one of the two given points to 0 and the other to some number x > 0. Thenumber p assigned to a point P by the ruler is called its coordinate.

What happens if we try to apply Postulate L-1 to the points on a circle? Does anything go wrong? How might we rephrase Postulate L-1 so as to be appropriate on a circle?

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    You have a function $L:\mathbb{R}^2\rightarrow \mathbb{R}^2$ taking two real numbers $a,b$ and returning $L(a,b)=(0,x),$ where $x$ doesn't depend on $a,b.$ This function is not injective. Or is this not what you meant? – 2017-02-16
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    As stated this Linear Ruler postulate doesn't seem very useful. I am assuming that it should have said "all points *on the line detemined by the two points*", but even that correction does not give it much power, as it puts no requirements on the comparative coordinates of the points and their position on the line. You could have point $B$ between $A$ and $C$, but $A$ has coordinate $0$, $C$ has coordinate $1$ and $B$ has coordinate $-10000$. – 2017-02-16

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