David Hilbert's Foundations of Geometry, section 9, Compatibility of the Axioms

Question

I am reading David Hilbert's Foundations of Geometry. In section 9, where he shows the "Compatibility of the Axioms" he begins with the following:

Let us consider the domain $\Omega$ consisting of all those algebraic numbers which may be obtained by beginning with the number one and applying to it a finite number of times the four arithmetical operations (addition, subtraction, multiplication, and division) and the operation $\sqrt{1+\omega^2}$, where $\omega$ represents a number arising from the five operations already given. Let us regard a pair of numbers $(x, y)$ of the domain $\Omega$ as defining a point and the ratio of three such numbers $(a : b : c) \in\Omega$, where $a, b$ are not both equal to zero, as defining a straight line. Furthermore, let the existence of the equation $ax + by +c = 0$ express the condition that the point $(x, y)$ lies on the straight line $(a : b : c)$.

My question is, what is a ratio of three numbers? Is it just a way to express the values that define a line? I have tried to look up a ratio for more that two numbers and found a mix of odd answers. Is he just using an older definition of the word "ratio"?

Answer 1

By "ratio" he means an equivalence class of triples $(a,b,c)$ (with $a,b,c$ not all $0$) where two triples $(a,b,c)$ and $(a',b',c')$ are considered equivalent if there exists a nonzero number $\lambda$ such that $a'=\lambda a$, $b'=\lambda b$, and $c'=\lambda c$. This is just a formalization of the informal notion of a three-way ratio that is commonly used in everyday life, e.g. as described here. The point here is if you are describing a line with an equation $ax+by+c=0$ then the line does not actually uniquely determine the triple $(a,b,c)$, since scaling them by a nonzero constant gives an equivalent equation.