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

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.
