The problem I have with that version of Euclid's definition is that direction is not defined. Playfair interprets Euclid as follows, I am using my own words he

The problem I have with that version of Euclid's definition is that direction is not defined. Playfair interprets Euclid as follows, I am using my own words here, a straight line is that figure which has the property that if it intersects its moved copy at 2 points it necessarily coincides with it everywhere. "Movement" is undefined, it has to be an isometry. Hilbert's is more abstract and based upon sets. Line is a primitive (undefined name) that interacts with two other undefined names (points and planes) according defined relations (lies on, lies between and is_congruent). More here https://en.wikipedia.org/wiki/Hilbert%27s_axioms https://en.wikipedia.org/wiki/Hilbert%27s_axioms