(related to Axiom: 1.1: Straight Line Determined by Two Distinct Points)
Note: If we have constructed two points \(A\) and \(B\) on a sheet of paper, and if we construct a segment from \(\overline{AB}\), this segment will have some irregularities due to the spread of ink or slight flaws in the paper, both of which introduce some height and width. Hence, it will not be a true geometrical segment no matter how nearly it may appear to be one.
This is the reason that Euclid postulates the construction of segments, rays, and straight lines from one point to another (where our choice of paper, application, etc., is irrelevant). For if a segment could be accurately constructed, there would be no need for Euclid to ask us to take such an action for granted. Similar observations apply to the other postulates. It is also worth nothing that Euclid never takes for granted the accomplishment of any task for which a geometrical construction, founded on other problems or on the foregoing postulates, can be provided.
