Euclid's first postulate is
- to draw a straight line from any point to any point,
and in the Elements this is taken as including uniqueness.
In a commentary on Euclid by David Joyce, I recently came across the following complaint about Proposition I.1, to construct an equilateral triangle with a given side:
Zeno of Sidon criticized the proof because it was not shown that the sides do not meet before they reach the vertices. Suppose AC and BC meet at E before they reach C, that is, the straight lines AEC and BEC have a common segment EC. Then they would contain a triangle ABE which is not equilateral, but isosceles.
Zeno recognized that in order to destroy his counterexample it was necessary to assume that straight lines cannot have a common segment. Proclus relates a supposed proof of that statement, the same one found in proposition XI.1, but it is faulty. Proclus and Posidonius quoted properties of lines and circles that were never proven and never explicitly assumed as postulates.
The possibilities that haven’t been excluded are much more numerous than Zeno’s example. The sides could meet numerous times and the region they contain could look like a necklace of bubbles. What needs to be shown (or assumed as a postulate) is that two infinitely extended straight lines can meet in at most one point.
[figure by Joyce]
There is a similar discussion in Beeson et al., "Proof-checking Euclid," https://doi.org/10.1007/s10472-018-9606-x : --
Outer connectivity was discussed already by Proclus, who stated it as “two straight lines cannot have a common segment” [38], p. 168-9, Section 216-17. Proclus says it is implicit in Euclid’s line extension axiom. Neverthless, Proclus considers some possible proofs of it–but not the ingenious proof offered by Potts in the commentary to Prop. I.11 in [37], p. 14, which shows that outer connectivity follows from perpendiculars and the fact that an angle cannot be less than itself. The latter, however, is a difficult theorem, if it (or a close equivalent) is not assumed as an axiom.
I find this all confusing. It seems to me that Proclus is right. Suppose two lines have a common segment. Then they have at least two points in common. But if they have two points in common, then by postulate 1 (which includes uniqueness) they are the same line.
Is there something wrong with this argument?