# Axiom: 1.5: Parallel Postulate

### (Postulate 5 from Book 1 of Euclid's “Elements”)

And that if a straight line falling across two (other) straight lines makes internal angles on the same side (of itself whose sum is) less than two right angles, then the two (other) straight lines, being produced to infinity, meet on that side (of the original straight line) that the (sum of the internal angles) is less than two right angles (and do not meet on the other side).1

if $\alpha+\beta < 180^\circ,$

then

### Modern (Equivalent)2 Formulation

In a plane, through a given point $$A$$ lying not on a given straight line $$g$$, exactly one straight line $$h$$ can be drawn, which is parallel to the given straight line.

### Note from the founder of BookofProofs

In my opinion, from the standpoint of a modern axiomatic method, the 5th postulate could be removed from the “Elements” without changing their validity! As far as I'm aware of, this observation is new and has never been dealt with in literature. I would appreciate receiving any comment, please contact me.

This is why:

• According to the Corollary to Proposition 1.16 of the “Elements”, the existence of straight lines in the plane not meeting at any point is ensured.
• The construction of the lines in the proof of this corollary uses two points for each straight line, through which the parallel lines have to go.
• Therefore, according to postulate 1, this kind of construction also ensures the uniqueness of the constructed parallel lines.
• Moreover, up to the corollary to the 16th proposition of the First Book of "Elements", the 5th postulate is never used by Euclid.
• In other words, all propositions up to the 16th proposition of the First Book do not use the 5th postulate, while the existence and uniqueness of parallel straight lines in the plane is already proven using the remaining four axioms!

Parts: 1
Proofs: 2 3 4 5

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### References

2. The formulations are equivalent, since only if the straight lines $$g$$ and $$h$$ are parallel, the segment $$AB$$ is perpendicular to both straight lines (i.e. the angles $$\alpha,\beta$$ are right angles). In other words, every other straight line drawn through the point $$A$$ would cause the angle $$\beta$$ to become either obtuse or acute. If it became acute, we would have the situation of the first formulation. If it became obtuse, the same formulation would apply for the supplemental angles of $$\alpha$$ and $$\beta$$.