Euclid himself already treats it with gloves, it has an unusually precise formulation, and is not used in the first 28 propositions of the Elements. Why? Did he doubt it? It's not like Euclid was a formalist, in the very first proof of proposition I.1 (construction of equilateral triangle), he freely concludes from the diagram that two circles with common interior points intersect at two points. This is called circle-cirle axiom by Greenberg and is not a postulate. Why is intersection of two circles "self-evident", but (effectively) intersection of two convergent lines is not? Nor does Euclid complicate the first postulate by stipulating uniqueness of a line through two points, even though it is clear from his proofs that he assumes it. The second postulate is also weaker than what Euclid actually uses.

In contrast, with parallels Euclid is very pedantic, as if trying to attract attention to them: "If a line segment intersects two straight lines forming two interior angles on the same side that sum to less than two right angles, then the two lines, if extended indefinitely, meet on that side on which the angles sum to less than two right angles."

Some textbooks blame other instances on Euclid's "flaws" and "gaps", but this is apparently a modernization based on reading Pasch and Hilbert into Euclid. Euclid's method was not axiomatic, it was synthetic. In other words, when he gives a geometric construction he accepts non-deductive conclusions "from the diagram", as long as the conclusions would follow from any possible diagram consistent with the construction. Listed postulates were the ones most commonly used and/or most "subtle" perhaps? In principle there was no need to list the parallel postulate at all, like Euclid didn't list the circle-circle or the uniqueness of the line.

Even if Euclid did want to list something about parallels he could opt for I.30 instead, two lines parallel to a third are parallel to each other. Much simpler. The contrapositive to this (two intersecting lines can not both be parallel to a third) is now well known and used as "Playfair's axiom". Even assuming that Euclid didn't know that it was equivalent, Ptolemy showed it explicitly later, according to Proclus, but that was not deemed satisfactory.

Proof attempts continued for over a thousand years. What proof were geometers looking for? Surely not deducing the fifth postulate "from the other four", since not much of anything can be deduced from the other four, or all five for that matter, not even I.1. As for synthetic proofs relying on "more self-evident" claims, they were offered starting with Archimedes apparently, but never accepted. Why?

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    $\begingroup$ You can see in this post a similar discussion. $\endgroup$ Commented Dec 18, 2014 at 13:14
  • $\begingroup$ Many people did try to prove the parallel postulate from the other four. $\endgroup$
    – Jack M
    Commented Dec 19, 2014 at 15:04
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    $\begingroup$ @Jack M This is due to sloppy writing in elementary textbooks. Hartshorne in Euclid and Beyond gives detailed account of many attempted proofs, not one of them attempts to prove "from the other four". Most authors explicitly introduce extra axioms, even those who don't, like Saccheri, use first 28 propositions of Elements with all the synthetic inferences "from the diagram" about congruence, betweenness and intersections in their proofs. $\endgroup$
    – Conifold
    Commented Dec 21, 2014 at 2:04
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    $\begingroup$ @Willemien Since Ptolemy it was known that Euclid's mouthful can be replaced with "two lines parallel to a third are parallel to each other". They could replace it with that and put the matter to rest, but no. Saccheri showed that existence of a single rectangle, no matter how small, implies the parallel postulate. If that is not "self-evident" then neither is any of the 4 postulates. $\endgroup$
    – Conifold
    Commented Dec 21, 2014 at 2:09
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    $\begingroup$ I think Euclid is happy to infer from the diagram if it is intuitively clear that a small change in the diagram wouldn't change the relevant property (i.e. two circles that meet will still meet if one of them is moved slightly). But if an arbitrarily small change in the configuration (i.e. one of two parallel lines being 'turned' ever-so-slightly) could change the property (e.g. make them meet), then he is unwilling to infer from the diagram and wants to deduce (or assume) explicitly and verbally. $\endgroup$ Commented Sep 30, 2017 at 22:39

1 Answer 1


The reasons are "simple". All other axioms and postulates appeal to our "everyday experience", at least in principle. The straight lines correspond to light rays in everyday experience. However it was probably already recognized by Euclid, that the parallel postulate is different from the other axioms. Of course, a degree of mathematical sophistication is required to understand this. But probably Euclid already understood this. Even when stated accurately, the V postulate looks much more complicated than the other axioms and postulates.

How indeed would you verify it experimentally? The postulate itself says that through every point A not on the line L one can draw only one line parallel to L. How would you propose to check this experimentally? There are clearly many lines through A which intersect L so far away that you cannot see this. So they do not intersect within your field of view.

Or one can try to verify any of its consequences. One of the simplest consequences is that the sum of the angles of a triangle is equal to two right angles. How can you verify that this holds in real life? No measurement, no matter how accurate will show you this.

Gauss and Lobachevski, who recognized that the postulate does not follow from the rest of the axioms, indeed discuss its possible experimental verification. One has to measure the angles of a very large triangle to do this. And any result that you obtain will have some error in the measurement, and leave the possibility that if you take a larger triangle, you will see that the sum is not equal to two right angles.

EDIT. To obtain a better intuitive understanding what the axioms mean, imagine that you live in a world in which one of the axioms is not satisfied, and explore how different this word looks. For example, suppose that two light rays can intersect at two points. This means that under certain conditions you will see the same object as two objects. Our everyday experience shows that this is not the case in our world.

Now imagine a world where the parallel postulate does not hold. Will you see something peculiar in your everyday life? The answer is "no". Everything will look more or less the same. Until you start measuring the angles of large triangles. But we do not really have everyday experience with measuring angles of large triangles.

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    $\begingroup$ That's exactly what I don't understand: what makes it so different? Can we verify that there is only one line between any two points, no matter how far apart? That any line extends indefinitely far? That circles really intersect? That convergent lines intersect is no less or more intuitive or testable. Spherical geometry is consistent with Euclid's postulates as written (parallel postulate is vacuously satisfied), he rules it out using synthetic arguments by I.16, which equally do not "follow" from the rest of the axioms and postulates in the modern sense. $\endgroup$
    – Conifold
    Commented Dec 18, 2014 at 20:27
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    $\begingroup$ Quine points out that straight lines correspond not just to light rays but to several different phenomena of everyday life: they are the shape of the edge of a folded paper (because a straight line is the intersection of two planes) and, most pertinently, they are the shape of a string stretched between two points. (The name “line” derives from this last; it literally means a string, as in fishing line, and is cognate with the linen from which it is made.) The line as a stretched string and as a light ray both derive from the fact that it is the shortest distance between two points. $\endgroup$ Commented Dec 20, 2014 at 20:50
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    $\begingroup$ Indeed, Democritus credits Egyptian "rope-stretchers" with teaching geometry to Greeks. Apollonius talks about straight lines as abstractions from walls and roads, and how the idea of length comes from that, see p.210 in Lucio Russo's link.springer.com/article/10.1007/s004070050016. Archimedes explicitly postulates that straight line is the shortest distance between two points in one of his books. $\endgroup$
    – Conifold
    Commented Dec 21, 2014 at 2:34
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    $\begingroup$ Walls and roads are not a good standards for straight lines. Everyone knows that walls and especially roads can be curved. How can we tell than a wall is straight (flat). Only by comparing it to the light rays. How can you tell that your ruler is straight? By comparing it with the light ray. $\endgroup$ Commented Dec 21, 2014 at 9:28

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