I am reading this book, Gödel's Proof, by James R. Newman, at location 117 (Kindle), it says,

For various reasons, this axiom, (through a point outside a given line only one parallel to the line can be drawn), did not appear "self-evident" to the ancients.

Any idea what the various reasons might be? It's self-evident enough to me.


Sorry, my bad, right after the sentence, (the above quote), there is a footnote, says that:

The chief reason for this alleged lack of self-evidence seems to have been the fact that the parallel axiom makes an assertion about infinitely remote regions of space. Euclid defines parallel lines as straight lines in a plane that, "being produced indefinitely in both directions," do not meet. Accordingly, to say that two lines are parallel is to make the claim that the two lines will not meet even "at infinity." But the ancients were familiar with lines that, though they do not intersect each other in any finite region of the plane, do meet "at infinity." Such lines are said to be "asymptotic." Thus, a hyperbola is asymptotic to its axes. It was therefore not intuitively evident to the ancient geometers that from a point outside a given straight line only one straight line can be drawn that will not meet the given line even at infinity.

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    $\begingroup$ I think the choice of words in't good. It's not that they doubted it's self-evidentess, it's that they were convinced it could be proved from the other axioms. $\endgroup$ – Git Gud May 20 '14 at 13:17
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    $\begingroup$ In the first place, the ancients didn't have that version of Euclid's 5th postulate --- that version came much later. But what all versions have in common is that to see them you have to see the whole infinite plane at once, instead of just some finite piece of it, and anything involving the infinite was problematic. $\endgroup$ – Gerry Myerson May 20 '14 at 13:29
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    $\begingroup$ "Euclid undoubtedly knew that any such axiom [the parallel lines axiom] states explicitly or implicitly what must happen in the infinite reaches of space and that any pronouncement on what must be true in infinite space is physically dubious because man's experiences are limited"- Morris Kline, Mathematical Thought from Ancient to Modern Times, Vol. 1, pg. 87. $\endgroup$ – David Mitra May 20 '14 at 13:29
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    $\begingroup$ I turned this into a comment; seems more appropriate as a comment but I had to make it fit somehow. However "self-evident" the parallel postulate may seem to be, it is now well established that there are models of geometry in which the parallel postulate is false, most importantly the hyperbolic plane. The existence of these models was put on solid mathematical ground in the 19th century... $\endgroup$ – Lee Mosher May 20 '14 at 13:38
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    $\begingroup$ Regarding the added footnote, it is interesting because point at the "difficulty" ancient Greeks had with "actual" infinity. Euclid is posterior to Aristotle, who denied the "actual" infinity and asserted only the existence of the "potential" one [we may say : the unlimited iteration of a process]. Thus Euclid - aware of this - takes care to say : "being produced indefinitely". The line is "potentially" infinite. $\endgroup$ – Mauro ALLEGRANZA May 20 '14 at 13:57

I think that it is not correct to say that "ancient hate the parallel postulate".

For sure, it is not so "self-evident" as others [but please, think at Common notion n°5 : "The whole is greater than the part"; until Cantor it was "absolutely" self-evident].

The possible explanation, as per Gerry's comment, is that it involves the infinite, and the infinite is not so easy to manage ...

According to Boris Rosenfeld, A History of Non-euclidean Geometry (original ed.1976), page 36, Euclid was "aware" of this :

Euclid tries to prove as many theorems as possible without using the fifth postulate. The first 28 propositions of Book I are so proved.

According to Rosenfeld [page 40] :

it seems that the first work devoted to this question was Archimedes' lost treatise On parallel lines that appeared a few decades after Euclid's Elements.

The title of this work in known only through the list of Archimedes' works by ibn al-Nadim (ca.990), and

it is possible that one of Ibn Qurra's preserved treatises on parallel lines represents an edited version of Archimedes' treatise.

[...] it is very likely that Archimedes used a definition of parallel lines different from Euclid's. [...] it is possible that Archimedes based his definition of parallel lines on distance.


As finely remarked by mau, the original definition and postulate are [see Thomas Heath, The Thirteen Books of Euclid's Elements . Volume 1 : Introduction & Books I and II (1908 - Dover reprint) ] :

Def 23: Parallel straight lines are straight lines which, being in the same plane and being produced indefinitely in both directions, do not meet one another in either direction.[page 154]

Postulate 5: That, if a straight line falling on two straight lines make the interior angles on the same side less than two right angles, the two straight lines, if produced indefinitely, meet on that side on which are the angles less than the two right angles.[page 155]

Heath's edition comments at lenght definitions and postulates: the comment to P5 span from page 202 to page 220, with a lot of informations about the recorded attempt to prove it, from Proclus on.

Page 220 lists the most common alternatives to Euclid's version of the postulate; among them :

(I) Through a given point only one parallel can be drawn to a given straight line or, Two straight lines which intersect one another cannot both be parallel to one and the same straight line.

This is commonly known as "Playfair's Axiom" - from John Playfair (10 March 1748 – 20 July 1819) - , but it was of course not a new discovery. It is distinctly stated in Proclus' note to Eucl.I.31.


Actually the standard definition of the Fifth Postulate does not involve (of course) infinite: Euclid says that that two lines which cross a third one will eventually meet on the side where the angles made with the third one add to less than two right angles.

Ancient Greeks were uneasy with it because they thought that it could be derived from the other postulates. Euclid's formulation, by the way, is really awkward, since an equivalent statement is "the angles of a triangle sum to two right angles"; I often wonder if he chose the other one so to warn future mathematicians.

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    $\begingroup$ (I wonder why someone downvoted it without a comment explaining why...) $\endgroup$ – mau Jul 7 '14 at 17:35

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