# Is Euclid's Postulate 5 really a postulate of parallels ?

Is Euclid's Postulate 5 really a "postulate of parallels"?

Euclid's Postulate 5 is widely regarded as synonymous with the Postulate of Parallels. Now, I am a philosopher by training and not a mathematician. In fact I failed 'O' Level Maths with the lowest possible grade. But I'd be interested to know, has this ever been an issue in the mathematical literature? Because this judgement appears to me to be mistaken, for at least three reasons:

1. Postulate 5 is worded to deal exclusively with the case where the interior angles add up to less than two right angles. That is to say, Euclid has chosen (and presumably not accidentally) a form of words which expressly excludes parallel lines from consideration.

2. Any triangular figure that we can conceptualise in conformance with Postulate 5 can, in principle, be fully described by a set of finite dimensions (since the angles will be of a definite magnitude and distance apart on the baseline, and the lines produced therefrom will intersect at a definite point). But parallel lines do not intersect, even if they are produced indefinitely; so the lines which could be measured under Postulate 5, are not measurable under the Postulate of Parallels. Parallel lines therefore have a different logical status to the lines drawn under Postulate 5; they are 'complete' lines, as opposed to 'line segments'. This moves the logical goalposts and, if we wanted to make Postulate 5 embrace the parallel case, we would need to add something to it to account for the nature of this shift.

3. If Postulate 5 is synonymous with the Postulate of Parallels, then Definition 23 is redundant. But Euclid, obviously, did not think it redundant. I take this to imply that he himself did not consider Postulate 5 to be synonymous with the Postulate of Parallels.

Of course, when we read Definition 23 and Postulate 5 together, we are irresistibly led to infer the Postulate of Parallels; but an inference is not the same thing as a logical equivalence.

Why did Euclid not clearly and unambiguously enunciate the Postulate of Parallels, since he presupposes it throughout the Elements? Well, the parallel postulate was a hot potato, even in his time. Was he trying, perhaps, to avoid leaving his system hostage to an issue he knew could not be resolved by the mathematics of his time?

lavinia
Gold Member

Euclid's Postulate 5 is widely regarded as synonymous with the Postulate of Parallels. Now, I am a philosopher by training and not a mathematician. In fact I failed 'O' Level Maths with the lowest possible grade. But I'd be interested to know, has this ever been an issue in the mathematical literature? Because this judgement appears to me to be mistaken, for at least three reasons:

1. Postulate 5 is worded to deal exclusively with the case where the interior angles add up to less than two right angles. That is to say, Euclid has chosen (and presumably not accidentally) a form of words which expressly excludes parallel lines from consideration.

2. Any triangular figure that we can conceptualise in conformance with Postulate 5 can, in principle, be fully described by a set of finite dimensions (since the angles will be of a definite magnitude and distance apart on the baseline, and the lines produced therefrom will intersect at a definite point). But parallel lines do not intersect, even if they are produced indefinitely; so the lines which could be measured under Postulate 5, are not measurable under the Postulate of Parallels. Parallel lines therefore have a different logical status to the lines drawn under Postulate 5; they are 'complete' lines, as opposed to 'line segments'. This moves the logical goalposts and, if we wanted to make Postulate 5 embrace the parallel case, we would need to add something to it to account for the nature of this shift.

3. If Postulate 5 is synonymous with the Postulate of Parallels, then Definition 23 is redundant. But Euclid, obviously, did not think it redundant. I take this to imply that he himself did not consider Postulate 5 to be synonymous with the Postulate of Parallels.

Of course, when we read Definition 23 and Postulate 5 together, we are irresistibly led to infer the Postulate of Parallels; but an inference is not the same thing as a logical equivalence.

Why did Euclid not clearly and unambiguously enunciate the Postulate of Parallels, since he presupposes it throughout the Elements? Well, the parallel postulate was a hot potato, even in his time. Was he trying, perhaps, to avoid leaving his system hostage to an issue he knew could not be resolved by the mathematics of his time?

I am a little confused by your second point so won't comment on it.

My impression - and this is not based on any knowledge of history at all - is that there is a principle of symmetry at work here. When I learned geometry in high school we were given the postulate that a line divides a plane into two half planes. Implicit was the idea that the half planes are identical in all of their geometric properties. From this it would seem to follow that if two lines intersect a third a right angles they must be parallel - for otherwise the symmetry of the two half planes that the third line determines would be violated. Something would distinguish one half plane from the other.This is because two lines can only intersect in one point and that point can lie in only one of the half planes. So the existence of parallels does not seem to be in doubt.

What is in doubt is whether the parallels are unique. The fifth postulate says that they are.

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lavinia
Gold Member

Here is a shot at your second point.

Another postulate that I was taught is that two points determine a unique line. Therefore if I can construct a perpendicular I can construct parallel lines. They are as you say - line segments - just like any other line.

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mathwonk
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To (a mathematician like) me #1 is entirely irrelevant. Postulate 5 gives conditions under which two lines are not parallel.

Since only one configuration fails to satisfy the hypotheses of his postulate, namely the hypotheses of Proposition I.27, and since one can construct a perpencidular to a line from any point on or off it, by Prop I.11, I.12, his postulate implies the statement that there is at most one line parallel to a given line L through a point P off L. Namely if M is a line perpendicular to L through P, the only line through P that could be parallel to L would be the perpendicular to M through P.

I do not know what version of the "postulate of parallels" you are referring to, but this statement we have just derived from Euclid's postulate, that there is at most one line parallel to a line L through a point P off L, or Playfair's postulate, is a common version.

Since also by Prop I.17, two common perpendiculars to the same line are indeed parallel, and we have remarked that perpendiculars exist, it follows that in fact there is also at least one parallel to a given line L through any point P off it, namely the line described at the end of the sentence before the previous one.

These statements also show that playfair's postulate implies Euclid's postulate 5. I.e. since there do exist parallel lines violating the hypotheses of Euclid's postulate, if there is at most one parallel, then the ones satisfying Euclid's hypotheses must not be parallel.

It is also clear then that in the presence of the other postulates of Euclid (modulo filling the well known topological gaps in his proofs, and possibly taking Prop. I.4 as an axiom), the three following statements are logically equivalent: 1) Euclid's postulate, 2) Playfair's postulate, 3) the postulate I learnt in school, namely there exists a unique line parallel to a given line and passing through a given point.

To me that is essentially the end of the matter, except to explore the cases where all these postulates fail, i.e. hyperbolic geometry. But I am not a philosopher.

AlephZero
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This was indeed a hot topic in Euclid's time, since people were trying to prove the existence of unique parallels from other more "obvious" postulates.

However the whole debate is now dead, in the sense that Gauss and others showed that the three possible postulates that "there are zero, one, or many lines parallel to a given line through a given point not on the line" are all equally self-consistent with the other Euclidean axioms.

The historical significance of Euclid's attempt a creating a formal axiomatic system of Geometry is of course immense, but the "Elements" is as full of logical holes as a Swiss cheese, so there is not much to be gained from trying to criticise it piecemeal. If you want to see a more modern attempt at a rigorous set of axioms for elementary geometry, read Hilbert's "Foundations of Geometry" (written round about 1900).

lavinia
Gold Member

However the whole debate is now dead, in the sense that Gauss and others showed that the three possible postulates that "there are zero, one, or many lines parallel to a given line through a given point not on the line" are all equally self-consistent with the other Euclidean axioms.

The zero parallel geometry is not consistent with Euclid's postulates The only other possibility is more than one parallel.

If you agree that a line separates a plane into two disjoint half planes and accept the postulate that two lines intersect in exactly one point, then you must have parallels as has been shown in this thread already.

If you do not agree that a line separates a plane into two disjoint half planes - and this would give you a non-Euclidean geometry - then you could construct a geometry where two lines intersect always and in only a single point. But this geometry violates Euclid's postulates.

mathwonk
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The swiss cheese comparison is somewhat subjective, so could be correct, and I respect that opinion, but after teaching through Euclid last year, I decided that in my own opinion there is really very little logical flaw in Euclid.

The only flaws I saw are the failure to make explicit as axioms, a couple of assumptions which are clearly used in some proofs.

The two main gaps are:
1) in the proof if Prop 1.4 (SAS) he uses "rigid motions", so one should either assume Prop 1.4 as Hilbert does as an axiom, or assume the existence of rigid motions, as I myself prefer to do. (These choices are equivalent.)

2) Although he speaks in several places of the "sides" of a line in the plane, he forgot to say explicitly that a line does divide the plane into two sides, so one should say this, as Hilbert does.

As Bubba says in Forrest Gump, that's about it, and I think those are very minimal "flaws". The propaganda about Euclid has been so pervasive that it has obscured in modern times the wonderful merits of this book, which is still by far the best geometry book in existence.

Modern books which spend scores or hundreds of pages belaboring the consequences of the axiom that a line has two sides, make the subject unbearably boring to students and essentially unteachable.

For a marvellous treatment that combines Euclid and Hilbert, see the beautiful book Geometry: Euclid and beyond, by Hartshorne. This book might be called "handbook to Euclid" as it requires one to read Euclid simultaneously.

AlephZero
Homework Helper

The swiss cheese comparison is somewhat subjective, so could be correct, and I respect that opinion, but after teaching through Euclid last year, I decided that in my own opinion there is really very little logical flaw in Euclid.

The only flaws I saw are the failure to make explicit as axioms, a couple of assumptions which are clearly used in some proofs.

The two main gaps are:
1) in the proof if Prop 1.4 (SAS) he uses "rigid motions", so one should either assume Prop 1.4 as Hilbert does as an axiom, or assume the existence of rigid motions, as I myself prefer to do. (These choices are equivalent.)

2) Although he speaks in several places of the "sides" of a line in the plane, he forgot to say explicitly that a line does divide the plane into two sides, so one should say this, as Hilbert does.
Sure, like any book, you have to read it in the spirit in which it was written. It seems to me Euclid's "zeroth axiom" is that there is some Platonic Form of "real world geometry" which he is describing. As such, the idea that there might be alternative geometries with different axioms is completely alien, and it seems the greek geometers' main question about the parallel postulate was how best to make it seem "obvious", not whether or not it was "true".

Another aspect of the Elements is that it does not appear to be a treatise on "the whole of Greek geometry", but a rather minimal set of theorems building up to the construction of the Platonic solids in the final book. For example he tends to proves the converses of theorems only if he is going to use the results, and not just because the converse happens to be true. (I'm talking about the original here in so far as we have it, not the many additions made over the centuries by translators and commentators). It may be that this approach led to his particular choice of axioms relating to parallels, though the reasoning behind the choice is not in the book.

There are many places where the reasoning implicitly depends on the appearance of the figure, often in choosing whether to add or subtract quantities. You probably know the spoof proof that all triangles are isosceles, based on drawing a plausible but impossible figure, and it is hard to see how Euclid would have refuted it, except by accurate drawing (but you can't accurately draw every possible triangle, of course).

For pedants, the problems start with Book 1 Proposition 1, to draw an equilateral triangle: Given one side, he draws two circles from the end points, in the obvious way, and then assumes (1) the circles intersect somewhere and (2) it doesn't matter which intersection point you choose to complete the triangle. (In fact I don't think he even mentions that there are two intersection points).

As Bubba says in Forrest Gump, that's about it, and I think those are very minimal "flaws". The propaganda about Euclid has been so pervasive that it has obscured in modern times the wonderful merits of this book, which is still by far the best geometry book in existence.

Modern books which spend scores or hundreds of pages belaboring the consequences of the axiom that a line has two sides, make the subject unbearably boring to students and essentially unteachable.
I entirely agree that pedantry is not the same as rigor (and vice versa!). I thnk the key is the same as in the modern concept of computer software design: information hiding. Package up the rigor so you know it is still there, but you can forget the pedantry surrounding it.

For a marvellous treatment that combines Euclid and Hilbert, see the beautiful book Geometry: Euclid and beyond, by Hartshorne. This book might be called "handbook to Euclid" as it requires one to read Euclid simultaneously.
I might put that on my next Christmas/birthday present list.

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AlephZero
Homework Helper

The two main gaps are:
1) in the proof if Prop 1.4 (SAS) he uses "rigid motions", so one should either assume Prop 1.4 as Hilbert does as an axiom, or assume the existence of rigid motions, as I myself prefer to do. (These choices are equivalent.)

This is interesting, if you look at Props 1.1 and 1.2.

If you take it that Euclids 'compasses' were like a modern pair of dividers with two points, for drawing on a sand table, then in 1.1 he is starting each circle with the compasses in identical positions, and rotating about the two ends to draw the two circles.

However the reason for putting 1.1 first is not because he wants to start the book with theorems about equilataral triangles, but because he needs it to prove 1.2, which is the apparently pedantic statement that a line of given length can be drawn with one end at a given point. Clearly if you are allowed to set your compasses to the length (defined by the length of a given line) and perform a rigid motion, this is self evident, but Euclid avoids the rigid motion assumption by a constructing an equilateral triangle and doing some arithmetic.

lavinia
Gold Member

I wonder why the parallel postulate was the method for defining euclidean geometry historically. Similar triangles seems more intuitive and also suggests alternative geometries.

lavinia
Gold Member

The fifth postulate seems to give more information than just the uniqueness of parallels. It tells you which half plane the two lines intersect in. I wonder if this is necessary. If one just postulated that when the interior angles sum to less than pi that the lines intersect somewhere is sufficient.

mathwonk
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Lavinia, similarity seems to me a much more sophisticated notion than congruence. It is usually the first non trivial equivalence relation one meets in high school, and confusion seems to begin there for many students.

Aleph, of course it is true that Props. 1.1 and 1.2 (?) have the flaw you mention, but it is of the same nature as the one about sides of a line, and I was concentrating attention on the two most significant gaps to me, 1) SAS and 2) topology of separation.

I still maintain that if one reads Euclid carefully one will find very little to quibble about and infinitely more to praise.

As for taking non trivial separation phenomena for granted without mention, look at a modern book like Arnol'd's wonderful Ordinary differential equations. In the first few pages he "solves" a problem in dynamical systems by reducing it to the fact that two continuous paths inside a plane rectangle, each joining diagonally opposite corners, must meet.

He gives no proof of this rather non trivial fact, which is quite similar to Euclid's assumptions about circles meeting, and yet he says he is finished.

Last time I taught from Euclid, we made it a game of finding holes in the proofs. As we read, we tried to spot logical gaps, and as we did so, we wrote them down, augmenting his axioms with our additional ones as we went along. Of course most of them occur near the beginning. (I have attached below our list, surely not a perfect one, for your amusement.)

After some time we compared to Hilbert's axioms and found that we had noticed the need for pretty much all of them. I feel strongly that compared to what is there, the parts that are missing are truly minuscule.

And it was reassuring to note that Hilbert too made a mistake (in the 1902 version I have). He asserted without full proof that his axioms were independent, without noticing, possibly because he had divided them into axioms about the line and axioms about the plane, that his separation axiom for the plane was so strong that it implied his separation axiom for the line. This was first pointed out by E.H.Moore.

In later editions of his book he rectified this. If you read Hartshorne carefully and compare to Hilbert you will also notice there are some differences between Hartshorne's version of Hilbert's axioms and Hilbert's own version. Indeed there are many versions of Hilbert's own book, so no perfect way to say definitively exactly what his axioms are.

Most things in life are somewhat in flux, even in mathematics.

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To (a mathematician like) me #1 is entirely irrelevant. Postulate 5 gives conditions under which two lines are not parallel.

Since only one configuration fails to satisfy the hypotheses of his postulate, namely the hypotheses of Proposition I.27, and since one can construct a perpencidular to a line from any point on or off it, by Prop I.11, I.12, his postulate implies the statement that there is at most one line parallel to a given line L through a point P off L. Namely if M is a line perpendicular to L through P, the only line through P that could be parallel to L would be the perpendicular to M through P.

I do not know what version of the "postulate of parallels" you are referring to, but this statement we have just derived from Euclid's postulate, that there is at most one line parallel to a line L through a point P off L, or Playfair's postulate, is a common version.

Since also by Prop I.17, two common perpendiculars to the same line are indeed parallel, and we have remarked that perpendiculars exist, it follows that in fact there is also at least one parallel to a given line L through any point P off it, namely the line described at the end of the sentence before the previous one.

These statements also show that playfair's postulate implies Euclid's postulate 5. I.e. since there do exist parallel lines violating the hypotheses of Euclid's postulate, if there is at most one parallel, then the ones satisfying Euclid's hypotheses must not be parallel.

It is also clear then that in the presence of the other postulates of Euclid (modulo filling the well known topological gaps in his proofs, and possibly taking Prop. I.4 as an axiom), the three following statements are logically equivalent: 1) Euclid's postulate, 2) Playfair's postulate, 3) the postulate I learnt in school, namely there exists a unique line parallel to a given line and passing through a given point.

To me that is essentially the end of the matter, except to explore the cases where all these postulates fail, i.e. hyperbolic geometry. But I am not a philosopher.

Mathwonk, I thank you for the courtesy of your reply, and the intellectual rigour which you have imported to the argument. Unfortunately, the absurd "time out" restrictions imposed by this site have obliterated the reply which I spent an evening elaborating.

So I'll self-impose a two-minute limit and say that:

1) arguments from non-Euclidean geometry are irrelevant to my thesis and need not be addressed.

2) because, as Hegel said, you cannot defeat the enemy where he is not. Mathwonk, this applies to your arguments also. Everything you say is true and full of point, but if you wish to rebut my arguments, you must argue on the basis of what Euclid says up to and including Postulate 4. Arguments from specific propositions are arguments "post facto" and can be disregarded. By arguing from propositions, you concede my argument that Euclid assumes Playfair's Axiom, but does not state it explicitly.

3) Thank you Lavinia for being the first to answer!

I wonder why the parallel postulate was the method for defining euclidean geometry historically. Similar triangles seems more intuitive and also suggests alternative geometries.

Very interesting point. I think it is because of all Euclid's postulates and common notions, it seems to be the most logically complex. Somehow one feels that it just has to be a theorem. When you think that a proposition so apparently simple as "the shortest distance between two points is a straight line" is not an axiom, but a theorem, you can begin to understand why the postulate 5 has exercised such a fascination.

lavinia
Gold Member

Alan as you have not directly responded to the answers, I am going to give my take one more try.

1. Postulate 5 is worded to deal exclusively with the case where the interior angles add up to less than two right angles. That is to say, Euclid has chosen (and presumably not accidentally) a form of words which expressly excludes parallel lines from consideration.

It seems to me that the existence of parallels is not in question. Only their uniqueness. That is why it makes sense that these words were chosen. Further it would not surprise me if Euclid's goal was to show that the properties of space ,including its metrical properties ,could be derived from primitive notions of line, plane and, point. The existence of parallels is easily derived from primitive notions. The uniqueness was not known but probably was suspected to be a theorem whose proof had not been found yet.

And it is almost correct that the geometry of a plane can be derived from primitive notions. This is what astounds me. There is only one other possibility. To me the whole idea is that there is a unique intrinsic geometry.

Saying things this way makes me take back what I said about similar triangles. This idea requires metrical relations and so to derive geometry from them one must have metrical relations already - outside of the constructs of the geometry. But what is desired it to show that metrical relations are intrinsic and therefore a necessary consequence of primitive notions.

2. Any triangular figure that we can conceptualise in conformance with Postulate 5 can, in principle, be fully described by a set of finite dimensions (since the angles will be of a definite magnitude and distance apart on the baseline, and the lines produced therefrom will intersect at a definite point). But parallel lines do not intersect, even if they are produced indefinitely; so the lines which could be measured under Postulate 5, are not measurable under the Postulate of Parallels. Parallel lines therefore have a different logical status to the lines drawn under Postulate 5; they are 'complete' lines, as opposed to 'line segments'. This moves the logical goalposts and, if we wanted to make Postulate 5 embrace the parallel case, we would need to add something to it to account for the nature of this shift.

I still don't see this point. I'm not sure what you mean by the lines that care measurable or not measurable under postulate 5. What are the logical goalposts?

3. If Postulate 5 is synonymous with the Postulate of Parallels, then Definition 23 is redundant. But Euclid, obviously, did not think it redundant. I take this to imply that he himself did not consider Postulate 5 to be synonymous with the Postulate of Parallels.[/QUOTE]

Postulate five is equivalent to the postulate of parallels, again because it guarantees uniqueness. It seems like a logical equivalence to start with the postulate of parallels and then show that two lines that cut a third at different angles must intersect, or to postulate that two lines that cut a third at different angles must intersect then prove the uniqueness of parallels. I do not see a logical difference.

Of course, when we read Definition 23 and Postulate 5 together, we are irresistibly led to infer the Postulate of Parallels; but an inference is not the same thing as a logical equivalence.

So can explain you explain the philosophical point again?

mathwonk
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Alan, forgive me, I did not mean to try to rebut your arguments, as I respect them too much. i suspect we have very different perspectives and you probably understand mine better than I do yours. i just wanted to present mine for your possible use. I am an old man, and have lost interest in intellectual jousting. I don't want to win any arguments, I just like to discuss and learn. Thank you for the courtesy of your response as well. You are fortunate that you have hooked the brilliant lavinia in your net.

My apology for my discourtesy in not replying to anybody in over a week. It was not from personal choice.

Thank you for your remarks, Mathwonk. I guess perspective does have a lot to do with it. My own is the perspective of mathematical philosophy rather than mathematics per se, for which (alas) I do not have much ability, though a great deal of interest. But to comment:

"The only flaws I saw are the failure to make explicit as axioms, a couple of assumptions which are clearly used in some proofs."

I agree that one should not place too much emphasis on this. I doubt that a rigorously comprehensive set of axioms can ever be achieved for a deductive system like Euclid's geometry; no matter how many axioms you have, it is always possible to think of one more; eg "knowledge is possible", or "axiomatic statements can lead to deductive knowledge". When Descartes embarked on an enterprise not so very different to Euclid's, he began with "I think, therefore I am"! Euclid would probably have thought that was going a bit far. I'm sure that after stating his definitions and primitive propositions, he would (metaphorically speaking) have said "right guys, that gives you enough to see where I'm coming from, now let's just get on with it, shall we? If I've missed anything, supply it for yourself."

With one proviso, my philosophical quibble is not primarily with what Euclid said or didn't say, but with what so many of his commentators seem to think he said. My interest was kicked off by a text (unfortunately I didn't make a note of the reference) where the commentator - a university mathematician - began his explication of Postulate 5 with the sentence "This is, of course, the famous Axiom of the Parallels"! Now, in informal discussion it might be useful to treat the two as though they were synonymous, but to put such a statement into a textbook is to take conversational shorthand too far.

The proviso I refer to is that in my opinion, to subject a primary source to a detailed, sentence-by-sentence critical analysis is not (or should not be) a negative exercise; to be read with the same kind of attention that he or she gave to the writing, is no more than an author's due.

Thanks for the Hartshorne reference, Mathwonk - I will try to source that.

AlephZero, I think you are very right to draw attention to the Platonic connection. Euclid was a Platonist, except in a very few particulars not inconsistent with Platonism - for example, his definition of a point is clearly influenced by Atomism. Everybody knows that Plato caused to be written above the doors of the Academy, "Do not enter if you have not studied geometry" (or words to that effect). Conversely, it is not possible to get the full flavour of Euclid without knowing something about Plato's theory of Forms (or Ideas). All of the figures which Euclid discusses must be thought of as Ideas in the world of ultimate reality (which is not the world in which WE live). This is why we have to be very careful about explicating Euclid in terms and analogies drawn from the world of our everyday experience

Lavinia, I still owe you a reply, but not here - this is already too long.

To (a mathematician like) me #1 is entirely irrelevant. Postulate 5 gives conditions under which two lines are not parallel.

Since only one configuration fails to satisfy the hypotheses of his postulate, namely the hypotheses of Proposition I.27, and since one can construct a perpencidular to a line from any point on or off it, by Prop I.11, I.12, his postulate implies the statement that there is at most one line parallel to a given line L through a point P off L. Namely if M is a line perpendicular to L through P, the only line through P that could be parallel to L would be the perpendicular to M through P.

I do not know what version of the "postulate of parallels" you are referring to, but this statement we have just derived from Euclid's postulate, that there is at most one line parallel to a line L through a point P off L, or Playfair's postulate, is a common version.

Since also by Prop I.17, two common perpendiculars to the same line are indeed parallel, and we have remarked that perpendiculars exist, it follows that in fact there is also at least one parallel to a given line L through any point P off it, namely the line described at the end of the sentence before the previous one.

These statements also show that playfair's postulate implies Euclid's postulate 5. I.e. since there do exist parallel lines violating the hypotheses of Euclid's postulate, if there is at most one parallel, then the ones satisfying Euclid's hypotheses must not be parallel.

It is also clear then that in the presence of the other postulates of Euclid (modulo filling the well known topological gaps in his proofs, and possibly taking Prop. I.4 as an axiom), the three following statements are logically equivalent: 1) Euclid's postulate, 2) Playfair's postulate, 3) the postulate I learnt in school, namely there exists a unique line parallel to a given line and passing through a given point.

To me that is essentially the end of the matter, except to explore the cases where all these postulates fail, i.e. hyperbolic geometry. But I am not a philosopher.

I've been amusing myself by spending time editing the Elements.

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