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Alan1000

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**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?