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In the Elements things that are equal can coincide perfectly with one another. However we cannot show things to be equal by "applying" one to another.

If two lines are equal, why can't we map one to the other and logically assume they share the same points afterwards, such as in the case of Book 1, Proposition 4 of The Elements

From what I've read, it seems the actual transformations required are not given as axioms which is the problem. But, ruler and compass transformations can map a line to any point on the plane, and its only left to assume that if two lines are equal, and share a point, that they must share the other point ( with some rotation. )

Read this statement from the link above: " If the triangle ABC is superposed on the triangle DEF, and if the point A is placed on the point D and the straight line AB on DE, then the point B also coincides with E, because AB equals DE."

The first part sounds much like it could be derived from I.2, but isn't. It seems the main problem from what I've read is that the required transformations do not have postulates, and that even if that doesn't matter. If two lines are congruent, it needs to be proved they may coincide and share points like Euclid describes.

Can someone verify this? Thank you!

soc3id
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    There are several modern axiom systems meant to give rigor to Euclid. One is an axiom system of Hilbert, which you may want to google, or just google on "axiom systems for plane geometry". At any rate many of the issues you mention have been studied, and you might like to look at some. – coffeemath Apr 29 '23 at 21:46
  • The question is, "Why is superposition not rigorous in Geometry?" and why is what I suggested not good enough.

    Someone with knowledge about it can probably give me at least a little bit of information about it before I try to read a chapter of a book about it.

     – soc3id Apr 29 '23 at 21:53
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    Superposition is a mapping from points to points, also known as a function. The ancient Greeks did not have the function concept so they could not use it in Geometry. – Somos Apr 29 '23 at 22:35
  • @Somos Do you mean it wasn't possible or stated that two points or lines could share the same space? – soc3id Apr 29 '23 at 22:38
  • Can you please elaborate on this? – soc3id Apr 29 '23 at 23:01
  • See here. – Jean Marie Apr 29 '23 at 23:16
  • Actually, a quick search shows (as you mentioned) that Euclid did rarely use superposition, but it is very controversial. I am guessing that is because it is not clear how it rigorously fits into the rest of geometry. To use it rigorously would require introducing mappings or functions and this the Greeks did not have. – Somos Apr 29 '23 at 23:46
  • @JeanMarie I have seen this post. It is not helpful, partially because its basically asking me to look at other axiomatic systems, which does not directly address my question, and secondly because you literally cannot view where the OP gets their proof from ( after scavenging the internet for it. ) – soc3id Apr 30 '23 at 00:02
  • @Somos If that is the case, Euclid could apply ruler and compass transformations to construct a line equal to a given one, on a given point. How is this different from a mapping? – soc3id Apr 30 '23 at 00:03
  • If Euclid could have just used ruler and compass, then there would be no need for superposition and yet he reluctantly rarely used it. Constructing a line is not a mapping. Greeks did not have the function concept. – Somos Apr 30 '23 at 00:19

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