Euclids Elements

[CharlieEuclid]

Was reflection allowed in Euclid's Elements? I seem to remember my high school teacher in geometry saying that it was not allowed. Yet it seems to me that the side-angle-side rule would allow two reflected triangles to be considered to be congruent. The reason that this question concerns me is that I read that a renowned mathematician said that the proof of the pythagorean theorem that uses reflected right triangles (they are arranged in two squares of sides "a" + "b" in the proof), is invalid because reflected triangle are not considered the same, i.e. not congruent because one can't fit them together.

 

[HallsofIvy]

In Euclid's elements, "reflection" was not a valid operation to prove congruence. Only translation and rotation were used.

 

[Dr.Peterson]

It's hard to give a definitive answer, because Euclid had no explicit concept of transformations or of congruence. See the note here, for example: https://mathcs.clarku.edu/~djoyce/elements/bookI/propI4.html .

The question would be, in the proof you refer to, how is "congruence" proved? Euclid wouldn't explicitly use reflection in a proof, but if SAS applies, the proof could be valid. Can you give a reference for the statement you have in mind?

 

[Dr.Peterson]

Why are you posting this again, rather than responding to what was already said to you?

 

[CharlieEuclid]

In Euclid's elements, "reflection" was not a valid operation to prove congruence. Only translation and rotation were used.

Thanks for clearifying this. In view of this, the mathematician was right.

It's hard to give a definitive answer, because Euclid had no explicit concept of transformations or of congruence. See the note here, for example: https://mathcs.clarku.edu/~djoyce/elements/bookI/propI4.html .

The question would be, in the proof you refer to, how is "congruence" proved? Euclid wouldn't explicitly use reflection in a proof, but if SAS applies, the proof could be valid. Can you give a reference for the statement you have in mind?

Thanks for your response. I'm new to this web site and after posting, I saw that a title to my question was required. I wrote a title and posted again. The reference you gave is excellent.

 

[CharlieEuclid]

Thanks for clearifying this. In view of this, the mathematician was right.

Thanks for your response. I'm new to this web site and after posting, I saw that a title to my question was required. I wrote a title and posted again. The reference you gave is excellent.

The journal where I read that that particular proof of the Pythagorean theorem was not considerd correct by the mathematician was the bulletin of the american mathematical society. I will see if I can find the particular issue. The proof in question can be found in Issaics College Geometry text. I remember when I first saw this proof in a popular paper book on geometry. It struck me as much easier to understand than the usual proof by Euclid, but seemed less concise. For this reason I like not accepting reflection and inversion and to specify that side-angle-side, etc. should only apply to a certain class of triangles (this class can be mathematically specified).