Euclids Elements

CharlieEuclid

New member
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

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

Dr.Peterson

Elite Member
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

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

CharlieEuclid

New member
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

New member
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).