Euclid in "Elements" wrote that "things which equal the same thing also equal

Euclid in "Elements" wrote that "things which equal the same thing also equal

Euclid in "Elements" wrote that "things which equal the same thing also equal one another." Is this true in all cases? I've read that it is only true for "absolute entities," but not to "relations," although I do not understand this exemption. Are there any examples of things that are equal to the same thing but not to one another? Are relations really exempt from Euclid's axiom, and if so, why?

Read another response by Stephen Maitzen
Read another response about Mathematics