1. "Things which are equal to the same thing are also equal to one another.
  2. If equals be added to equals, the wholes are equal.
  3. If equals be subtracted from equals, the remainders are equal.
  4. Things which coincide with one another are equal to one another.
  5. The whole is greater than the part.

Euclid's Common Notions 1 and 4 define what we now call an equivalence relation, which is not necessarily the equality relation

For any a, b and c ( \cong is an equivalence relation):

a\cong a (reflexive)
a\cong b \Rightarrow b\cong a (
symmetric)
a\cong b and b\cong c \Rightarrow a\cong c (
transitive)"

Two exercises to which I have (really) no clue.

Explain how Common Notions 1 and 4 may be interpreted as the transitive and reflexive properties. Note that the natural way to write Common Notion 1 symbolically is slightly different from the statement of transitivity above.
Show that the symmetric property follows from Common Notions 1 and 4.




The book goes on to say:
"Hilbert took advantage of Common Notions 1 and 4 in his rectification of Euclid's axiom system. He defined equality of length by postulating a transitive and reflexive relation on line segments, and stated transitivity in the style of Euclid, so that the symmetric property was a consequence."
Could someone outline what he'd done (or point to other source)?