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

Euclid's Common Notions1and4define what we now call anequivalence relation, which is not necessarily the equality relation

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

$\displaystyle a\cong a $ (reflexive)

$\displaystyle a\cong b \Rightarrow b\cong a$ (symmetric)

$\displaystyle a\cong b$ and $\displaystyle b\cong c \Rightarrow a\cong c$ (transitive)"

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

Explain howCommon Notions 1and4may be interpreted as the transitive and reflexive properties. Note that the natural way to writeCommon Notion 1symbolically is slightly different from the statement of transitivity above.Show that the symmetric property follows fromCommon Notions 1and4.

The book goes on to say:

"Hilbert took advantage ofCommon Notions 1and4in his rectification of Euclid's axiom system. Hedefinedequality 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)?