Postulates:

  1. (Closure) Addition produces a unique sum. (There is only one center of mass.)
  2. (Commutativity) $nP + mQ = mQ + nP$. (Just view the "teeter-totter" from the other side.)
  3. (Associativity) $nP + (mQ + kR) = (nP + mQ) + kR = nP + mQ + kR$. (This sum is called the center of mass or centroid of the system. The propery is equivalent to the theorem of Menelaus.)
  4. (Scalar multiplication) $m(nP) = (mn)P = mnP$.
  5. (Idempotent) $nP +mP = (n+m)P$
  6. (Homogeneity) $k(nP + mQ )=knP + kmQ$.
  7. (Subtraction) If $n>m$ then $nP = mQ + xX$ may be solved for the unknown mass point $xX$. Namely, $xX = (n-m)R$ where $P$ on $\overline{RQ}$ and $RP:PQ=m:(n-m)$.