The basic principle of the law of the lever
and - possibly - the concept of center of gravity
were established on a mathematical basis
by scholars earlier than Archimedes.
Archimedes proved the law of the lever, starting with these three assumptions.
Assumption 1. Equal weights at equal distances from the fulcrum balance. Equal weights at unequal distance from the fulcrum do not balance, but the weight at the greater distance will tilt its end of the lever down.
Assumption 2. If, when two weights balance, we add something to one of the weights, they no longer balance. The side holding the weight we increased goes down.
Assumption 3. If, when two weights balance, we take something away from one, they no longer balance. The side holding the weight we did not change does down.