the definition of speed: v_{av} ≡ Δd / Δt v_{av} x Δt = Δd 
to get an average (assuming uniform acceleration) v_{av} = v_{initial} + v_{final} / 2 if v_{initial} = 0, then v_{av} = 1/2 v_{final} 
(1) (2) 
substituting for v_{av} from (2) 1/2 v_{final} x Δt = Δd 
(3) 

according to Galileo, acceleration is... a ≡ Δv / Δt 
but change is always the difference: Δv = v_{final}  v_{initial} 
(4) 
so substituting Δv from (4), a = v_{final}  v_{initial} / Δt if v_{initial} = 0, then a = v_{final} / Δt a x Δt = v_{final} 
(5) (6) (7) 

substuting (7) into (3) 1/2 ( a x Δt ) x Δt = Δd 1/2 a Δt^{2} = Δd 
(9) (10) 
So the distance a falling object transverses, Δd, is proportional to the SQUARE of the elapsed time, Δt, multiplied by a constant, 1/2 a,
presuming acceleration is constant as Galileo claims, and measurements of t, d and v_{initial} are made from rest.
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