Galileo's Rejection of the Possibility of Velocity Changing Uniformly with Respect to Distance

TN his "Two New Sciences" Galileo discusses the question as to whether in luniformly accelerated motion the velocity increases uniformly with time or with distance. The topic at issue is the motion of falling bodies. Simplicius, the Aristotelian, admits that he is one of those who believe that "a falling body" has a "velocity . . . increasing in proportion to the space." Salviati then presents an argument to show that Simplicius' belief is "false and impossible, as that motion should be completed instantaneously," and then gives "a very clear demonstration of it" as follows: