On the Acceleration of Falling Bodies

OF THE LETTERS

ARTICLES IX. X. XI. XII. Regarding the fallacy which is charged against Galileo as he attacks a false definition.

Galileo attacks the definition approved by the Reverend Father Pierre Cazré, a Jesuit critic of Galileo's work. This definition suggests that if velocities were proportional to the spaces traveled, then a double space would be traversed at twice the velocity of half that space. If this were true, it would follow that the double and the half, or the whole and its part, would be traversed in the same or equal amount of time. Indeed, whether the motion is uniform or uniformly accelerated, the speed cannot be double across a double space without that speed being double at every point. If it were double everywhere, then double the distance would be covered in any given time. Thus, the end of the double distance and the end of the half distance would be reached at the exact same moment. The Reverend Father contends that a fallacy paralogismo: a piece of false or illogical reasoning is committed here. However, he offers no reason for this claim other than what is contained in these words:

If a heavy body descending through AB takes any amount of time, for instance, a quarter of an hour; and then traverses BC (which is equal to AB) in half of that quarter: who would deny that at C the velocity is double what it was at B? And yet the same heavy body would not traverse the whole distance AC and its half AB in the same amount of time.

In this argument, he assumes the very thing that is being disputed and treats it as a proven principle. Specifically, he assumes that the second part of the distance is traversed in half the time of the first part. This position leads to a significant inconvenience. In the same way, the third part would have to be traversed in half the time of the second, and the fourth part in half the time of the third, and so on. As the second unit of time passes, an infinite space would have to be traversed, since all those halves of halves, or fragments of time, cannot