On the Acceleration of Falling Bodies

...[veloci]ties acquired are in the same ratio as the spaces traversed:

Therefore, in the accelerated motion of falling heavy bodies, it is necessary for the spaces traversed in equal times to be in a continually double ratio. original: "ratione continuò dupla." This refers to a geometric progression like 1, 2, 4, 8, which Gassendi argues is the logical result of Cazré's (incorrect) premise.

From page 93 to 97.

Clearly, as is evident from the sequence of the previous Letter, a beginning was made from the confirmation of the Assumption A term for a premise in a logical argument while the argument against Galileo was conducted. Then, a step was taken toward the proof of that same Assumption while that notable Experiment of the Balance was recounted. Finally, it came to proving the consequence of the Proposition while the effort concerning the double ratio of spaces was explained, which was the same conclusion of the Demonstration.

TO ARTICLES VI, VII, VIII. On the Definition of Uniformly Accelerated Motion.

From page 97 to 110.

The definition which Galileo handed down always stands firm. For degrees of velocity cannot be represented by the bases of triangles if the spaces are represented by parts of the sides, as the Reverend Father Father Pierre Cazré, the Jesuit mathematician wishes. This is both because a strange irregularity would be understood without any mention of time, and because a degree of velocity once acquired must either perish or do nothing. Nor can it be double that which is acquired. Furthermore, it would follow that in the second interval of time, nearly eight spaces equal to the first would be traversed. The Reverend Father confuses Uniformity with Proportion when he maintains that geometric progressions are no less uniform than arithmetic ones. He falls again into the same disadvantages when, in a non-uniform triangle, he repeatedly wishes spaces to be represented by parts of the sides rather than by the intercepted triangles. Gassendi is critiquing Cazré's use of geometry to map motion. In the Galilean method, the area of a triangle represents distance, but Cazré attempts to use the lengths of the sides instead.