On the Acceleration of Falling Bodies

LETTERS.

ARTICLES XXXIX. XL. XLI. XLII.

Regarding the continuously double ratio, by which the Reverend Father R.P. refers to Pierre Cazré, the Jesuit mathematician who was Gassendi's intellectual rival contends that spaces are traversed in equal times.

Having divided the same space of the fall into any number of equal parts, and having subdivided the first part into two halves, the Reverend Father takes that time in which the lower half is traversed as the first time. He wants the second part to be traversed in a second equal time, which is certainly double that half. In a third time, he says the third and fourth parts are traversed, which together are double the second part. In a fourth time, he includes the fifth, sixth, seventh, and eighth parts, which together are double the third and fourth. In a fifth time, the succeeding eight parts are covered, and in a sixth, the following sixteen. This continues in a continuously double ratio original: "ratione continuò dupla," a geometric progression where each term is twice the previous one: 1, 2, 4, 8, 16, etc..

But here also, the first half of the first part is passed over without reason. This is a problem because we need to know the ratio of acceleration original: "accelerationis ratio," the mathematical rule governing how speed increases from the very beginning of the motion, not from the midpoint of the first part. The Reverend Father also overturns the arithmetic progression original: "Arithmeticam progressionem," a sequence of numbers where the difference between each term is constant that he used earlier. When he previously established that velocities original: "velocitates" are proportional to the spaces, he wanted one degree of speed original: "celeritatis gradum" to belong to one whole part, not its half, two degrees to two parts, and so on.

Once again, this leads to the conclusion that a whole and a part are traversed in the same amount of time. It would mean that a third and a fourth part together are equal to the whole. Likewise, the fifth, sixth, seventh, and eighth parts would be equal to the whole, and so on for the rest. It is also concluded that spaces are traversed not only in a double ratio, but also in a triple or quadruple ratio. Finally, the same consequence follows that was objected to before: once the first time has passed, a second equal time cannot elapse without an infinite space original: "spatium infinitum" having been traversed.