BRIEF SUMMARIES.

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The Reverend Father R.P. refers to Pierre Cazré takes the lowest point of the first part, and from that point measures upward a half, a third, a quarter of that same first part, and generally as many fragments as there are equal parts below. Then he contends that in whatever amount of time the lower half of the first part is traversed, the second equal part is subsequently traversed in that same time. In whatever time a third is traversed, the third part is traversed; in whatever time a quarter, the fourth, and so on. It would follow, then, that all the individual times in which all the remaining equal parts are traversed are contained within the lower half of the first part.

Yet, unreasonably and entirely without cause, he neglects the upper half and takes no account of it. However, the motion begins at the start of that upper half, not at its end, and the object is already accelerating through it, since it contains no fewer parts than the lower half. He also assumes the lower half without proof and ties the fate of all subsequent parts to it. For when he argues that the second part is double this half, and therefore the velocity is double and the time equal, he is doing nothing other than begging the question original: "quæsitum petit", a logical fallacy where the conclusion is assumed in the premises.

Furthermore, it is established by various other arguments that, once any first unit of time is assumed, both the lower half and the second part are traversed in shorter and shorter times into infinity. This occurs by subdividing the previous half into two others, and the first of those into others again, and so on. It also follows that both the lower half and the second part would be traversed in half the time it takes for the whole first part. Further, the time for the second part would be in a sesquialter ratio original: "sesqui-alterum", a mathematical term for a ratio of 3 to 2, or one and a half, not double, compared to the time in which the lower half is traversed. It would even follow that the first part alone, and the first and second parts together, which is to say the part and the whole, are traversed in the same or equal time. These and other such contradictions can be raised by the same proportion regarding the third, the fourth, and other fragments assumed from the first part.