On the Acceleration of Falling Bodies

A demonstration by which the ratio, measure, method, and power of the acceleration of motion in the natural descent of heavy bodies are determined. This is directed against the pseudo-science original: "Pseudo-scientiam" concerning the same motion recently devised by Galileo Galilei, the Florentine philosopher and mathematician. This, therefore, is the letter to which the first response is here provided. From page 1 to 3.

ART. II. III. IV. V. The Status of the Controversy.

The question, indeed, is whether, when a stone (for example) is falling from a height, one may take a certain first space, such as one fathom orgyia: a classical unit of length roughly equal to six feet, or the span of a man's outstretched arms, and consider as the moment, or first unit of time, that which passes until the stone falls through this fathom. One would then take as the first degree of speed that speed which is acquired at the end of this first unit of time. Whether, I say, after equal spaces, times, and degrees have been subsequently taken, the stone or its motion should be thought to have acquired two degrees of speed when the second fathom has been covered, even if the second unit of time has not yet fully passed. In the same way, whether it should be three degrees when the third fathom is covered, four with the fourth, and so on, so that velocities are always in the same ratio as the spaces. This is what the Reverend Father Pierre Cazré contends. Or rather, whether it should be thought to have acquired two degrees of speed when the second unit of time has passed, even if more than two spaces have been covered. In the same way, whether it should be three degrees when the third unit of time has passed, four when the fourth, and so on, so that velocities are always in the same ratio as the times. This is Galileo's opinion. From this it follows that in equal units of time, the spaces covered are in accordance with the sequence of odd numbers starting from one. For instance, if in the first unit of time the stone falls through one fathom, it falls through three in the second, five in the third, seven in the fourth, and so on. Therefore, the total spaces covered from the beginning to the end of each time period are in the same ratio as the squares of the times. That is, the spaces covered at the end of one