Various Questions and Inventions

conclusion, that where there is greater velocity in the ball fired violently through the air, there is less gravity, and conversely, that is, that where there is less velocity in it, there is greater gravity in it.
S.D. It is the truth.
N. I also say that where there is greater gravity in it, there is a greater stimulation of it in pulling the said ball toward the center of the world, that is, toward the earth.
S.D. It is a credible thing.
N. Now, to conclude our proposal, we shall suppose that the entire transit, or journey, that the ball fired by the aforementioned culverin must make, or has made, is the whole line a.b.c.d. And if it is possible that there is any part in it that is perfectly straight, let us suppose that this is the whole part a.b., which is divided into two equal parts at point e. And because the ball will transit faster through the space a.e. (by the third proposition of the first of our new science) than it will through the space e.b., therefore the said ball will travel more straightly, for the reasons adduced above, through the space a.e. than it will through the space e.b. Whence the line a.e. would be straighter than e.b., which thing is impossible, because if the whole a.b. is supposed to be perfectly straight, the half of it cannot be either more or less straight than the other half. And if, indeed, one half were straighter than the other, it follows necessarily that the other half is not straight, and therefore it follows of necessity that the part e.b. is not perfectly straight.

A woodcut illustration shows a cannon firing a cannonball. The trajectory is marked with letters: 'A' at the muzzle, 'F' and 'E' along the initial path, 'B' where the curve begins, 'C' further along, and 'D' at the point of impact. The segment from A to B is depicted as nearly straight, while the path from B to D is a distinct curve.

And if anyone still had the opinion that the part a.e. was nonetheless perfectly straight, such an opinion would be reproved as false by the same means and ways, that is, by dividing the said part a.e. into two equal parts at point f. And by the same reasons adduced above, it will be manifest that the part a.f. is straighter than the part f.e.; therefore, the said part f.e. of necessity will not be perfectly straight. Similarly, if one were to divide the a.f. into two equal parts, by the same reasons it is manifest that the half of it toward a is straighter than that toward f. And thus, whoever were to divide that half into other two equal parts, the same will follow, that is, the part terminating at a is straighter than the other. And because this procedure is infinite, it follows of necessity that not only is the whole a.b. not perfectly straight, but that no minimal part of it can be perfectly straight, which is the proposal. One sees, therefore, how the ball fired by the said culverin in such a direction does not travel any minimal part of its motion, or transit, in a perfectly straight line (let it exit with whatever great velocity one wishes), because the velocity (however great it may be) is never sufficient, in similar directions, to make it go in a straight line. It is true that the faster it goes in similar directions, the more its motion approaches straight motion, that is, going by a straight line; yet it can never arrive at such a mark. And therefore it is more appropriate to say in a similar case that the more the said ball goes fast, the less curved it makes...