My demonstration is this, _in the parallelogram A B C D, (Fig. 11). Let the side A B be conceived to be moved uniformly till it lie in C D; and let the time of that motion be A C, or B D. And in the same time let it be conceived that A C is moved with uniform acceleration, till it lie in B D._ To which you object, _that then the acceleration last acquired must be far greater than that wherewith A B is moved uniformly: else it shall never come to the place you would have it in the same time_. What proof bring you for this? None here. Where then? Nowhere that I remember. On the contrary I have proved (Art. 9 of the chapter) that the line described by the concourse of those two motions, namely, uniform from A B to C D, and uniformly accelerated from A C to B D, is the crooked line of the semiparabola A H D. And though I had not, yet it is well known that the same is demonstrated by Galileo. And seeing it is manifest that in what proportion the motion is accelerated in the line A B, in the same proportion the impetus beginning from rest in A is increased in the same times (which impetus is designed all the way by the ordinate lines of the semiparabola), the greatest impetus acquired must needs be the base of the semiparabola, namely B D, equal to A C, which designs the whole time. I cannot therefore imagine what should make you say without proof, that the greatest acquired impetus is greater than that which is designed by the base B D. Next you say, “you see not to what end I divide A B in the middle at E.” No wonder; for you have seen nothing all the way. Others would see it is necessary for the demonstration; as also that the point F is not to be taken arbitrarily; and likewise that the thirteenth article, which you admit not for proof, is sufficiently demonstrated, and your objections to it answered. By the way you advise me, where I say _percursam eodem motu uniformi, cum impetu ubique_, &c. to blot out _cum_; because the _impetus_ is not a _companion_ in the way, but the _cause_. Pardon me in that I cannot take your learned counsel; for the word _motu uniformi_ is the ablative of the _cause_, and _impetu_ the ablative of the _manner_. But to come again to your objections, you say, I make “_a greater space run over in the same time by the slower motion than by the swifter_.” How does that appear? _because there is no doubt, but the swiftness is greater where the greatest impetus is always maintained, than where it is attained to in the same time from rest_. True, but that is, when they are considered asunder without concourse, but not then when by the concourse they debilitate one another, and describe a third line different from both the lines, which they would describe singly. In this place I compare their effects as contributing to the description of the parabolical line A H D. What the effects of their several motions are, when they are considered asunder, is sufficiently shown before in the first article.