Then again, running on in the same blindness of passion, you pretend I make the proportion of B I to F K double to that of A B to A F, and then confute it; when you knew I made the proportion of F K to B I, double to that of F N, to B I, that is, of A F to A B; and this was it you should have confuted. That which followeth is but a triumphing in your own ignorance, wherein you also say, “_that all that I afterwards build upon this doctrine is false_.” You see whether it be like to prove so or not. As for your _Arithmetica Infinitorum_, I shall then read to you a piece of a lesson on it when I come to your objections against the next Chapter. In the mean time let me tell you, it is not likely you should be great geometricians, that know not what is quantity, nor measure, nor straight, nor angle, nor homogeneous, nor heterogeneous, nor proportion, as I have already made appear in this and the former lessons. To the first corollary of this fourth article your exception I confess is just, and (which I wonder at) without any incivility. But this argues not ignorance, but security. For who is there that ever read any thing in the Conics, that knows not that the parts of a parabola cut off by lines parallel to the base, are in triplicate proportion to their bases? But having hitherto designed the time by the diameter, and the impetus by the base; and in the next chapter (where I was to calculate the proportion of the parabola, to the parallelogram) intending to design the time by the base, I mistook and put the diameter again for the time; which any man but you might as easily have corrected as reprehended. To the second corollary, which is this, _that the lengths run over in equal times by motion so accelerated, as that the impetus increase in double proportion to their times, are as the differences of the cubic numbers beginning at unity, that is, as seven, nineteen, thirty-seven, &c._ you say it is false. But why? “_Because_” say you “_portions of the parabola of equal altitude, taken from the beginning, are not as those numbers seven, nineteen, thirty-seven, &c._” Does this, think you, contradict any thing in this proposition of mine? Yes, because, you think, the lengths gone over in equal times, are the same with the parts of the diameter cut off from the vertex, and proportional to the numbers one, two, three, &c. Whereas the lengths run over, are as the aggregates of their velocities, that is, as the parts of the parabola itself, that is, as the cubes of their bases, that is, as the numbers one, eight, twenty-seven, sixty-four, &c., and consequently the lengths run over in equal times, are as the differences of those cubic numbers, one, eight, twenty-seven, sixty-four, whose differences are seven, nineteen, thirty-seven, &c. The cause of your mistake was, that you cannot yet, nor perhaps ever will, contemplate time and motion (which requireth a steady brain) without confusion.