perpendicular should be taken near to the end C, and in the inclination, far from it.
Salv.You see then, that the Proposition which saith, that the motion by the perpendicular is more swift than by the inclination, holds not true universally, but onely of the motions, which begin from the extremity, namely from the point of rest: without which restriction, the Proposition would be so deficient, that its very direct contrary might be true; namely, that the motion in the inclining plane is swifter than in the perpendicular: for it is certain, that in the said inclination, we may take a space past by the moveable in less time, than the like space past in the perpendicular. Now because the motion in the inclination is in some places more, in some less, than in the perpendicular; therefore in some places of the inclination, the time of motion of the moveable, shall have a greater proportion to the time of the motion of the moveable, by some places of the perpendicular, than the space passed, to the space passed: and in other places, the proportion of the time to the time, shall be less than that of the space to the space. As for example: two moveables departing from their quiescence, namely, from the point C, one by the perpendicular C B, [in Fig. 4.] and the other by the inclination C A, in the time that, in the perpendicular, the moveable shall have past all C B, the other shall have past C T lesser. And therefore the time by C T, to the time by C B (which is equal) shall have a greater proportion than the line C T to C B, being that the same to the less, hath a greater proportion than to the greater. And on the contrary, if in C A, prolonged as much as is requisite, one should take a part equal to C B, but past in a shorter time; the time in the inclination shall have a less proportion to the time in the perpendicular, than the space to the space. If therefore in the inclination and perpendicular, we may suppose such spaces and velocities, that the proportion between the said spaces be greater and less than the proportion of the times; we may easily grant, that there are also spaces, by which the times of the motions retain the same proportion as the spaces.
Sagr.I am already freed from my greatest doubt, and conceive that to be not onely possible, but necessary, which I but now thought a contradiction: but nevertheless I understand not as yet, that this whereof we now are speaking, is one of these possible or necessary cases; so as that it should be true, that the time of descent by C A, to the time of the fall by C B, hath the same proportion that the line C A hath to C B; whence it may without contradiction be affirmed, that the velocity by the inclination C A, and by the perpendicular C B, are equal.
Salv.Content your self for this time, that I have removedyour
