Page:Mathematical collections and translations, in two tomes - Salusbury (1661).djvu/229

 had in the centre, successively until it come to total extinction, do carry the moveable in such a time such a certain space, as it had gone in such a like quantity of time, by the acquist of velocity from the total privation of it until it came to that its greatest degree; it seemeth very reasonable, that if it should move always with the said greatest degree of velocity it would pass, in such another quantity of time, both those spaces: For if we do but in our mind successively divide those velocities into rising and falling degrees, as v. g. these numbers in the margine; so that the first sort unto 10 be supposed the increasing velocities, and the others unto one 1, be the increasing; and let those of the time of the descent, and the others of the time of the ascent being added all together, make as many, as if one of the two sums of them had been all of the greatest degrees, and therefore the whole space passed by all the degrees of the increasing velocities, and decreasing, (which put together is the whole diameter) ought to be equal to the space passed by the greatest velocities, that are in number half the aggregate of the increasing and decreasing velocities. I know that I have but obscurely expressed my self, and I wish I may be understood.

I think I understand you very well; and also that I can in a few words shew, that I do understand you. You had a mind to say, that the motion begining from rest, and all the way increasing the velocity with equal augmentations, such as are those of continuate numbers beginning at 1, rather at 0, which representeth the state of rest, disposed as in the margine: and continued at pleasure, so as that the least degree may be 0, and the greatest v. g. 5, all these degrees of velocity wherewith the moveable is moved, make the sum of 15;but if the moveable should move with as many degrees in number as these are, and each of them equal to the biggest, which is 5, the aggregate of all these last velocities would be double to the others, namely 30. And therefore the moveable moving with a like time, but with uniform velocity, which is that of the highest degree 5, ought to pass a space double to that which it passeth in the accelerate time, which beginneth at the state of rest.

According to your quick and piercing way of apprehending things, you have explained the whole business with more plainness than I my self; and put me also in mind of adding something more: for in the accelerate motion, the augmentation being continual, you cannot divide the degrees of velocity, which continually increase, into any determinate number, because changing every moment, they are evermore infinite. Therefore we shall be the better able to exemplifie our intentions by describing a Triangle, which let be this ABC, [in Fig. 8.] taking in the