Page:Mathematical collections and translations, in two tomes - Salusbury (1661).djvu/230

 Page 205 side AC, as many equal parts as we please, AD, DE, EF, FG, and drawing by the points D, E, F, G, right lines parallel to the 〈◊〉 BC. Now let us imagine the parts marked in the line AC, to be equal times, and let the parallels drawn by the points D, E, F, G, represent unto us the degrees of velocity accelerated, and increasing equally in equal times; and let the Point A be the state of rest, from which the moveable departing, hath v. g. in the time AD, acquired the degree of velocity DH, in the second time we will suppose, that it hath increased the velocity from DH, as far as to EI, and so supposing it to have grown greater in the succeeding times, according to the increase of the lines FK, GL, &c. but because the acceleration is made continually from moment to moment, and not disjunctly from one certain part of time to another; the point A being put for the lowest moment of velocity, that is, for the state of rest, and AD for the first instant of time following; it is manifest, that before the acquist of the degree of velocity DH, made in the time AD, the moveable must have past by infinite other lesser and lesser degrees gained in the infinite instants that are in the time DA, answering the infinite points that are in the line DA; therefore to represent unto us the infinite degrees of velocity that precede the degree DH, it is necessary to imagine infinite lines successively lesser and lesser, which are supposed to be drawn by the infinite points of the line DA, and parallels to DH, the which infinite lines represent unto us the superficies of the Triangle AHD, and thus we may imagine any space passed by the moveable, with a motion which begining at rest, goeth uniformly accelerating, to have spent and made use of infinite degrees of velocity, increasing according to the infinite lines that begining from the point A, are supposed to be drawn parallel to the line HD, and to the rest IE, KF, LG, the motion continuing as far as one will.

Now let us compleat the whole Parallelogram AMBC, and let us prolong as far as to the side thereof BM, not onely the Parallels marked in the Triangle, but those infinite others imagined to be drawn from all the points of the side AC; and like as BC, was the greatest of those infinite parallels of the Triangle, representing unto us the greatest degree of velocity acquired by the moveable in the accelerate motion, and the whole superficies of the said Triangle, was the mass and sum of the whole velocity, wherewith in the time AC it passed such a certain space, so the parallelogram is now a mass and aggregate of a like number of degrees of velocity, but each equal to the greatest BC, the which mass of velocities will be double to the mass of the increasing velocities in the Triangle, like as the said Parallelogram is double to the Triangle: and therefore if the moveable, that falling did make use