Page:Mathematical collections and translations, in two tomes - Salusbury (1661).djvu/203

 motion of descent, diminished in infinitum by the approach of the moveable to the first state of rest, which approximation is augmentable in infinitum. Now let us find the other diminution of velocity, which likewise may proceed to infinity, by the diminution of the gravity of the moveable, and this shall be represented by drawing other lines from the point A, which contein angles lesse than the angle BAE, which would be this line AD, the which intersecting the parallels KL, HI, FG, in the points M, N, and O, represent unto us the degrees FO, HN, KM, acquired in the times AF, AH, AK, lesse than the other degrees FG, HI, KL, acquired in the same times; but these latter by a moveable more ponderous, and those other by a moveable more light. And it is manifest, that by the retreat of the line EA towards AB, contracting the angle EAB (the which may be done in infinitum, like as the gravity may in infinitum be diminished) the velocity of the cadent moveable may in like manner be diminished in infinitum, and so consequently the cause that impeded the projection; and therefore my thinks that the union of these two reasons against the projection, diminished to infinity, cannot be any impediment to the said projection. And couching the whole argument in its shortest terms, we will say, that by contracting the angle EAB, the degrees of velocity LK, IH, GF, are diminished; and moreover by the retreat of the parallels KL, HI, FG, towards the angle A, the same degrees are again diminished; and both these diminutions extend to infinity: Therefore the velocity of the motion of descent may very well diminish so much, (it admitting of a twofold diminution in infinitum) as that it may not suffice to restore the moveable to the circumference of the wheel, and thereupon may occasion the projection to be hindered and wholly obviated.

Again on the contrary, to impede the projection, it is necessary that the spaces by which the project is to descend for the reuniting it self to the Wheel, be made so short and close together, that though the descent of the moveable be retarded, yea more, diminished in infinitum, yet it sufficeth to reconduct it thither: and therefore it would be requisite, that you find out a diminution of the said spaces, not only produced to infinity, but to such an infinity, as that it may superate the double infinity that is made in the diminution of the velocity of the descending moveable. But how can a magnitude be diminished more than another, which hath a twofold diminution in infinitum? Now let Simplicius observe how hard it is to philosophate well in nature, without Geometry. The degrees of velocity diminished in infinitum, as well by the diminution of the gravity of the moveable, as by the approxination to the first term of the motion, that is, to the state