time.) for it passeth every following line or side in the same space of time yt it doth the first from whence it fell. (as being let fall from X, in the same time that it passeth from X to H3), in the same it passeth from *h* to *f*, from *f* to *d*, from *d* to *b*, and from *b* to H.) Because the first velocity is alwayes in proportion to the distance. as X*h* being the fifth side from H X, X*j* is to *ba* as 5 to 1, and consequently the Bullets velocity at X being let fall there, is to the Bullets velocity at *b*, if thence let fall, as 5 to 1. and the retardation arising from the inclination of the following lines or sides, is still equall to the accelleration arising from the continuance of the motion. (as X*h* having five degrees of velocity, *hf* has but foure because X*j* is to *hg* as 5 to 4, and so one degree of velocity is lost by the greater inclination of the side *hf*: but because there is one moment of time past since the motion begun, when the Bullet is fallen to *h* from X, therefore one degree of velocity is acquired from that continuance of motion or second impulse; and therefore one degree onely being lost by the position of the side, and an other got in lieu thereof by the continuance of the motion; the velocity remaines still the same at *h* that it had at X, and the like of all the rest.) And so in a Circle the versed sines of small arches, equally increasing are so very neare that they may well bee said to have the proportion of squares; and therefore their differences as odd numbers, that is to encrease equally, or in an arithmeticall progression. And because the first difference, if the
arches bee infinitely small, or lesse then any assignable quantity, is not onely absolutely, but in respect to those arches also, lesse then any assignable quantity or infinitely small, as is easie to demonstrate and hath been don already in my paper of recoyling. therefore after an infinite progression, as in the curve H X C, and not before, these differences do equally increase vnto an equallity with the first arch. And indeed this Curve is no other then what results from the continuance of that series vnto which at the beginning of thè quadrant (where the vibrations are at least physically equall) there is so great a neerenesse that in respect to themselves the difference is lesse then any assignable quantity.

I proceed therefore and say next that this Curve is a Cicloid: For in the triangle MDE, MD, & DE, being equall to DC and divided into as many equall parts E*q*, *qs*, *su* &c, as there are sides in the Curve HXC and the Radiaus of the Quadrant DPF, which is also equall therevnto, divided in the same manner: the sides of the triangles E*pq*, E*rs*, E*tu*, &c. are proportionall to the triangles H*ba*, *bdc*, *dfe*, &c., and therefore *pq. μq* (=E*q*.)∷ H*a. ab* and *rs. λs* (=E*s*) ∷ *bc. cd*, &c. so of the rest. And therefore HI = *ab* + *cd* + *ef*. I*f* = H*a* + *bc* + *de* ∷ *μq* + *λs* + *ϰu. pq* + *rs* + *tu* that is, the number being infinit, HI the intercepted diameter is to I*f* the ordinate ∷ as the respective triangle E*ϰu*, to the respective portion F *tu*E. And therefore AH. HI ∷ Δ EMD. Δ E*ϰu*. But