Page:The Mathematical Principles of Natural Philosophy - 1729 - Volume 1.djvu/105

 both ways, it will pas through the centre of force.

If the chords AB, BC, and DE, EF, of arcs decrib'd in equal times, in paces void of reitance, are compleated into the parallelograms ABCD, DEFZ; the forces in B and E are one to the other in the ultimate ratio of the diagonals BV, EZ, when thoe arcs are diminihed in infinitum. For the motions B; and EF of the body (by cor. 1. of the laws) are compounded of the motions Bc, BV and Ef, EZ: but BV and EZ, which are equal to Cc and Ff in the demontration of this propoition, were generated by the impules of the centripetal force in B and E, and are therefore proportional to thoe impules.

The forces by which bodies, in paces void of reitance, are drawn back from rectilinear motions, and turned into curvilinear orbits, are one to another as the vers'd ines of arcs decribed in equal times; which vered ines tend to the centre of force, and biect the chords when thoe arcs are diminihed to infinity. For uch vers'd ines are the halfs of the diagonals mentioned in cor. 3.

And therefore thoe forces are to the force of gravity, as the aid vers'd ines to the vers'd ines perpendicular to the horizon of thoe parabolic arcs which projectiles decribe in the ame time.

And the ame things do all hold good (by cor. 5. of the laws) when the planes in which the bodies are mov'd, together with the centres of force which are placed in thoe planes, are not at ret but move uniformly forward in right lines.