Page:The Mathematical Principles of Natural Philosophy - 1729 - Volume 2.djvu/24

14 and the ſpace which any body can deſcribe in the ſame time AD, with the firſt velocity AB, in a nonreſiſing medium, by the rectangle AB⨯AD.

Hence the ſpace deſcribed in a reſiſting medium is given, by taking it to the ſpace deſcribed with the uniform velocity AB in a non-reſiſting medium, as the hyperbolic area, ABGD to the rectangle AB⨯AD.

The reſiſtance of the medium is alſo given, by making it equal, in the very beginning of the motion, to an uniform centripetal force, which could generate, in a body falling thro a non-reſiſting medium, the velocity AB, in the time AC. For if BT be drawn touching the hyperbola in B, and meeting the aſymptote in T; the right line AT will be equal to AC, and will expreſs the time, in which the firſt reſiſtance uniformly continued, may take away the whole velocity AB.

And thence is alſo given the proportion of this reſiſtance to the force of gravity, or any other given centripetal force.

And vice verſa, if there is given the proportion of the reſiſtance to any given centripetal force; the time AC is alſo given, in which a centripetal force equal to the reſiſtance may generate any velocity as AB; and thence is given the point B, through which the hyperbola, having CH, CD for its aſymptotes, is to be deſcribed; as alſo the ſpace ABGD, which a body, by beginning its motion with that velocity AB, can deſcribe in any time AD, in a ſimilar reſiſting medium.