rot. XVI.] PHILOSOPHICAL TRAVSACTIOKfS. 359 



the curve described be a circle passing through the centre of tendency ; then 

 the force or tendency towards that centre is in all points as the 5th power, or 

 squared-cube, of the distance from it reciprocally : if in the proportional spiral, 

 reciprocally as the cube of the distance : if in an ellipse about the centre of it, 

 directly as the distance. If in any of the conic sections about the focus ; then 

 he demonstrates that the vis centripeta, or tendency towards that focus, is in 

 all places reciprocally as the square of the distance from it ; and that according 

 to the velocity of the impressed motion, the curve described is an hyperbola ; if 

 the body moved be swift to a certain degree, than a parabola ; if slower, an 

 ellipse, or a circle in one case. From this sort of tendency or gravitation it 

 follows likewise, that the squares of the times of the periodical revolutions, are 

 as the cubes of the radii or transverse axes of the ellipses. All which being 

 found to agree with the phsenomena of the celestial motions, as discovered by 

 the great sagacity and diligence of Kepler, our author extends himself upon the 

 consequences of this sort of vis centripeta ; showing how to find the conic 

 section which a body shall describe when projected with any velocity in a given 

 line, supposing the quantity of the said force known : and laying down several 

 neat constructions to determine the orbs, either from the focus given, and two 

 points or tangents ; or, without it, by 5 points or tangents, or any number of 

 points and tangents, making together 5. He then shows how, from the time given, 

 to find the point in a given orbit answering to it ; which he performs accurately 

 in the parabola, and, by concise approximations, comes as near as he pleases in 

 the ellipse and hyperbola : all which are problems of the highest concern in 

 astronomy. Next he lays down the rules of the perpendicular descent of bodies 

 towards the centre, particularly in the case where the tendency to it is recipro- 

 cally as the square of the distance ; and generally in all other cases, supposing a 

 general quadrature of curve lines : upon which supposition likewise he delivers a 

 general method of discovering the orbits described by a body moving in such a 

 tendency towards a centre, increasing or decreasing in any given relation to the 

 distance from the centre ; and the*" with great subtilty he determines in all cases 

 the motion of the apses, or of the joints of greatest distance from the centre, 

 in all these curves, in such orbits as are nearly circular. Showing the apses 

 fixed, if the tendency be reciprocally as the square of the distance ; direct in 

 motion, in any ratio between the square and the cube ; and retrograde, if under 

 the square: which motion he determines exactly from the rule of the increase 

 or decrease of the vis centripeta. 



Next the motion of bodies in given surfaces is considered, as likewise the 

 oscillatory motion of pendules ; where it is shown how to make a pendulum 

 vibrate always in equal limes, though the centre or point of tendency be never 



