C 55* ] 
Direction of the Gravity, that refults from the Com- 
petition of the centripetal Forces, then (hall its Ve- 
locity, and its Diftance from the Point where the Per- 
pendicular from the Centre of Curvature meets that 
right Line, flow proportionally , /. e. the Fluxion of 
the Velocity (or of the right Line that meafures it) 
{hall be to the Velocity as the Fluxion of that Diflancc 
is to the Diftance. When the Velocity and Direction 
of the Motion is the fame in the Line of fwifteft De- 
feent as in the Trajectory, their Curvature is the fame. 
Thus in the common Hypothefis of Gravity, the Cur- 
vature in the Cycloid, the Line of fwifteft Defcent, 
is the fame as the ^Parabola deferibed by a Projectile, 
if the Velocities in thofe Lines be equal, and their 
Tangents be equally inclined to the Horizon. In 
order to find the Nature of the Catenaria in any 
Hypothefis of Gravity, fuppofe the Gravity to be in- 
creafed or diminifhed in the fame Proportion as the 
Thicknefs of the Chain varies, and to have its Di- 
rection changed into the oppofite Direction; then 
imagine a Body to fet out with a juft Velocity from 
a given Point in the Chain, and to deferibe the Curve. 
The Tenfion of the Chain at any Point will be always 
as the Square of the Velocity acquired at that Point, 
and if a Body be projected with this Velocity in the 
Direction of the Tangent, the Curvature of the Tra- 
jectory deferibed by it will be one Half of the Curva- 
ture of the Chain at that Point. We muft refer to 
the Book for a fuller Account of thefe and of other 
Theorems. 
In the XHIth Chapter, the Problems concerning 
the Lines of fwifteft Defcent, the Figures which 
amongft all thofe that have equal Perimeters produce 
Ma- 
