C 4H ] 
and general, (which were defcribed above) are again 
demonftrated. Next follow the Algebraic Rules for 
finding the Centre of Curvature, and determining 
the Cauftics by Reflexion and Refradion, and the 
centripetal Forces. The Conftrudion of the Tra- 
jedory is given, which is defcribed by a Force that is 
inverfely as the Fifth Power of the Diftancc from the 
Centre, becaufe this Conftrudion requires Hyper- 
bolic and Elliptic Arcs, and becaufe a remarkable 
Ciicumftance takes place in this Cafe, (and indeed in 
an Infinity of other Cafes) which could not obtain 
in thofe that have been already conftruded by others, 
viz. That a Body may continually defeend in a fpiral 
Line towards the Centre, and yet never approach fo 
near to it as to defeend to a Circle of a certain Radius ; 
and a Body may recede for ever from the Centre, 
and yet never arife to a certain finite Altitude. The 
Conftrudion of the Cafes wherein this obtains is per- 
formed by Logarithms or Hyperbolic Areas, the 
Angles defcribed about the Centre being always pro- 
portional to the Hyperbolic Sedors, while the Di- 
ftances from the Centre are diredly or inverfely as 
the Tangents of the Hyperbola at its Vertex. The 
Circle is an Afymptote to the Spiral ; and this can 
never be, unlefs the Velocities requifite to carry 
Bodies in Circles increafe while theDiftances decreafe, 
(or decreafe while the Diftances increafe) in a higher 
Proportion than the Velocity in the Trajedory ; that 
is, unlefs the Force be inverfely as a higher Power 
of the Diftance than the Cube. Next follow The- 
orems for computing the Time of Defcent in any Arc 
of a Curve, for finding the Refiftance and Denfity of 
the Medium when the Trajedory and centripetal 
. Force 
