[ 345 3 
defcribes the Cycloid, is fuch as would be acquired by 
a heavy Body defcending along the cycloidal Arc, 
the Axis of the Figure being fuppofed perpendicular 
to the Horizon. In the next place, the Cauftics, by 
Reflexion and Refra&ion, are determined. If Per- 
pendiculars be always drawn from the radiating Point 
to the Tangents of the Curve, and a new Curve be 
fuppofed to be the Locus of the Interfedions of the 
Perpendiculars and Tangents, then the Line, by the 
Evolution of which that new Curve can be defcribed, 
is fimilar and fimilarly fituated to the Cauftic by Re- 
flexion. The Doctrine of centripetal Forces is treated 
at length from Art. 416.10493. 
Firft, a Body is fuppofed to defeend freely by its 
Gravity in a vertical Line j and becaufe the Gravity 
is the Power which accelerates the Motion of the 
Body, it mult be meafured by the Fluxion of its 
Velocity, or the Second Fluxion of the Space de- 
fcribed by it. When the vertical Line is fuppofed . 
to move parallel to itfelf with an uniform Motion, 
the Body will defeend in it in the fame manner as 
before s and the Gravity 'will beftill meafured by the 
Second Fluxion of theDefcent, or the Second Fluxion 
of the Ordinate of the Curve that is traced ' in this 
Cafe by the Body on an immoveable Plain, and there- 
fore is as the Square of the Velocity (which is mea- 
fured by the Fluxion of the Curve) diredly, and the 
Chord of the Circle of Curvature that is in the Di- 
rection of the Gravity inverfely, by a Propofition - 
mentioned above. When the Gravity ads uniformly, 
and in parallel Lines; the Projedile, in deferibing any 
Arc, falls below the Tangent drawn at the Beginning 
of the Arc, as much as if it had fallen perpend icu- 
lari 
