NEWTON S FRINCIPIA/ \0l 



parallel to the axis, and we suppose the body to begin its 

 motion in the double curve P g H, with the same velocity 

 as that given or central velocity, with which it would 

 describe P p h, the double curve is found by taking the 

 ordinates p g in a given proportion to the square of 



P p- 

 the circular archPp, or as ; and consequently Pg H 



is a species of quadratrix described on a cylinder. 



The motion of pendulums is evidently a case of motion 

 in a curve surface by a force directed towards a point in 

 the axis of the solid, of which solid the curve described 

 by the pendulous body is a section ; and Sir Isaac Newton 

 discusses this subject fully. As subservient to this in 

 quiry, he gives some important properties of the cycloid, 

 or rather of the hypocycloid and hypercycloid : For he is 

 not satisfied with the investigation, which is sufficiently 

 easy, of the ordinary cycloid s properties, the curve de 

 scribed by a point in a circle or wheel running along 

 a straight line, but examines what is more difficult, the 

 properties of the hypercycloid and hypocycloid, or the 

 curves described by a wheel moving on the convex, and 

 the concave great circle of a sphere respectively. Of these 

 properties the most important is this. If D be the dia 

 meter of the sphere, and d that of the wheel, the length 



of the hypercycloid is equal to four times yy x (D + d), 



or four times the length of a fourth proportional to the sum 

 of the two diameters, the wheel s diameter and the sphere s. 

 It is then shown how a pendulum may be made to vi 

 brate in a given cycloid, or rather hypocycloid, namely, by 

 taking a distance, which is a third proportional to the 

 part of D, which the hypocycloid cuts off (that is, the 

 distance of the hypercycloid from the centre of the 



H 3 



