44 LECTURE V. 



Hence is derived the idea conveyed by the term living or ascending force; 

 for since the height, to which a body will rise perpendicularly, is as the square 

 of its velocity, it will preserve a tendency to rise to a height which is as the 

 square of its velocity, whatever may be the path into which it is directed, 

 provided that it meet with no abrupt angle, or that it rebound at each angle 

 in a new direction, without losing any velocity. The same idea is somewhat 

 more concisely expressed by the term energy, which indicates the tendency 

 of a body to ascend or to penetrate to a certain distance, in opposition to a 

 retarding force. 



The most important cases of the motion of bodies, confined to given sur- 

 fiices, are those which relate to the properties of pendulums. Of these the 

 simplest is the motion of a body in a cycloidal path. The cycloid is a curve 

 which has many peculiarities; we have already seen that it is described by 

 marking the path of a given point in the circumference of a circle which rolls 

 on a right line. Galileo was the first that considered it with attention, but 

 he failed in his attempts to investigate its properties. It is singular enough, 

 that the principal cause of his want of success was an inaccurate experiment: 

 in order to obtain some previous information respecting the area included by 

 it, he cut a board into a cycloidal form, and weighed it, and he inferred from 

 the experiment, that the area bore some irrational proportion to that of the 

 describing circle, while in fact it is exactly triple. In the same manner it has 

 happened in later times, that Newton, in his closet, determined the figure of 

 the earth more accurately, than Cassini from actual measurement. It was 

 Huygcns that first demonstrated the properties of the cycloidal pendulum, 

 which are of still more importance in the solution of various mechanical pro- 

 blems, than for the immediate purposes of timekeepers, to which that emi- 

 nent philosopher intended to apply them. (Plate I. Fig 5.) 



If a body be suspended by a thread playing between two cycloidal cheeks, 

 it will describe another equal cycloid by the evolution of the thread, and the 

 time of vibration will be the same, in whatever part of the curve it may begin 

 to descend. Hence the vibrations of a body moving in a cycloid are denomi- 

 nated isochronous, or of equal duration. This equality may be shown by let- 

 ting go two pendulous balls at tlie same instant, at different points of the curve, 

 and observing that they meet at the lowest point. (Plate II. Fig. 24.) 



