A6 ^ECTUUE V. 



* of a wheel revolving uniformly on its axis, or rolling uniformly on a horizontal . 

 l^laue. 



The cycloid is the curve in which a body may descend, in the shortest 

 possible time, from a given point, to another obliquely below it. It mav 

 easily be shown that a body descends more rapidly in a cycloid than in the 

 right line joining the two points. This property is of little practical uti- 

 lity; the proposition was formerly considered as somewhat difficult to be 

 demonstrated, but of late, from the invention of new modes of calculation, 

 theorems of a similar nature have been much extended with great facility. The 

 experiment 'naturally suggests a familiar proverb, which cautions us against 

 being led away too precipitately by an appearance of brevity and facility. 

 (Plate II. Fig. '25.) 



It has been found that the inconveniences, resulting from the complicated 

 apparatus necessary to introduce a cycloidal motion, for the pendulums of 

 clocks, are more than equivalent to the advantage of perfect isochronism' in 

 theory. For since, in small cycloidal arcs, the curvature is nearly constant, the 

 time of vibration of a simple circular pendulum must be ultimately the same/ 

 as tliat of a cycloidal pendulum of the same length; but in larger arcs, the time 

 must be somewhat greater, because the circular arc falls without the cycloidal, 

 and is less inclined to the horizon at e(}ual distances from the lowest point. • 

 This may be shown by a comparison of two equal pendulums, vibrating in arcs 

 of different extent : if may also be observed, by an experiment with two simple 

 pendulums of different lengths, that their times of vibration, like those of cy- 

 cloidal pendulums, are proportional to the square roots of their lengths; a 

 half second pendulum being only one fourth as long as a pendulum vibrating 

 seconds. 



We have been obliged to suppose the weight, as well as the inertia, of a pen- 

 dulum, to be referred to one point, since we are not at present prepared to ex- 

 amine the effect of tlie slight difference between the situations, and the velocities 

 of the different parts of the substances, employed in our experiments. The na- 

 ture of rotatory motion requires to be more fully understood, before we can 

 attend to the determination of the centres of oscillation of bodies of various 



