MECHANICS. 1S5 



214. If / be the length of a pendulum vibrating 

 in a cycloid, the time of a vibration, whether the 



vibration be great or small, will be #y - 



o 



215. The line in which a heavy body descends 

 in the least time from one given point to another, 

 not in the same horizontal plane, nor in the same 

 vertical line with it, is an arch of a cycloid, having 

 for its base a horizontal line drawn through the 

 uppermost of the given points. 



Hence the cycloid is called the line of swiftest de- 

 scent. 



The problem of finding the line of swiftest descent was 

 first proposed by JOHN BERNOUILLI, Acta EruAito- 

 rum, 1697. Though the solution of it requires the 

 assistance of the higher geometry, elementary demon- 

 strations have been given by several authors, parti- 

 cularly by MACLAURIN, Fluxions, vol. n. 574, 

 &c. ; and by FRISIUS, Cosmographia, Introd. 22 f 

 and 23. 



SECT. 



