34 LECTURE V. 



find, in all cases, that the body ascends to a height equal to that from 

 which it descended, with a small deduction on account of friction. (Plate 

 II. Fig. 23.) 



Hence is derived the idea conveyed hy 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- 

 faces, 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. [p. 19.] Galileo was the first that con- 

 sidered 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.t It was Huygens ^ 

 that first demonstrated the properties of the cycloidal pendulum, which are 

 of still more importance in the solution of various mechanical problems, 

 than for the immediate purposes of timekeepers, to which that eminent 

 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 denominated isochronous, or of equal duration. This 

 equality may be shown by letting go two pendulous balls at the same in- 



* On the authority of Toricelli, Op. 1644. Consult Wallis, Op. 3 vols. fol. Oxf. 

 1699, i. 543, and Ph. Tr. xix. Ill, 561. The cycloid was known to Cusanus 1454, 

 and to Bovillus 1500, a century before it was considered by Galileo. See Leibnitz, 

 Op. iii. 95, and British Magazine for 1800. 



f Cassini, from his father's and M. Picard's measurements, proved that the earth 

 must be a spheroid, whose axis is greater than its equatorial diameter. Newton de- 

 duced the contrary from theory; and it is so in fact. See Mem. de 1'Acad. 1713, 

 1718. Newton's Principia, and Ph. Tr. 1725, pp. 33, 201, 239, 277, 344. Against 

 Mairan, Mem. de 1'Acad. 1720. 



J Horologium Oscillatorium, fol. Paris, 1673. 



Ibid. Compare Part I. with Prop. 25, Part II. In Birch's History of the 

 Royal Soc. is found an investigation of the same property by Lord Brouncker, 

 registered Jan. 22, 1662. The president was ordered to send a copy of it to Huy- 

 gens. 



