HISTORY OF SCIENCE. 



and forwards within a few degrees (say three or four) of its vertical 

 position, its oscillations are performed in times which are practically 

 equal, whatever may be the extent of each oscillation, provided it does 

 not exceed some such small extent as we have named. The oscilla- 

 tions of the pendulum are communicated, by the arrangement shown 

 in the figure, to the pallets, m n, which enter alternately into the 

 spaces of the toothed wheel R in such a manner that at each oscillation 

 one tooth of the wheel is allowed to pass the pallets. The wheel R is 

 connected with a series of others, by which its motion is registered on 

 the dial of the clock, and the whole train of wheels is actuated either 

 by a weight or by a coiled spiral spring. 



As the time of the oscillations of a pendulum moving in a circular 

 arc is strictly not independent of the length of that arc, Huyghens in- 

 vestigated the kind of curve in which the suspended body must move 

 in order that the oscillations might have absolutely equal periods 

 whatever the extent of the motion. He found that the required con- 

 dition was fulfilled when the pendulous body moved in a cycloid. The 

 cycloid is the curve traced by a point in the circumference of a circle 

 which rolls in a straight line. Thus if the circle p c, Fig. 92, rolls along 

 the straight line P A, the point p in the circumference of the circle will 

 trace out the curve P c A, which is called the cycloid. Thus a nail fixed 

 in the circumference of a cart wheel might be made to trace a cycloid 

 on an upright wall as the cart moved along the ground. The cycloid 

 is one of the most interesting of geometrical curves from its properties 

 and from its history, for while the former are very remarkable, no other 

 geometrical line has given rise to so many disputes. The French 

 geometer P. Mersenne and Galileo appear to have been the first to 

 notice this curve, and it became the object of the profound study of 

 several eminent geometers. Roberval in 1 634 discovered that the area 

 of the cycloid is exactly three times that of the generating circle. 

 Descartes, Pascal, Torricelli, Wallis, Wren, and others, laboured suc- 

 cessfully to solve various problems connected with this curve. Now, 

 Huyghens discovered a property of the cycloid which we may thus illus- 

 trate : Let a board A D (Fig. 93) have a portion, E B F, cut out in the 

 shape of an inverted cycloid, and let a groove be formed in which a 

 marble may run by the action of gravity when the board is fixed in an 

 upright position. Let B be the lowest point of the curve, then from what- 

 ever part of the curve the marble is allowed to commence its descent, it 

 will arrive at B always after the same interval of time. It follows, 

 therefore, that if the weight of a pendulum were constrained to follow 

 a cycloidal path in the descent, it also would accomplish its oscillation 

 in periods precisely equal, whatever might be their extent. Huyghens 

 succeeded in contriving an arrangement by which this might be effected, 

 and in doing so he discovered another curious property of the cycloid, 

 and introduced a new idea into geometry, namely, that of involutes. 

 Let c A, c D, Fig. 94, be two semi-cycloids produced by the equal 



