1919] on Clock Escapements 447 



but, in spite of efforts made after his death to claim priority for him 

 in the invention of the pendulum clock, the evidence has not 

 convinced historians of his title to that honour. 



Huygens, being aware of the fact that the motion of a particle 

 under gravity was only isochronous, independently of the extent of 

 the arc of swing, when the body describes a cycloid, and knowing the 

 property of that curve to reproduce itself as an involute of an equal 

 cycloid, attempted to secure the desired isochronism by suspending 

 his pendulum from a silk thread which swung between two cheeks of 

 brass cut to the shape of the cycloid, thus ol)liging the bol) to trace 

 an involute. But the silk was so affected by the weather that no 

 good result ensued. 



Another objection to the verge escapement was the large arc of 

 swing necessary to permit the escapement to unlock itself. Huygens 

 attempted to overcome this difficulty by making the verge the axis, 

 not of the pendulum-crutch, but of a pinion gearing into a larger 

 wheel to the arbor of which the crutch was attached. This 

 construction permitted the angle of swing to be reduced at pleasure, 

 but more friction was introduced, and little improvement was effected. 



The calculation of the time of swing of a free pendulum describing 

 a circular arc can only be made approximately, but the approximation 

 can be carried as far as desired, and as the arc of swing is never large, 

 a few terms suffice. This is the formula : — 



_ TT/C/, i.„a y.a \ 



T= .^ T^l l+TSin^- + ^.sin-^- + . . .) 



r^ / 1 . „a 9 



from which, by differentiation. 



ilT Tr/csina/ ,,0-2^, "\ 



Here T is the time of swing of the pendulum from its highest position 

 to the vertical, and a is the semi-angle — that is, the angle turned 

 through from the highest to the lowest position. Now of the factors 

 making up the expressions on the right-hand side of these equations, 

 only 77 and (/ and the numerical coefficients can really be considered 

 as constant. It has been suggested that even (/ may one day be 

 shown to be variable. As for h and A;— that is, the distance from 

 the axis of motion to the centre of gravity and the radius of gyration 

 respectively — these are well known to be dependent on temperature, 

 and an interesting account might be given, if time permitted, of the 

 evolution of the compensated pendulum. The recent discovery of 

 alloys of iron and nickel, the coefficient of expansion of which is very 

 low, has much facilitated this. 



The factor which has most influence on the value of T is a, the 

 ans:le of swinj;. The formulae show us two things : first, that the 



