EVOLUTION OF QUARTZ CRYSTAL CLOCK 517 



where ( and g are respectively the length and gravity expressed in the same 

 system of units, for example, the c.g.s. system. 



When any such resonant element is strained from its rest condition, and 

 released, it will oscillate with gradually decreasing amplitude until all of 

 the stored energy has been dissipated in internal friction or resistance, and 

 in the friction or resistance of the coupling with the supports. In general, 

 the resulting amplitude of free oscillation may be given as 



A = AQe~ sin pt 



the graph of which is a damped sine wave. The rate of free oscillation, p, 

 is dependent chiefly on the effective mass and stiffness and to a small degree 

 on the effective resistance of the element, while the rate of loss of amplitude, 

 that is, the logarithmic decrement, k, is dependent on the ratio of effective 

 resistance to effective mass. 



If the resistance could be made exactly zero, such a motion once started 

 would continue forever and its rate would be controlled wholly by the 

 effective mass and stiffness of the resonant element. Actually, of course, 

 such a condition cannot be realized in practice but, by the selection of 

 suitable materials and environment, and by special control means, it is 

 possible to approach very closely to the ideal condition by causing the 

 oscillation to be maintained almost as though there were no damping. 



The evolution of precision timekeeping, whether consciously or not, has 

 centered around the study and development of these two ideas: to discover 

 resonant elements whose rate-determining properties are inherently stable, 

 and to discover means for sustaining them in oscillation as though they had 

 no effective resistance; or in employing means to circumvent or to com- 

 pensate for any such resistance. The high precision of rate control that 

 can now be obtained has been the result largely of developments in these 

 two categories. 

 The Pendulum 



The gravity pendulum was the first truly resonant element to be used to 

 regulate the rate of a clock and for nearly three centuries maintained the 

 supremacy for precision measurements of time. The pendulum was more 

 a discovery than an invention, the popular story of its origin being that, 

 while still a youth of seventeen years, Gallileo Galilei chanced to notice 

 that a hanging lamp in the Cathedral of Pisa seemed to swing at the same 

 rate regardless of amplitude. This he confirmed approximately by com- 

 parison with his pulse, and later made an extensive study of the isochronism 

 of swinging bodies. These studies were in progress as early as 1583. Near- 

 ly sixty years later Gallileo described to his son Vincenzio how a pendulum 

 could be used to control a clock, but no concrete result of this advice is 

 known to have been made at that time. A working model of this clock. 



