518 BELL SYSTEM TECHNICAL JOURNAL 



made subsequently from the original drawings, is on exhibition in the South 

 Kensington Science Museum, London. The first authentic record of the 

 actual use of a pendulum in a clock is attributed to the great Dutch scientist. 

 Christian Huygens, who produced his first pendulum clock in 1657. This 

 was described by him in the Horologium in 1658.- 



The performance of pendulum clocks was so good that almost immedi- 

 ately clocks of all other types were modified to include a pendulum. So 

 complete was this transformation that very few unmodified clocks are now 

 in existence which antedate the first application of the pendulum to time- 

 keeping. This, as a matter of fact, is one of the major reasons that so 

 little is known about the actual mechanisms used in mechanical clocks 

 that were made before the introduction of the pendulum. 



The subsequent history of pendulum clock development is well described 

 in numerous books and papers and covers a wide field. Only those factors 

 that relate the pendulum to other means of rate control will be discussed in 

 the following. 



The properties of a pendulum which make it such a good timekeeper are 

 easily seen from a study of the forces on the bob as illustrated in Fig. 3. 

 Since these forces must be in equilibrium at all times we may write (as- 

 suming no friction) 



Mg sin e = Ml 





The nearl^^ isochronous property of the pendulum is contained in this 

 relationship since the period, on solution, is 



vi( 



1+4^'" 2+64^'" 2 + 



where B is the maximum semi-amplitude of swing expressed in radians* 

 When this arc is small the period approaches a minimum. For smal^ 

 angles the natural period depends almost wholly on the ratio of / to g and 

 the stability of T depends chiefly upon the constancy of ( and g. Figure 4 

 shows the relation between period and the arc of swing, expressed as seconds 

 per day departure from the theoretical rate for zero arc. 



The sum of all the terms that depend upon powers of sin 6/2 is known as 

 the circular error, relating to the fact that the bob is constrained to move 

 on the arc of a circle. It was shown theoretically by Christian Huygens'' 

 that if the bob could be constrained to move on the arc of an epicycloid it 

 would be truly isochronous, that is, the period would be completely in- 

 dependent of its amplitude of motion. It is of interest to note at this 

 point that in no other resonator used for precision timekeeping is there 



