MEASUREMENT OF VARIABLE PROPERTIES 117 



two states of equilibrium are possible — viz. a and b ; in other 

 words, two events a and b may be observed.^ The cause is 

 chance (combined cause), just as in the example of the ball. 

 The variation of this cause is vmlimited, but it has only two 

 different effects : a ov b. This depends on certain properties 

 of the coin ; in other words, on its specific energy. (If the 

 coin were a hving being, we should say that there is alternative 

 variation with regard to a pair of primordia a and b.Y 



The two events a and b are equally possible, because one 

 caimot discover any cause (factor) by which one of both would 

 be more easily realized than the other. ^ 



Let us now have recourse to experiment. If several series 

 of experiments are made, each series consisting of a rather 

 small number of tosses (for instance, twenty), the order of 

 succession of the events a and b is quite irregular in each series 

 and the ratio in which a and b are observed is very variable 

 from one series to another. It seems as if there were no rule 

 whatever. But if we persevere, repeating the experiment, for 

 instance, 500, 1000, 1500 . . . times, we observe that the 

 number of events a and the nimaber of events b approach more 

 and more to equaHty, in proportion as the total number of 

 experiments (tosses) is increased. If this nimiber is, for in- 

 stance, 10,000, each of the events a and b occurs about 5000 

 times.* 



In general, if the number of tosses is n (a large niunber), the 

 event a (head) occurs approximately f times. 



We call probability (frequency) of the event a the ratio 

 between the number of tosses in which a is observed and the 

 total number of observations (tosses); thus ^•.n = \. The 

 same value is, of course, the approximate expression of the 

 frequency of b. 



The above expression of frequency is drawn from experience : 

 the value \ means that in a second series of n' tosses [n' being a 

 large number) the event a wiU be observed approximately |.' 

 times {n' x the frequency of a). 



On the other hand, the probability (frequency) may be 



1 In reality a third state of equilibrium (event) is possible, in which the coin 

 would be in a vertical position, resting on the plane p by one point of its border. 

 This event practically never occurs and may therefore be left out of account. 



2 When the coin has reached its final state of equilibrium (adult state I ) 

 three primordia may be discerned (G = centre of gravity ; P = centre of the 

 side of the coin on which it is resting): (i) the length GP (invariable); 

 (2 ) the direction of GP (always vertical ) ; (3 ) the sign of GP (head = a = + and 

 tail= 6= - ). Alternative variation exists here in respect of the primordium 

 sign. 



' The two states of equilibrium a and 6 are equally stable. 



* Contrary to the belief of many gamblers, the order of succession is always 

 irregular, however great the number of tosses may be, because a given toss 

 has no influence upon the next one. 



