116 THE QUANTITATIVE METHOD IN BIOLOGY 



any other way, and to find an exact expression for the dis- 

 covered facts. This is independent of any biological theory. 

 In Part VI. I wish to expound some principles of the theory 

 of chance (frequency) in a simple form by means of a series of 

 examples. The following pages are written for biologists, who 

 prefer concrete facts to abstract considerations. 



§ 91.— FIRST EXAMPLE : THE POSITION OF EQUI- 

 LIBRIUM OF A SPHERE ON A HORIZONTAL PLANE.— 



Let us suppose that a ball (for instance, a billiard ball), exactly 

 spherical and homogeneous, is thrown at random and falls 

 upon a horizontal plane surface p. After a certain time all 

 motion ceases ; the ball is then at rest on the plane. Its state 

 of equilibrium is characterized by two measurable values : 

 (i) the direction of the straight line which joins the centre (of 

 gravity) G of the ball and the point of contact P (this line is 

 vertical) ; (2) the distance GP (radius of the ball). 



Which are, in this experiment, the relations between cause 

 and effect ? 



The cause is, in reality, a combination or resultant of an 

 enormous mmiber of forces or factors, such as the direction in 

 which the ball has been thrown, its initial velocity, the initial 

 rotatory motion, its successive positions while rolling over the 

 plane before it came to rest, etc. Since it is practically im- 

 possible to determine or even to enumerate the factors which 

 are in play, the cause is called chance : we may call it a com- 

 bined cause. If the above experiment is repeated several 

 times the cause is, of course, different from one throw to another, 

 for a given combination of factors practically never occurs a 

 second time. In each experiment the cause may be called 

 accidental ox fortuitous. 



The effect, on the other hand, is always the same ; it is the 

 above described state of (indifferent) equilibrium of the ball. 

 This effect is invariable, because it depends on certain prop- 

 erties of the object. If the ball were an animal or a plant, we 

 should say that it always reacts in the same way, according to its 

 specific energy, and that its primordia {direction of GP and 

 length of GP) are invariable. 



In this example chance, in spite of its unlimited variation, 

 produces an invariable effect. (Compare § 104.) 



§ 92.— SECOND EXAMPLE : THE POSITIONS OF EQUI- 

 LIBRIUM OF A COIN. HEAD OR TAIL. In the above 

 experiment (§ 91) the spherical ball may be replaced by any 

 other object, all the conditions being the same. Let us throw, 

 for instance, a very thin disc (practically a thin coin), one of its 

 sides being called head or a and the other side tail or h. Here 



