xxu.J QUANTITATIVE INDUCTION. 48C 



A similar question actually occurs in the case of the 

 moon's motion. We have no record that any other por- 

 tion of the moon was ever visible to men than such as we 

 now see. This fact sufficiently proves that within the 

 historical period the rotation of the moon on its own axis 

 has coincided with its revolutions round the earth. Does 

 this coincidence prove a relation of cause and effect to 

 exist ? The answer must be in the negative, because 

 there might have been so slight a discrepancy between 

 the motions that there has not yet been time to produce 

 any appreciable effect. There may nevertheless be a high 

 probability of connection. 



The whole question of the relation of quantities thus 

 resolves itself into one of probability. When we can 

 only rudely measure a quantitative result, we can assign 

 but slight importance to any correspondence. Because 

 the brightness of two stars seems to vary in the same 

 manner, there is no considerable probability that they have 

 any relation with each other. Could it be shown that 

 their periods of variation were the same to infinitely 

 small quantities it would be certain, that is infinitely pro- 

 bable, that they were connected, however unlikely this 

 might be on other grounds. The general mode of esti- 

 mating such probabilities is identical with that applied 

 to other inductive problems. That any two periods of 

 variation should by chance become absolutely eqnal is in- 

 finitely improbable ; hence if, in the case of the moon or 

 other moving bodies, we could prove absolute coincidence 

 we should have certainty of connection. 1 With approximate 

 measurements, which alone are within our power, we must 

 hope for approximate certainty at the most. 



The principles of inference and probability, according 

 to which we treat causes and effects varying in amount, 

 are exactly the same as those by which \\Q treated simple 

 experiments. Continuous quantity, however, affords us 

 an infinitely more extensive sphere of observation, because 

 every different amount of cause, however little different, 

 ought to be followed by a different amount of effect. 

 If we can measure temperature to the one-hundredth part 

 of a degree centigrade, then between o and 100 we have 



1 Laplace, System of the World, translated by Harte, vol. ii. p. 366. 



