380 THE PRINCIPLES OF SCIENCE. [chap. 



exceed three inches ; and the odds are 1022 to 2 against 

 the occurrence of the greatest possible error of five inches. 



If any case should arise in which the observer knows 

 the number and magnitude of the chief errors which 

 may occur, he ought certainly to calculate from the Arith- 

 metical Triangle the special Law of Error which would 

 apply. But the general law, of which we are in search, 

 is to be used in the dark, when we have no knowledge 

 whatever of the sources of error. To assume any special 

 number of causes of error is then an arbitrary proceeding, 

 and mathematicians have chosen the least arbitrary course 

 of imagining the existence of an infinite number of in- 

 finitely small errors, just as, in the inverse method of 

 probabilities, an infinite numl:)er of infinitely improbable 

 hypotheses were submitted to calculation (p. 255). 



The reasons in favour of this choice are of several 

 different kinds. 



1. Jt cannot be denied that there may exist infinitely 

 numerous causes of error in any act of obsei'vation. 



2. The law resulting from the hypothesis of a moderate 

 number of causes of error, does not appreciably differ from 

 that given by the hypothesis of an infinite number of 

 causes of error. 



3. We gain by the hypothesis of infinity a general law 

 capable of ready calculation, and applicable by luiiform 

 rules to all problems. 



4. This law, when tested by comparison with extensive 

 series of observations, is strikingly verified, as will Ije 

 shown in a later section. 



When we imagine the existence of anv large number of 

 causes of error, for instance one hundred, the numbers of 

 combinations become impracticably large, as may be seen 

 to be the case from a glance at the Arithmetical Triangle, 

 which proceeds only up to the seventeenth line. Quetelet, 

 by suitable abbreviating processes, calculated out a table 

 of probability of errors on the hypothesis of one thousand 

 distinct causes ; ^ but mathematicians have generally 

 proceeded on tlie hypothesis of infinity, and then, by tlie 

 devices of analysis, have substituted a general law of easy 



^ Letters on the Theory of Probabilities, Letter XV. and Appendix, 

 note pp. 256 266. 



