200 THE PRINCIPLES OF SCIENCE. 



of all these important profes ior-s is but to form reasonable 

 conjectures concerning the several objects which engage 

 their attention, and all wise conjectures are the results of 

 a just and careful examination of the several different 

 effects that may possibly arise from the causes that are 

 capable of producing them/ c 



Distinction of Combinations and Permutations. 



We must at once consider the deep difference which 

 exists between Combinations and Permutations ; a dif- 

 ference involving important logical principles, and in- 

 fluencing the form of all our mathematical expressions. 

 In permutation we recognise varieties of order or arrange- 

 ment, treating AB as a different group from BA. In 

 combination we take notice only of the presence or 

 absence of a certain thing, and pay no regard to its 

 place in order of time or space. Thus the four letters 

 a, e, m, n can form but one combination, but they occur 

 in language in several permutations, as name, amen, 

 mean, mane. 



We have hitherto been dealing with purely logical 

 questions, involving only combination of qualities. I have 

 fully pointed out in more than one place that, though our 

 symbols could not but be written in order of place and 

 read in order of time, the relations expressed had no 

 regard to place or time (pp. 40, 131). The Law of Com- 

 mutativeness, in fact, expresses the condition that in logic 

 we deal with combinations, and the same law is true 

 of all the processes of algebra. In nature and art, order 

 may be a matter of indifference ; it makes no difference, 

 for instance, whether gunpowder is a mixture of sulphur, 

 carbon and nitre, or carbon, nitre and sulphur, or nitre, sul- 

 phur and carbon, provided that the substances are present in 



c James Bernouilli, ' De Arte Conjectandi,' translated by Baron 

 Maseres. London, 1795, pp. 35-36. 



