ix.] COMBINATIONS AND PERMUTATIONS. 177 



lead to the object of our inquiry, deserves to be considered 

 as most eminently useful and worthy of our highest esteem 

 and attention. And this is the business of the art or 

 doctrine of combinations. Nor is this art or doctrine to be 

 considered merely as a branch of the mathematical sciences. 

 For it has a relation to almost every species of useful know- 

 ledge that the mind of man can be employed upon. It 

 proceeds indeed upon mathematical principles, in calculat- 

 ing the number of the combinations of the things proposed : 

 bat by the conclusions that are obtained by it, the sagacity 

 of the natural philosopher, the exactness of the historian, 

 the skill and judgment of the physician, and the prudence 

 and foresight of the politician may be assisted ; because 

 the business of all these important professions 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." x 



Distinction of Combinations and Permutations. 



We must first consider the deep difference which exists 

 between Combinations and Permutations, a difference in- 

 volving important logical principles, and influencing the 

 form of mathematical expressions. In permutation we re- 

 cognise varieties of order, 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 ques- 

 tions, 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. 33, 114). The Law of Commutativeness, 

 in fact, expresses the condition that in logic we deal with 



1 James Bernoulli, De Arte Conjectandi, translated by Baron 

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



