173 THE PRINCIPLES OF SCIENCE. [CHAP. 



combinations, and the same law is true of all the processes 

 of algebra. In some cases, 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, sulphur, and carbon, 

 provided that the substances are present in proper propor- 

 tions and well mixed. But this indifference of order does 

 not usually extend to the events of physical science or the 

 operations of art. The change of mechanical energy into 

 heat is not exactly the same as the change from heat into 

 mechanical energy ; thunder does not indifferently precede 

 and follow lightning ; it is a matter of some importance 

 that we load, cap, present, and fire a rifle in this precise 

 order. Time is the condition of all our thoughts, space of 

 all our actions, and therefore both in art and science we 

 are to a great extent concerned with permutations. 

 Language, for instance, treats different permutations of 

 letters as having different meanings. 



Permutations of things are far more numerous than 

 combinations of those things, for the obvious reason that 

 each distinct thing is regarded differently according to 

 its place. Thus the letters A, B, C, will make different 

 permutations according as A stands first, second, or third ; 

 having decided the place of A, there are two places 

 between which we may choose for B; and then there 

 remains but one place for C. Accordingly the permuta- 

 tions of these letters will be altogether 3x2x1 or 6 in 

 number. With four things or letters, A, B, C, D, we 

 shall have four choices of place for the first letter, three 

 for the second, two for the third, and one for the fourth, 

 so that there will be altogether, 4x3x2x1, or 24 

 permutations. The same simple rule applies in all cases ; 

 beginning with the whole number of things we multiply 

 at each step by a number decreased by a unit. In general 

 language, if n be the number of things in a combination, 

 the number of permutations is 



n (n- i) (TI- 2) 4.3.2.1. 



If we were to re-arrange the names of the days of 

 the week, the possible arrangements out of which we 

 should have to choose the new order, would be no less 

 than 7'. 6 . 5 . 4 . 3 . 2 . i, or 5040, or, excluding the 

 existing order, 5039. 



