“It must therefore be acknowledged, that that art which affords a cure to this weakness, or defect, of our understandings, and teaches us so to enumerate all the possible ways in which a given number of things may be mixed and combined together, that we may be certain that we have not omitted any one arrangement of them that can 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 knowledge that the mind of man can be employed upon. It proceeds indeed upon mathematical principles, in calculating the number of the combinations of the things proposed: but 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.”[96] [96] James Bernoulli, *De Arte Conjectandi*, translated by Baron Maseres. London, 1795, pp. 35, 36. *Distinction of Combinations and Permutations.* We must first consider the deep difference which exists between Combinations and Permutations, a difference involving important logical principles, and influencing the form of mathematical expressions. In *permutation* we recognise 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*.