On Combinatorics (-1899)

“So there arise two kinds of variation: complexion [combinations] and situs [permutations]. And viewed in themselves, both complexion and situs belong to metaphysics, or to the science of whole and parts. If we look at their variability, however, that is, at the quantity of variation, we must turn to numbers and to arithmetic. I am inclined to think that the science of complexions pertains more to pure arithmetic, and that of situs to an arithmetic of figure.” (Gottfried Leibniz, “Dissertatio de arte combinatoria” [“Dissertation on the Art of Combinations”], Leipzig, 1666)

“Even the wisest and most prudent people often suffer from what Logicians call insufficient enumeration of cases.” (Jacob Bernoulli , 1692)

“[This subject] has a relation to almost every species of useful knowledge that the mind of man can be employed upon.” (Jacob Bernoulli, “Ars Conjectandi”[“The Art of Conjecturing”], 1713)

“It is easy to perceive that the prodigious variety which appears both in the works of nature and in the actions of men, and which constitutes the greatest part of the beauty of the universe, is owing to the multitude of different ways in which its several parts are mixed with, or placed near, each other. But, because the number of causes that concur in producing a given event, or effect, is oftentimes so immensely great, and the causes themselves are so different one from another, that it is extremely difficult to reckon up all the different ways in which they may be arranged, or combined together, it often happens that men, even of the best understandings and the greatest circumspection, ale guilty of that fault in reasoning which the writers on logic call the insufficient, or imperfect enumeration of parts, or cases: insomuch that I will venture to assert, that this is the chief, and almost the only, source of the vast number of erroneous opinions, and those too very often in matters of great importance, which we are apt to form on all the subjects we reflect upon, whether they relate to the knowledge of nature, or the merits and motives of human actions. 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.” (Jakob Bernoulli, “Ars Conjectandi” [“The Art of Conjecturing”], 1713)

"The Combinatorial Analysis is a branch of mathematics which teaches us to ascertain and exhibit all the possible ways in which a given number of things may be associated and mixed together; so that we may be certain that we have not missed any collection or arrangement of these things, that has not been enumerated. [...] Besides its uses in mathematical investigations, it not only enables us to form our ideas of the elegant compositions of design, but to contemplate the prodigious variety which constitutes the beauties of nature, and which arises from the combinations of objects, by their number, forms, color, and positions. It has a relation to every species of useful knowledge upon which the mind of man can be employed." (Peter Nicholson, "Essays on the Combinatorial Analysis", 1818)

"Partitions constitute the sphere in which analysis lives, moves, and has its being; and no power of language can exaggerate or paint too forcibly the importance of this till recently almost neglected, but vast, subtle, and universally permeating, element of algebraical thought and expression." (James J Sylvester, "On the Partition of Numbers", 1857)

"We have continually to make our choice among different courses of action open to us, and upon the discretion with which we make it, in little matters and in great, depends our prosperity and our happiness. Of this discretion a higher philosophy treats, and it is not to be supposed that Arithmetic has anything to do with it; but it is the province of Arithmetic, under given circumstances, to measure the choice which we have to exercise, or to determine precisely the number of courses open to us." (William A Whitworth, "Choice and Chance", 1870)

"The combinatory analysis in my opinion holds the ground between the theory of numbers and algebra, and is the proper passage between the realms of discontinuous and continuous quantity. It would appear advisable [...] to consider the theory of partitions an important part of combinatory analysis." (Percy A MacMahon, "Combinatory Analysis: A Review of the Present State of Knowledge", Proceedings of the London Mathematical Society Vol. s1-28 No. 1, 1896)