observed. To determine exactly how many words might exist in the English language under these circumstances, would be an exceedingly complex problem, the solution of which has never been attempted. The number of existing English words may perhaps be said not to exceed one hundred thousand, and it is only by investigating the combinations presented in the dictionary, that we can learn the Laws of Euphony or calculate the possible number of words. In this example we have an epitome of the work and method of science. The combinations of natural phenomena are limited by a great number of conditions which are in no way brought to our knowledge except so far as they are disclosed in the examination of nature. It is often a very difficult matter to determine the numbers of permutations or combinations which may exist under various restrictions. Many learned men puzzled themselves in former centuries over what were called Protean verses, or verses admitting many variations in accordance with the Laws of Metre. The most celebrated of these verses was that invented by Bernard Bauhusius, as follows: 1 "Tot tibi sunt dotes, Virgo, quot sidera cœlo." One author, Ericius Puteanus, filled forty-eight pages of a work in reckoning up its possible transpositions, making them only 1022. Other calculators gave 2196, 3276, 2580 as their results. Wallis assigned 3096, but without much confidence in the accuracy of his result. It required the skill of James Bernoulli to decide that the number of transpositions was 3312, under the condition that the sense and metre of the verse shall be perfectly preserved. 2 In approaching the consideration of the great Inductive problem, it is very necessary that we should acquire correct notions as to the comparative numbers of combinations which may exist under different circumstances. The doctrine of combinations is that part of mathematical science which applies numerical calculation to determine the numbers of combinations under various conditions. It is a part of the science which really lies at the base not only of other sciences, but of other branches of mathe 1 Montucla, Histoire, &c., vol. iii. p. 388. matics. The forms of algebraical expressions are determined by the principles of combination, and Hindenburg recognised this fact in his Combinatorial Analysis. The greatest mathematicians have, during the last three centuries, given their best powers to the treatment of this subject; it was the favourite study of Pascal; it early attracted the attention of Leibnitz, who wrote his curious essay, De Arte Combinatoria, at twenty years of age; James Bernoulli, one of the very profoundest mathematicians, devoted no small part of his life to the investigation of the subject, as connected with that of Probability; and in his celebrated work, De Arte Conjectandi, he has so finely described the importance of the doctrine of combinations, that I need offer no excuse for quoting his remarks at full length. "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 greatest circumspection, are 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. 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.” 1 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. 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. 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. N 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 proportions 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 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 permutations of these letters will be altogether 3 x 2 × I 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, 4 × 3 × 2 × I, 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 .... 2 4.3 I. n (n − 1) (n − 2) .. 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 existing order, 5039. 2. I, or 5040, or, excluding the The reader will see that the numbers which we reach inquestions of permutation, increase in a more extraordinary manner even than in combination. Each new object or term doubles the number of combinations, but increases the permutations by a factor continually growing. Instead of 2 x 2 x 2 x 2 x ..... we have 2 x 3 X4 X5 X ..... and the products of the latter expression immensely exceed those of the former. These products of increasing factors are frequently employed, as we shall see, in questions both of permutation and combination. They are technically called factorials, that is to say, the product of all integer numbers, from unity up to any number n is the factorial of n, and is often indicated symbolically by in. I give below the factorials up to that of twelve : -- 479,001,600 = 12 The factorials up to 36 are given in Rees's 'Cyclopædia,' art. Cipher, and the logarithms of factorials up to 265 are to be found at the end of the table of logarithms published under the superintendence of the Society for the Diffusion of Useful Knowledge (p. 215). To express the factorial 265 would require 529 places of figures. Many writers have from time to time remarked upon the extraordinary magnitude of the numbers with which we deal in this subject. Tacquet calculated1 that the twenty-four letters of the alphabet may be arranged in more than 620 thousand trillions of orders; and Schott estimated 2 that if a thousand millions of men were employed for the same number of years in writing out these arrangements, and each man filled each day forty pages with forty arrangements in each, they would not have accomplished the task, as they would have written only 584 thousand trillions instead of 620 thousand trillions. 1 Arithmetica Theoria. Ed. Amsterd. 1704. P. 517. |