It may here be pointed out that there are two modes in which we can calculate the whole number of combinations of certain things. Either we may take the whole number at once as shown in the Logical Alphabet, in which case the number will be some power of two, or else we may calculate successively, by aid of permutations, the number of combinations of none, one, two, three things, and so on. Hence we arrive at a necessary identity between two series of numbers. In the case of four things we shall have
In a general form of expression we shall have
the terms being continued until they cease to have any value. Thus we arrive at a proof of simple cases of the Binomial Theorem, of which each column of the Logical Alphabet is an exemplification. It may be shown that all other mathematical expansions likewise arise out of simple processes of combination, but the more complete consideration of this subject must be deferred to another work.
Possible Variety of Nature and Art.
We cannot adequately understand the difficulties which beset us in certain branches of science, unless we have some clear idea of the vast numbers of combinations or permutations which may be possible under certain conditions. Thus only can we learn how hopeless it would be to attempt to treat nature in detail, and exhaust the whole number of events which might arise. It is instructive to consider, in the first place, how immensely great are the numbers of combinations with which we deal in many arts and amusements.
In dealing a pack of cards, the number of hands, of thirteen cards each, which can be produced is evidently 52 × 51 × 50 × ... × 40 divided by 1 × 2 × 3 ... × 13. or 635,013,559,600. But in whist four hands are simultaneously held, and the number of distinct deals becomes so vast that it would require twenty-eight figures to express it. If the whole population of the world, say one thousand millions of persons, were to deal cards day and night, for a hundred million of years, they would not in that time have exhausted one hundred-thousandth part of the possible deals. Even with the same hands of cards the play may be almost infinitely varied, so that the complete variety of games at whist which may exist is almost incalculably great. It is in the highest degree improbable that any one game of whist was ever exactly like another, except it were intentionally so.
The end of novelty in art might well be dreaded, did we not find that nature at least has placed no attainable limit, and that the deficiency will lie in our inventive faculties. It would be a cheerless time indeed when all possible varieties of melody were exhausted, but it is readily shown that if a peal of twenty-four bells had been rung continuously from the so-called beginning of the world to the present day, no approach could have been made to the completion of the possible changes. Nay, had every single minute been prolonged to 10,000 years, still the task would have been unaccomplished.12 As regards ordinary melodies, the eight notes of a single octave give more than 40,000 permutations, and two octaves more than a million millions. If we were to take into account the semitones, it would become apparent that it is impossible to exhaust the variety of music. When the late Mr. J. S. Mill, in a depressed state of mind, feared the approaching exhaustion of musical melodies, he had certainly not bestowed sufficient study on the subject of permutations.
Similar considerations apply to the possible number of natural substances, though we cannot always give precise numerical results. It was recommended by Hatchett13 that a systematic examination of all alloys of metals should be carried out, proceeding from the binary ones to more complicated ternary or quaternary ones. He can hardly have been aware of the extent of his proposed inquiry. If we operate only upon thirty of