present, and the numbers of these classes immediately produce the numbers of the arithmetical triangle. 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 Hence we arrive at a necessary identity between two series of numbers. In the case of four things we shall have on. 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 x 51 x 50 x ... × 40 divided by I × 2 × 3.. x 13. or 635,013,559,600. But in whist four hands are simul taneously 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.1 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 Hatchett & 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 1 Wallis, Of Combinations, p. 116, quoting Vossius. inquiry. If we operate only upon thirty of the known metals, the number of binary alloys would be 435, of ternary alloys 4060, of quaternary 27,405, without paying regard to the varying proportions of the metals, and only regarding the kind of metal. If we varied all the ternary alloys by quantities not less than one per cent., the number of these alloys would be 11,445,060. An exhaustive investigation of the subject is therefore out of the question, and unless some laws connecting the properties of the alloy and its components can be discovered, it is not apparent how our knowledge of them can ever be more than fragmentary. The possible variety of definite chemical compounds, again, is enormously great. Chemists have already examined many thousands of inorganic substances, and a still greater number of organic compounds;1 they have nevertheless made no appreciable impression on the number which may exist. Taking the number of elements at sixty-one, the number of compounds containing different selections of four elements each would be more than half a million (521,855). As the same elements often combine in many different proportions, and some of them, especially carbon, have the power of forming an almost endless number of compounds, it would hardly be possible to assign any limit to the number of chemical compounds which may be formed. There are branches of physical science, therefore, of which it is unlikely that scientific men, with all their industry, can ever obtain a knowledge in any appreciable degree approaching to completeness. Higher Orders of Variety. The consideration of the facts already given in this chapter will not produce an adequate notion of the possible variety of existence, unless we consider the comparative numbers of combinations of different orders. By a combination of a higher order, I mean a combination of groups, which are themselves groups. The immense numbers of compounds of carbon, hydrogen, and oxygen, 1 Hofmann's Introduction to Chemistry, p. 36. described in organic chemistry, are combinations of a second order, for the atoms are groups of groups. The wave of sound produced by a musical instrument may be regarded as a combination of motions; the body of sound proceeding from a large orchestra is therefore a complex aggregate of sounds, each in itself a complex combination of movements. All literature may be said to be developed out of the difference of white paper and black ink. From the unlimited number of marks which might be chosen we select twenty-six conventional letters. The pronounceable combinations of letters are probably some trillions in number. Now, as a sentence is a selection of words, the possible sentences must be inconceivably more numerous than the words of which it may be composed. A book is a combination of sentences, and a library is a combination of books. A library, therefore, may be regarded as a combination of the fifth order, and the powers of numerical expression would be severely tasked in attempting to express the number of distinct libraries which might be constructed. The calculation, of course, would not be possible, because the union of letters in words, of words. in sentences, and of sentences in books, is governed by conditions so complex as to defy analysis. I wish only to point out that the infinite variety of literature, existing or possible, is all developed out of one fundamental difference. Galileo remarked that all truth is contained in the compass of the alphabet. He ought to have said that it is all contained in the difference of ink and paper. One consequence of successive combination is that the simplest marks will suffice to express any information. Francis Bacon proposed for secret writing a biliteral cipher, which resolves all letters of the alphabet into permutations of the two letters a and b. Thus A was aaaaa, В aaaab, X babab, and so on. In a similar way, as Bacon clearly saw, any one difference can be made the ground of a code of signals; we can express, as he says, omnia per omnia. The Morse alphabet uses only a succession of long and short marks, and other systems of telegraphic language employ right and left strokes. A single lamp obscured at various intervals, long or 1 Works, edited by Shaw, vol. i. pp. 141-145, quoted in Rees' Encyclopædia, art. Cipher. short, may be made to spell out any words, and with two lamps, distinguished by colour, position, or any other circumstance, we could at once represent Bacon's biliteral alphabet. Babbage ingeniously suggested that every lighthouse in the world should be made to spell out its own name or number perpetually, by flashes or obscurations of various duration and succession. A system like that of Babbage is now being applied to lighthouses in the United Kingdom by Sir W. Thomson and Dr. John Hopkinson. Let us calculate the numbers of combinations of different orders which may arise out of the presence or absence of a single mark, say A. In these figures Form them into a group we have four distinct varieties. of a higher order, and consider in how many ways we may vary that group by omitting one or more of the component parts. Now, as there are four parts, and any one may be present or absent, the possible varieties will be 2 × 2 × 2 × 2, or 16 in number. Form these into a new whole, and proceed again to create variety by omitting any one or more of the sixteen. The number of possible changes will now be 2.2.2.2.2.2.2.2.2.2.2.2.2.2.2.2, or 21, and we can repeat the process again and again. We are imagining the creation of objects, whose numbers are represented by the successive orders of the powers of two. At the first step we have 2; at the next 22, or 4; at the third 222, or 16, numbers of very moderate amount. 2 2 Let the reader calculate the next term, 22, and he will be surprised to find it leap up to 65,536. But at the next step he has to calculate the value of 65,536 two's multiplied together, and it is so great that we could not possibly compute it, the mere expression of the result requiring 19,729 places of figures. But go one step more and we pass the bounds of all reason. The sixth order of the powers of two becomes so great, that we could not even express the number of figures required in writing it down, without using about 19,729 figures for the purpose. The successive orders of the powers of two have then the |