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, 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, B aaaab, X babab, and so on.[108] 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 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
| A | A | A | A | A | A | A | A |
we have four distinct varieties. Form them into a group 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 216, 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 (22)2, or 16, numbers of very moderate amount. Let the reader calculate the next term, ((22)2)2, 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 following values so far as we can succeed in describing them:—
| First order | 2 | |
| Second order | 4 | |
| Third order | 16 | |
| Fourth order | 65,536 | |
| Fifth order, number expressed by | 19,729 | figures. |
| Sixth order, number expressed by figures, to express the number of which figures would require about | 19,729 | figures. |
It may give us some notion of infinity to remember that at this sixth step, having long surpassed all bounds of intuitive conception, we make no approach to a limit. Nay, were we to make a hundred such steps, we should be as far away as ever from actual infinity.
It is well worth observing that our powers of expression rapidly overcome the possible multitude of finite objects which may exist in any assignable space. Archimedes showed long ago, in one of the most remarkable writings of antiquity, the Liber de Arcnæ Numero, that the grains of sand in the world could be numbered, or rather, that if numbered, the result could readily be expressed in arithmetical notation. Let us extend his problem, and ascertain whether we could express the number of atoms which could exist in the visible universe. The most distant stars which can now be seen by telescopes—those of the sixteenth magnitude—are supposed to have a distance of about 33,900,000,000,000,000 miles. Sir W. Thomson has shown reasons for supposing that there do not exist more than from 3 × 1024 to 1026 molecules in a cubic centimetre of a solid or liquid substance.[109] Assuming these data to be true, for the sake of argument, a simple calculation enables us to show that the almost inconceivably vast sphere of our stellar system if entirely filled with solid matter, would not contain more than about 68 × 1090 atoms, that is to say, a number requiring for its expression 92 places of figures. Now, this number would be immensely less than the fifth order of the powers of two.