ix.J COMBINATIONS AND PERMUTATIONS. 193 



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 com- 

 bination 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 differ- 

 ence. 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 &. Thus A was 

 aaaaa, B aaadb, X bdbab, and so on. 1 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. 141145, quoted in Rees' 

 Encyclopedia, art. Cipher. 



O 



