COMBINATIONS AND PERMUTATIONS. 191 



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 pos- 

 sible 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 4 0,000 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 2 

 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. 1 1 6, quoting Vossius. 



2 Philosophical Transactions (1803), vol. xciii. p. 193. 



