COMBINATIONS AND PERMUTATIONS. 



195 



First order ... 2 



Second order . 



Third order '. 



Fourth order . 65,536 



Fifth order, number expressed by 19,729 figures. 

 feixtn order, number expressed by 

 figures, to express the number 

 of which figures would require 

 T , about . .... 19,729 figures. 



It may give us some notion of infinity to remember 

 that at this sixth step, having long surpassed all bounds 

 ot intuitive conception, we make no approach to a limit 

 JNay, 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 writing 

 of antiquity, the liber de Arence Numero, that the grains Sf 

 sand in the world could be numbered, or rather that if 

 numbered, the result could readily be expressed in arith- 

 metical 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 x io 24 to io 26 molecules in a cubic 

 centimetre of a solid or liquid substance. 1 Assumino- 

 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 x io 90 

 atoms, that is to say, a number requiring for its expression 

 92 places of figures. Now, this number would be im- 

 mensely less than the fifth order of the powers of two 

 In the variety of logical relations, which may exist 



1 Nature, vol. i. p. 553. 



o 2 



