COMBINATIONS AND PERMUTATIONS. 195 



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 Arence Numcro, that the grains of 

 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 

 3 3, 900,000,0x30,000,000 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 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 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. D. 553 



o 2 



