222 THE PRINCIPLES OF SCIENCE. 



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 : 



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 a powerful notion of infinity to remem- 

 ber that at this sixth step, having long surpassed all 

 bounds of conception, we have made no approach to the 

 goal 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 ex- 

 pression rapidly overcome the possible multitude of 

 finite objects which may exist in any assignable space. 

 Archimedes showed long ago, in one of the most won- 

 derful writings of antiquity, 1 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, again, has shown reasons for supposing 



4 ' Liber de Arenas Numero.' 



u Chambers's ' Astronomy ' ( 1 86 1 ), p. 272. 



